Compound Interest

Percentage Accurate: 27.8% → 94.4%

Time: 10.4s

Alternatives: 22

Speedup: 24.3×

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: 27.8% 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: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} - -1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 5e-306)
     (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ i n))
     (if (<= t_0 INFINITY)
       (* (* (/ (- (pow (- (/ i n) -1.0) n) 1.0) i) 100.0) n)
       (* 100.0 n)))))

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 <= 5e-306) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 5e-306) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (((Math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 5e-306:
		tmp = (math.expm1((math.log1p((i / n)) * n)) * 100.0) / (i / n)
	elif t_0 <= math.inf:
		tmp = (((math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n
	else:
		tmp = 100.0 * n
	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 <= 5e-306)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * n);
	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, 5e-306], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * n), $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))) < 4.99999999999999998e-306

   1. Initial program 28.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 98.0

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

   4. Applied rewrites 98.0%

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

   if 4.99999999999999998e-306 < (*.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 97.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 47.6

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

   4. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.6

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

   6. Applied rewrites 47.6%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 97.7

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

   8. Applied rewrites 97.7%

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

   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. Taylor expanded in i around 0

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

   5. Applied rewrites 72.1%

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

Alternative 2: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} - -1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 5e-306)
     (* (/ (expm1 (* (log1p (/ i n)) n)) i) (* n 100.0))
     (if (<= t_0 INFINITY)
       (* (* (/ (- (pow (- (/ i n) -1.0) n) 1.0) i) 100.0) n)
       (* 100.0 n)))))

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 <= 5e-306) {
		tmp = (expm1((log1p((i / n)) * n)) / i) * (n * 100.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 5e-306) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) / i) * (n * 100.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (((Math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 5e-306:
		tmp = (math.expm1((math.log1p((i / n)) * n)) / i) * (n * 100.0)
	elif t_0 <= math.inf:
		tmp = (((math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n
	else:
		tmp = 100.0 * n
	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 <= 5e-306)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * Float64(n * 100.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * n);
	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, 5e-306], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * n), $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))) < 4.99999999999999998e-306

   1. Initial program 28.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 97.4

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

   4. Applied rewrites 97.4%

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

   if 4.99999999999999998e-306 < (*.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 97.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 47.6

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

   4. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.6

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

   6. Applied rewrites 47.6%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 97.7

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

   8. Applied rewrites 97.7%

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

   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. Taylor expanded in i around 0

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

   5. Applied rewrites 72.1%

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

Alternative 3: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} - -1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 5e-306)
     (* (* 100.0 (/ (expm1 (* (log1p (/ i n)) n)) i)) n)
     (if (<= t_0 INFINITY)
       (* (* (/ (- (pow (- (/ i n) -1.0) n) 1.0) i) 100.0) n)
       (* 100.0 n)))))

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 <= 5e-306) {
		tmp = (100.0 * (expm1((log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 5e-306) {
		tmp = (100.0 * (Math.expm1((Math.log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (((Math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 5e-306:
		tmp = (100.0 * (math.expm1((math.log1p((i / n)) * n)) / i)) * n
	elif t_0 <= math.inf:
		tmp = (((math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n
	else:
		tmp = 100.0 * n
	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 <= 5e-306)
		tmp = Float64(Float64(100.0 * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * n);
	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, 5e-306], N[(N[(100.0 * N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * n), $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))) < 4.99999999999999998e-306

   1. Initial program 28.1%

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

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 28.0

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 97.4

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

   4. Applied rewrites 97.4%

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

   if 4.99999999999999998e-306 < (*.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 97.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 47.6

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

   4. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.6

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

   6. Applied rewrites 47.6%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 97.7

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

   8. Applied rewrites 97.7%

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

   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. Taylor expanded in i around 0

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

   5. Applied rewrites 72.1%

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

Alternative 4: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-306}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} - -1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 5e-306)
     (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
     (if (<= t_0 INFINITY)
       (* (* (/ (- (pow (- (/ i n) -1.0) n) 1.0) i) 100.0) n)
       (* 100.0 n)))))

