Compound Interest

Percentage Accurate: 28.2% → 81.7%

Time: 9.1s

Alternatives: 9

Speedup: 9.4×

• could not determine a ground truth

  1. i = 3.056337735276788e-208
  2. n = -2.1786918994891632e+27

Specification

Math

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

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: 28.2% accurate, 1.0× speedup

Math

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

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: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ t_1 := 100 \cdot \frac{i \cdot 1}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -9 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-207}:\\ \;\;\;\;100 \cdot \frac{n + -1 \cdot n}{i}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (i n)
  :precision binary64
  (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i)))
       (t_1 (* 100.0 (/ (* i 1.0) (/ i n)))))
  (if (<= n -9e-73)
    t_0
    (if (<= n -1.22e-166)
      t_1
      (if (<= n 3.8e-207)
        (* 100.0 (/ (+ n (* -1.0 n)) i))
        (if (<= n 5.6e-11) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double t_1 = 100.0 * ((i * 1.0) / (i / n));
	double tmp;
	if (n <= -9e-73) {
		tmp = t_0;
	} else if (n <= -1.22e-166) {
		tmp = t_1;
	} else if (n <= 3.8e-207) {
		tmp = 100.0 * ((n + (-1.0 * n)) / i);
	} else if (n <= 5.6e-11) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double t_1 = 100.0 * ((i * 1.0) / (i / n));
	double tmp;
	if (n <= -9e-73) {
		tmp = t_0;
	} else if (n <= -1.22e-166) {
		tmp = t_1;
	} else if (n <= 3.8e-207) {
		tmp = 100.0 * ((n + (-1.0 * n)) / i);
	} else if (n <= 5.6e-11) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	t_1 = 100.0 * ((i * 1.0) / (i / n))
	tmp = 0
	if n <= -9e-73:
		tmp = t_0
	elif n <= -1.22e-166:
		tmp = t_1
	elif n <= 3.8e-207:
		tmp = 100.0 * ((n + (-1.0 * n)) / i)
	elif n <= 5.6e-11:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	t_1 = Float64(100.0 * Float64(Float64(i * 1.0) / Float64(i / n)))
	tmp = 0.0
	if (n <= -9e-73)
		tmp = t_0;
	elseif (n <= -1.22e-166)
		tmp = t_1;
	elseif (n <= 3.8e-207)
		tmp = Float64(100.0 * Float64(Float64(n + Float64(-1.0 * n)) / i));
	elseif (n <= 5.6e-11)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(N[(i * 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9e-73], t$95$0, If[LessEqual[n, -1.22e-166], t$95$1, If[LessEqual[n, 3.8e-207], N[(100.0 * N[(N[(n + N[(-1.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.6e-11], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.0000000000000004e-73 or 5.6e-11 < n

   1. Initial program 28.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0%

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

   4. Applied rewrites 71.0%

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

   if -9.0000000000000004e-73 < n < -1.22e-166 or 3.8e-207 < n < 5.6e-11

   1. Initial program 28.2%

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

   2. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.4%

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 43.5%

      \[\leadsto 100 \cdot \frac{i \cdot \left(1 + i \cdot 0.5\right)}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot 1}{\frac{i}{n}} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.3%

       \[\leadsto 100 \cdot \frac{i \cdot 1}{\frac{i}{n}} \]

   if -1.22e-166 < n < 3.8e-207

   1. Initial program 28.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. sub-flip N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. add-to-fraction N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. div-flip-rev N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 22.7%

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

   3. Applied rewrites 22.7%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

      3. lower-*.f64 17.5%

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

   6. Applied rewrites 17.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n + -1 \cdot n}{i}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -8.6 \cdot 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-177}:\\ \;\;\;\;100 \cdot \frac{n + -1 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (i n)
  :precision binary64
  (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
  (if (<= n -8.6e-169)
    t_0
    (if (<= n 8e-177) (* 100.0 (/ (+ n (* -1.0 n)) i)) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -8.6e-169) {
		tmp = t_0;
	} else if (n <= 8e-177) {
		tmp = 100.0 * ((n + (-1.0 * n)) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -8.6e-169) {
		tmp = t_0;
	} else if (n <= 8e-177) {
		tmp = 100.0 * ((n + (-1.0 * n)) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -8.6e-169:
		tmp = t_0
	elif n <= 8e-177:
		tmp = 100.0 * ((n + (-1.0 * n)) / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -8.6e-169)
		tmp = t_0;
	elseif (n <= 8e-177)
		tmp = Float64(100.0 * Float64(Float64(n + Float64(-1.0 * n)) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -8.6e-169], t$95$0, If[LessEqual[n, 8e-177], N[(100.0 * N[(N[(n + N[(-1.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.5999999999999997e-169 or 7.9999999999999996e-177 < n

   1. Initial program 28.2%

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

   2. Taylor expanded in i around -inf

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

      1. lower-expm1.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 15.4%

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

   4. Applied rewrites 15.4%

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

   6. Applied rewrites 27.8%

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

   7. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-expm1.f64 75.8%

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

   9. Applied rewrites 75.8%

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

   if -8.5999999999999997e-169 < n < 7.9999999999999996e-177

   1. Initial program 28.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. sub-flip N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. add-to-fraction N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. div-flip-rev N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 22.7%

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

   3. Applied rewrites 22.7%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

      3. lower-*.f64 17.5%

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

   6. Applied rewrites 17.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n + -1 \cdot n}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (i n)
  :precision binary64
  (if (<=
     (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))
     INFINITY)
  (* 100.0 (* (expm1 i) (/ n i)))
  (* 100.0 n)))

C

double code(double i, double n) {
	double tmp;
	if ((100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n))) <= ((double) INFINITY)) {
		tmp = 100.0 * (expm1(i) * (n / i));
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))) <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (Math.expm1(i) * (n / i));
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))) <= math.inf:
		tmp = 100.0 * (math.expm1(i) * (n / i))
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n))) <= Inf)
		tmp = Float64(100.0 * Float64(expm1(i) * Float64(n / i)));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[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], Infinity], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]

Derivation

1. Split input into 2 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 28.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0%

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

   4. Applied rewrites 71.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. div-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-flip-rev N/A

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

      10. lower-/.f64 60.9%

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

   6. Applied rewrites 60.9%

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

   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)))

   1. Initial program 28.2%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.9%

      \[\leadsto 100 \cdot \color{blue}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 65.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(n \cdot \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, n\right) \cdot 100\\ \mathbf{if}\;n \leq -2.2 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;100 \cdot \frac{i \cdot 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-176}:\\ \;\;\;\;100 \cdot \frac{n + -1 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (i n)
  :precision binary64
  (let* ((t_0
        (* (fma (* n (fma 0.16666666666666666 i 0.5)) i n) 100.0)))
  (if (<= n -2.2e+32)
    t_0
    (if (<= n -1.22e-166)
      (* 100.0 (/ (* i 1.0) (/ i n)))
      (if (<= n 1.85e-176) (* 100.0 (/ (+ n (* -1.0 n)) i)) t_0)))))

C

double code(double i, double n) {
	double t_0 = fma((n * fma(0.16666666666666666, i, 0.5)), i, n) * 100.0;
	double tmp;
	if (n <= -2.2e+32) {
		tmp = t_0;
	} else if (n <= -1.22e-166) {
		tmp = 100.0 * ((i * 1.0) / (i / n));
	} else if (n <= 1.85e-176) {
		tmp = 100.0 * ((n + (-1.0 * n)) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(Float64(n * fma(0.16666666666666666, i, 0.5)), i, n) * 100.0)
	tmp = 0.0
	if (n <= -2.2e+32)
		tmp = t_0;
	elseif (n <= -1.22e-166)
		tmp = Float64(100.0 * Float64(Float64(i * 1.0) / Float64(i / n)));
	elseif (n <= 1.85e-176)
		tmp = Float64(100.0 * Float64(Float64(n + Float64(-1.0 * n)) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(n * N[(0.16666666666666666 * i + 0.5), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -2.2e+32], t$95$0, If[LessEqual[n, -1.22e-166], N[(100.0 * N[(N[(i * 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-176], N[(100.0 * N[(N[(n + N[(-1.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.2e32 or 1.8499999999999999e-176 < n