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 <= 5e-306) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 5e-306) {
		tmp = 100.0 * ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (((Math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 5e-306:
		tmp = 100.0 * ((math.expm1((math.log1p((i / n)) * n)) / i) * n)
	elif t_0 <= math.inf:
		tmp = (((math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n
	else:
		tmp = 100.0 * n
	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 <= 5e-306)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * n);
	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, 5e-306], 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], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * n), $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))) < 4.99999999999999998e-306

   1. Initial program 28.1%

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

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 28.1

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 97.3

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

   4. Applied rewrites 97.3%

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

   if 4.99999999999999998e-306 < (*.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 97.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 47.6

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

   4. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.6

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

   6. Applied rewrites 47.6%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 97.7

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

   8. Applied rewrites 97.7%

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

   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. Taylor expanded in i around 0

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

   5. Applied rewrites 72.1%

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

Alternative 5: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \mathbf{if}\;n \leq -7.6 \cdot 10^{-184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(n \cdot \frac{\mathsf{fma}\left(\log n, -1, \log i\right)}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ (* (expm1 i) 100.0) i) n)))
   (if (<= n -7.6e-184)
     t_0
     (if (<= n -2e-310)
       (* (/ (- 1.0 1.0) i) (* 100.0 n))
       (if (<= n 1.45e-99)
         (* (* (* n (/ (fma (log n) -1.0 (log i)) i)) 100.0) n)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) * 100.0) / i) * n;
	double tmp;
	if (n <= -7.6e-184) {
		tmp = t_0;
	} else if (n <= -2e-310) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.45e-99) {
		tmp = ((n * (fma(log(n), -1.0, log(i)) / i)) * 100.0) * n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) * 100.0) / i) * n)
	tmp = 0.0
	if (n <= -7.6e-184)
		tmp = t_0;
	elseif (n <= -2e-310)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.45e-99)
		tmp = Float64(Float64(Float64(n * Float64(fma(log(n), -1.0, log(i)) / i)) * 100.0) * n);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -7.6e-184], t$95$0, If[LessEqual[n, -2e-310], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.45e-99], N[(N[(N[(n * N[(N[(N[Log[n], $MachinePrecision] * -1.0 + N[Log[i], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.60000000000000033e-184 or 1.44999999999999993e-99 < n

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if -7.60000000000000033e-184 < n < -1.999999999999994e-310

   1. Initial program 74.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if -1.999999999999994e-310 < n < 1.44999999999999993e-99

   1. Initial program 11.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 59.6

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

   4. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 59.6

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

   6. Applied rewrites 59.6%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 81.2%

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

Alternative 6: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \mathbf{if}\;n \leq -7.6 \cdot 10^{-184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-99}:\\ \;\;\;\;\left(n \cdot \frac{\mathsf{fma}\left(\log n, -1, \log i\right)}{i}\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ (* (expm1 i) 100.0) i) n)))
   (if (<= n -7.6e-184)
     t_0
     (if (<= n -2e-310)
       (* (/ (- 1.0 1.0) i) (* 100.0 n))
       (if (<= n 1.45e-99)
         (* (* n (/ (fma (log n) -1.0 (log i)) i)) (* n 100.0))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) * 100.0) / i) * n;
	double tmp;
	if (n <= -7.6e-184) {
		tmp = t_0;
	} else if (n <= -2e-310) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.45e-99) {
		tmp = (n * (fma(log(n), -1.0, log(i)) / i)) * (n * 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) * 100.0) / i) * n)
	tmp = 0.0
	if (n <= -7.6e-184)
		tmp = t_0;
	elseif (n <= -2e-310)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.45e-99)
		tmp = Float64(Float64(n * Float64(fma(log(n), -1.0, log(i)) / i)) * Float64(n * 100.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -7.6e-184], t$95$0, If[LessEqual[n, -2e-310], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.45e-99], N[(N[(n * N[(N[(N[Log[n], $MachinePrecision] * -1.0 + N[Log[i], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.60000000000000033e-184 or 1.44999999999999993e-99 < n