   1. Initial program 28.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0%

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in i around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 57.2%

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(0.16666666666666666, i \cdot n, 0.5 \cdot n\right)\right) \]

   7. Applied rewrites 57.2%

      \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot \mathsf{fma}\left(0.16666666666666666, i \cdot n, 0.5 \cdot n\right)}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.2%

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

   9. Applied rewrites 57.2%

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

   if -2.2e32 < n < -1.22e-166

   1. Initial program 28.2%

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

   2. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.4%

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 43.5%

      \[\leadsto 100 \cdot \frac{i \cdot \left(1 + i \cdot 0.5\right)}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot 1}{\frac{i}{n}} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.3%

       \[\leadsto 100 \cdot \frac{i \cdot 1}{\frac{i}{n}} \]

   if -1.22e-166 < n < 1.8499999999999999e-176

   1. Initial program 28.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. sub-flip N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. add-to-fraction N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. div-flip-rev N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 22.7%

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

   3. Applied rewrites 22.7%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

      3. lower-*.f64 17.5%

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

   6. Applied rewrites 17.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n + -1 \cdot n}{i}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+43}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq -1.22 \cdot 10^{-166}:\\ \;\;\;\;100 \cdot \frac{i \cdot 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-176}:\\ \;\;\;\;100 \cdot \frac{n + -1 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot n, i, n\right) \cdot 100\\ \end{array} \]

FPCore

(FPCore (i n)
  :precision binary64
  (if (<= n -5e+43)
  (* 100.0 (/ (* i n) i))
  (if (<= n -1.22e-166)
    (* 100.0 (/ (* i 1.0) (/ i n)))
    (if (<= n 1.85e-176)
      (* 100.0 (/ (+ n (* -1.0 n)) i))
      (* (fma (* 0.5 n) i n) 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e+43) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= -1.22e-166) {
		tmp = 100.0 * ((i * 1.0) / (i / n));
	} else if (n <= 1.85e-176) {
		tmp = 100.0 * ((n + (-1.0 * n)) / i);
	} else {
		tmp = fma((0.5 * n), i, n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e+43)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= -1.22e-166)
		tmp = Float64(100.0 * Float64(Float64(i * 1.0) / Float64(i / n)));
	elseif (n <= 1.85e-176)
		tmp = Float64(100.0 * Float64(Float64(n + Float64(-1.0 * n)) / i));
	else
		tmp = Float64(fma(Float64(0.5 * n), i, n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e+43], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.22e-166], N[(100.0 * N[(N[(i * 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-176], N[(100.0 * N[(N[(n + N[(-1.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * n), $MachinePrecision] * i + n), $MachinePrecision] * 100.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.0000000000000004e43

   1. Initial program 28.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0%

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 49.8%

         \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]

   7. Applied rewrites 49.8%

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]

   if -5.0000000000000004e43 < n < -1.22e-166

   1. Initial program 28.2%

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

   2. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.4%

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 43.5%

      \[\leadsto 100 \cdot \frac{i \cdot \left(1 + i \cdot 0.5\right)}{\frac{i}{n}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot 1}{\frac{i}{n}} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.3%

       \[\leadsto 100 \cdot \frac{i \cdot 1}{\frac{i}{n}} \]

   if -1.22e-166 < n < 1.8499999999999999e-176

   1. Initial program 28.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. sub-flip N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. add-to-fraction N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. div-flip-rev N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 22.7%

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

   3. Applied rewrites 22.7%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

      3. lower-*.f64 17.5%

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

   6. Applied rewrites 17.5%

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

   if 1.8499999999999999e-176 < n

   1. Initial program 28.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0%

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot \left(\frac{1}{2} \cdot n + i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \mathsf{fma}\left(\frac{1}{24}, i \cdot n, \frac{1}{6} \cdot n\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \mathsf{fma}\left(\frac{1}{24}, i \cdot n, \frac{1}{6} \cdot n\right)\right)\right) \]

      7. lower-*.f64 58.8%

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right)\right) \]

   7. Applied rewrites 58.8%

      \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right)}\right) \]

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 54.6%

         \[\leadsto 100 \cdot \left(n + i \cdot \left(0.5 \cdot n\right)\right) \]

   10. Applied rewrites 54.6%

       \[\leadsto 100 \cdot \left(n + i \cdot \left(0.5 \cdot n\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 54.6%

          \[\leadsto \color{blue}{\left(n + i \cdot \left(0.5 \cdot n\right)\right) \cdot 100} \]

   12. Applied rewrites 54.6%

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

Alternative 6: 61.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot n, i, n\right) \cdot 100\\ \mathbf{if}\;n \leq -1.32 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-176}:\\ \;\;\;\;100 \cdot \frac{n + -1 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (i n)
  :precision binary64
  (let* ((t_0 (* (fma (* 0.5 n) i n) 100.0)))
  (if (<= n -1.32e-166)
    t_0
    (if (<= n 1.85e-176) (* 100.0 (/ (+ n (* -1.0 n)) i)) t_0))))

C

double code(double i, double n) {
	double t_0 = fma((0.5 * n), i, n) * 100.0;
	double tmp;
	if (n <= -1.32e-166) {
		tmp = t_0;
	} else if (n <= 1.85e-176) {
		tmp = 100.0 * ((n + (-1.0 * n)) / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(Float64(0.5 * n), i, n) * 100.0)
	tmp = 0.0
	if (n <= -1.32e-166)
		tmp = t_0;
	elseif (n <= 1.85e-176)
		tmp = Float64(100.0 * Float64(Float64(n + Float64(-1.0 * n)) / i));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(0.5 * n), $MachinePrecision] * i + n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -1.32e-166], t$95$0, If[LessEqual[n, 1.85e-176], N[(100.0 * N[(N[(n + N[(-1.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.32e-166 or 1.8499999999999999e-176 < n

   1. Initial program 28.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0%

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot \left(\frac{1}{2} \cdot n + i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \mathsf{fma}\left(\frac{1}{24}, i \cdot n, \frac{1}{6} \cdot n\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \mathsf{fma}\left(\frac{1}{24}, i \cdot n, \frac{1}{6} \cdot n\right)\right)\right) \]

      7. lower-*.f64 58.8%

         \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right)\right) \]

   7. Applied rewrites 58.8%

      \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right)}\right) \]

   8. Taylor expanded in i around 0

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

      1. lower-*.f64 54.6%

         \[\leadsto 100 \cdot \left(n + i \cdot \left(0.5 \cdot n\right)\right) \]

   10. Applied rewrites 54.6%

       \[\leadsto 100 \cdot \left(n + i \cdot \left(0.5 \cdot n\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 54.6%

          \[\leadsto \color{blue}{\left(n + i \cdot \left(0.5 \cdot n\right)\right) \cdot 100} \]

   12. Applied rewrites 54.6%

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

   if -1.32e-166 < n < 1.8499999999999999e-176

   1. Initial program 28.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. sub-flip N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. add-to-fraction N/A