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if -7.60000000000000033e-184 < n < -1.999999999999994e-310

   1. Initial program 74.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if -1.999999999999994e-310 < n < 1.44999999999999993e-99

   1. Initial program 11.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 59.6

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

   4. Applied rewrites 59.6%

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

   5. Taylor expanded in n around 0

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

   7. Applied rewrites 81.1%

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

Alternative 7: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{+148}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \mathbf{elif}\;i \leq 6 \cdot 10^{+225}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} - -1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log i - \log n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -3.2e+148)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i 2.5e+130)
     (* (/ (* (expm1 i) 100.0) i) n)
     (if (<= i 6e+225)
       (* (* (/ (- (pow (- (/ i n) -1.0) n) 1.0) i) 100.0) n)
       (* 100.0 (* (* n n) (/ (- (log i) (log n)) i)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -3.2e+148) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else if (i <= 2.5e+130) {
		tmp = ((expm1(i) * 100.0) / i) * n;
	} else if (i <= 6e+225) {
		tmp = (((pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((n * n) * ((log(i) - log(n)) / i));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -3.2e+148) {
		tmp = 100.0 * ((Math.pow((i / n), n) - 1.0) / (i / n));
	} else if (i <= 2.5e+130) {
		tmp = ((Math.expm1(i) * 100.0) / i) * n;
	} else if (i <= 6e+225) {
		tmp = (((Math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((n * n) * ((Math.log(i) - Math.log(n)) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -3.2e+148:
		tmp = 100.0 * ((math.pow((i / n), n) - 1.0) / (i / n))
	elif i <= 2.5e+130:
		tmp = ((math.expm1(i) * 100.0) / i) * n
	elif i <= 6e+225:
		tmp = (((math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((n * n) * ((math.log(i) - math.log(n)) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -3.2e+148)
		tmp = Float64(100.0 * Float64(Float64((Float64(i / n) ^ n) - 1.0) / Float64(i / n)));
	elseif (i <= 2.5e+130)
		tmp = Float64(Float64(Float64(expm1(i) * 100.0) / i) * n);
	elseif (i <= 6e+225)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) * Float64(Float64(log(i) - log(n)) / i)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -3.2e+148], N[(100.0 * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e+130], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[i, 6e+225], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -3.1999999999999999e148

   1. Initial program 99.1%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.1%

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

   if -3.1999999999999999e148 < i < 2.4999999999999998e130

   1. Initial program 14.9%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if 2.4999999999999998e130 < i < 6.000000000000001e225