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

      12. lower-/.f64 N/A

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

      13. *-lft-identity N/A

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

      14. lower-+.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-/.f64 N/A

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

      17. div-flip-rev N/A

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

      18. distribute-neg-frac N/A

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

      19. lower-/.f64 N/A

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

      20. lower-neg.f64 22.7%

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

   3. Applied rewrites 22.7%

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

   4. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

      3. lower-*.f64 17.5%

         \[\leadsto 100 \cdot \frac{n + -1 \cdot n}{i} \]

   6. Applied rewrites 17.5%

      \[\leadsto 100 \cdot \color{blue}{\frac{n + -1 \cdot n}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;i \leq 9.6 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \]

FPCore

(FPCore (i n)
  :precision binary64
  (if (<= i 9.6e-186) (* 100.0 n) (* 100.0 (/ (* i n) i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 9.6e-186) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (i <= 9.6d-186) then
        tmp = 100.0d0 * n
    else
        tmp = 100.0d0 * ((i * n) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 9.6e-186) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 9.6e-186:
		tmp = 100.0 * n
	else:
		tmp = 100.0 * ((i * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 9.6e-186)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 9.6e-186)
		tmp = 100.0 * n;
	else
		tmp = 100.0 * ((i * n) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 9.6e-186], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 9.6000000000000001e-186

   1. Initial program 28.2%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.9%

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

   if 9.6000000000000001e-186 < i

   1. Initial program 28.2%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0%

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 49.8%

         \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]

   7. Applied rewrites 49.8%

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.6% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (i n)
  :precision binary64
  (* (fma (* 0.5 n) i n) 100.0))

C

double code(double i, double n) {
	return fma((0.5 * n), i, n) * 100.0;
}

Julia

function code(i, n)
	return Float64(fma(Float64(0.5 * n), i, n) * 100.0)
end

Wolfram

code[i_, n_] := N[(N[(N[(0.5 * n), $MachinePrecision] * i + n), $MachinePrecision] * 100.0), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

2. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-expm1.f64 71.0%

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

4. Applied rewrites 71.0%

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

5. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot \left(\frac{1}{2} \cdot n + i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)}\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \mathsf{fma}\left(\frac{1}{24}, i \cdot n, \frac{1}{6} \cdot n\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(\frac{1}{2}, n, i \cdot \mathsf{fma}\left(\frac{1}{24}, i \cdot n, \frac{1}{6} \cdot n\right)\right)\right) \]

   7. lower-*.f64 58.8%

      \[\leadsto 100 \cdot \left(n + i \cdot \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right)\right) \]

7. Applied rewrites 58.8%

   \[\leadsto 100 \cdot \left(n + \color{blue}{i \cdot \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right)}\right) \]

8. Taylor expanded in i around 0

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

   1. lower-*.f64 54.6%

      \[\leadsto 100 \cdot \left(n + i \cdot \left(0.5 \cdot n\right)\right) \]

10. Applied rewrites 54.6%

    \[\leadsto 100 \cdot \left(n + i \cdot \left(0.5 \cdot n\right)\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 54.6%

       \[\leadsto \color{blue}{\left(n + i \cdot \left(0.5 \cdot n\right)\right) \cdot 100} \]

12. Applied rewrites 54.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot n, i, n\right) \cdot 100} \]
13. Add Preprocessing

Alternative 9: 48.9% accurate, 9.4× speedup

Math

\[100 \cdot n \]

FPCore

(FPCore (i n)
  :precision binary64
  (* 100.0 n))

C

double code(double i, double n) {
	return 100.0 * 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 * n
end function

Java

public static double code(double i, double n) {
	return 100.0 * n;
}

Python

def code(i, n):
	return 100.0 * n

Julia

function code(i, n)
	return Float64(100.0 * n)
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * n;
end

Wolfram

code[i_, n_] := N[(100.0 * n), $MachinePrecision]

Derivation

1. Initial program 28.2%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 48.9%

   \[\leadsto 100 \cdot \color{blue}{n} \]
5. Add Preprocessing

Developer Target 1: 34.7% accurate, 0.5× speedup

Math

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

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

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
    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 2025215 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform c (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))))