   1. Initial program 79.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.1

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

   6. Applied rewrites 50.1%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 80.3

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

   8. Applied rewrites 80.3%

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

   if 6.000000000000001e225 < i

   1. Initial program 41.0%

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

   3. Taylor expanded in n around 0

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

   5. Applied rewrites 74.4%

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

Alternative 8: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.2 \cdot 10^{+148}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \mathbf{elif}\;i \leq 6 \cdot 10^{+225}:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} - -1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left(n \cdot n\right)\right) \cdot \frac{\log i - \log n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -3.2e+148)
   (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
   (if (<= i 2.5e+130)
     (* (/ (* (expm1 i) 100.0) i) n)
     (if (<= i 6e+225)
       (* (* (/ (- (pow (- (/ i n) -1.0) n) 1.0) i) 100.0) n)
       (* (* 100.0 (* n n)) (/ (- (log i) (log n)) i))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -3.2e+148) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else if (i <= 2.5e+130) {
		tmp = ((expm1(i) * 100.0) / i) * n;
	} else if (i <= 6e+225) {
		tmp = (((pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = (100.0 * (n * n)) * ((log(i) - log(n)) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -3.2e+148) {
		tmp = 100.0 * ((Math.pow((i / n), n) - 1.0) / (i / n));
	} else if (i <= 2.5e+130) {
		tmp = ((Math.expm1(i) * 100.0) / i) * n;
	} else if (i <= 6e+225) {
		tmp = (((Math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n;
	} else {
		tmp = (100.0 * (n * n)) * ((Math.log(i) - Math.log(n)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -3.2e+148:
		tmp = 100.0 * ((math.pow((i / n), n) - 1.0) / (i / n))
	elif i <= 2.5e+130:
		tmp = ((math.expm1(i) * 100.0) / i) * n
	elif i <= 6e+225:
		tmp = (((math.pow(((i / n) - -1.0), n) - 1.0) / i) * 100.0) * n
	else:
		tmp = (100.0 * (n * n)) * ((math.log(i) - math.log(n)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -3.2e+148)
		tmp = Float64(100.0 * Float64(Float64((Float64(i / n) ^ n) - 1.0) / Float64(i / n)));
	elseif (i <= 2.5e+130)
		tmp = Float64(Float64(Float64(expm1(i) * 100.0) / i) * n);
	elseif (i <= 6e+225)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(100.0 * Float64(n * n)) * Float64(Float64(log(i) - log(n)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -3.2e+148], N[(100.0 * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.5e+130], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[i, 6e+225], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(100.0 * N[(n * n), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -3.1999999999999999e148

   1. Initial program 99.1%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 99.1%

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

   if -3.1999999999999999e148 < i < 2.4999999999999998e130

   1. Initial program 14.9%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if 2.4999999999999998e130 < i < 6.000000000000001e225

   1. Initial program 79.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.1

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

   6. Applied rewrites 50.1%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 80.3

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

   8. Applied rewrites 80.3%

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

   if 6.000000000000001e225 < i

   1. Initial program 41.0%

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

   3. Taylor expanded in n around 0

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

   5. Applied rewrites 74.4%

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

Alternative 9: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \mathbf{if}\;n \leq -7.6 \cdot 10^{-184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.42 \cdot 10^{-185}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-99}:\\ \;\;\;\;100 \cdot \frac{\log \left(\frac{i}{n}\right) \cdot n}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ (* (expm1 i) 100.0) i) n)))
   (if (<= n -7.6e-184)
     t_0
     (if (<= n 1.42e-185)
       (* (/ (- 1.0 1.0) i) (* 100.0 n))
       (if (<= n 1.45e-99) (* 100.0 (/ (* (log (/ i n)) n) (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) * 100.0) / i) * n;
	double tmp;
	if (n <= -7.6e-184) {
		tmp = t_0;
	} else if (n <= 1.42e-185) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.45e-99) {
		tmp = 100.0 * ((log((i / n)) * n) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) * 100.0) / i) * n;
	double tmp;
	if (n <= -7.6e-184) {
		tmp = t_0;
	} else if (n <= 1.42e-185) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.45e-99) {
		tmp = 100.0 * ((Math.log((i / n)) * n) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) * 100.0) / i) * n
	tmp = 0
	if n <= -7.6e-184:
		tmp = t_0
	elif n <= 1.42e-185:
		tmp = ((1.0 - 1.0) / i) * (100.0 * n)
	elif n <= 1.45e-99:
		tmp = 100.0 * ((math.log((i / n)) * n) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) * 100.0) / i) * n)
	tmp = 0.0
	if (n <= -7.6e-184)
		tmp = t_0;
	elseif (n <= 1.42e-185)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.45e-99)
		tmp = Float64(100.0 * Float64(Float64(log(Float64(i / n)) * n) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -7.6e-184], t$95$0, If[LessEqual[n, 1.42e-185], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.45e-99], N[(100.0 * N[(N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.60000000000000033e-184 or 1.44999999999999993e-99 < n

   1. Initial program 25.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if -7.60000000000000033e-184 < n < 1.42000000000000003e-185

   1. Initial program 47.3%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 76.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if 1.42000000000000003e-185 < n < 1.44999999999999993e-99

   1. Initial program 13.6%

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

   3. Taylor expanded in n around 0

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

   5. Applied rewrites 72.6%

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

   7. Applied rewrites 66.1%

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

Alternative 10: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -7.6 \cdot 10^{-184}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -1.15e-28)
     t_0
     (if (<= n -7.6e-184)
       t_1
       (if (<= n 5.8e-204)
         (* (/ (- 1.0 1.0) i) (* 100.0 n))
         (if (<= n 1.85e-9) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.15e-28) {
		tmp = t_0;
	} else if (n <= -7.6e-184) {
		tmp = t_1;
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.85e-9) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.expm1(i) * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.15e-28) {
		tmp = t_0;
	} else if (n <= -7.6e-184) {
		tmp = t_1;
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.85e-9) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) * n) / i)
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -1.15e-28:
		tmp = t_0
	elif n <= -7.6e-184:
		tmp = t_1
	elif n <= 5.8e-204:
		tmp = ((1.0 - 1.0) / i) * (100.0 * n)
	elif n <= 1.85e-9:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -1.15e-28)
		tmp = t_0;
	elseif (n <= -7.6e-184)
		tmp = t_1;
	elseif (n <= 5.8e-204)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.85e-9)
		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[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.15e-28], t$95$0, If[LessEqual[n, -7.6e-184], t$95$1, If[LessEqual[n, 5.8e-204], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-9], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.14999999999999993e-28 or 1.85e-9 < n

   1. Initial program 25.8%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 89.7%

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

   if -1.14999999999999993e-28 < n < -7.60000000000000033e-184 or 5.80000000000000018e-204 < n < 1.85e-9

   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 i around 0

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

   5. Applied rewrites 60.7%

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

   if -7.60000000000000033e-184 < n < 5.80000000000000018e-204

   1. Initial program 52.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 79.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

Alternative 11: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \mathbf{if}\;n \leq -7.6 \cdot 10^{-184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ (* (expm1 i) 100.0) i) n)))
   (if (<= n -7.6e-184)
     t_0
     (if (<= n 5.8e-204)
       (* (/ (- 1.0 1.0) i) (* 100.0 n))
       (if (<= n 2.9e-9) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) * 100.0) / i) * n;
	double tmp;
	if (n <= -7.6e-184) {
		tmp = t_0;
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 2.9e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) * 100.0) / i) * n;
	double tmp;
	if (n <= -7.6e-184) {
		tmp = t_0;
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 2.9e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) * 100.0) / i) * n
	tmp = 0
	if n <= -7.6e-184:
		tmp = t_0
	elif n <= 5.8e-204:
		tmp = ((1.0 - 1.0) / i) * (100.0 * n)
	elif n <= 2.9e-9:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) * 100.0) / i) * n)
	tmp = 0.0
	if (n <= -7.6e-184)
		tmp = t_0;
	elseif (n <= 5.8e-204)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 2.9e-9)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -7.6e-184], t$95$0, If[LessEqual[n, 5.8e-204], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.9e-9], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.60000000000000033e-184 or 2.89999999999999991e-9 < n

   1. Initial program 27.3%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 86.0

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

   7. Applied rewrites 86.0%

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

   if -7.60000000000000033e-184 < n < 5.80000000000000018e-204

   1. Initial program 52.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 79.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   if 5.80000000000000018e-204 < n < 2.89999999999999991e-9

   1. Initial program 11.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 60.7%

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

Alternative 12: 67.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-100}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, 0.041666666666666664, \left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e-100)
   (*
    100.0
    (fma
     (fma
      (fma
       (* n i)
       0.041666666666666664
       (*
        (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
        n))
      i
      (* (- 0.5 (/ 0.5 n)) n))
     i
     n))
   (if (<= n 5.8e-204)
     (* (/ (- 1.0 1.0) i) (* 100.0 n))
     (if (<= n 1.85e-9)
       (* 100.0 (/ i (/ i n)))
       (*
        100.0
        (*
         (fma
          (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
          i
          1.0)
         n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e-100) {
		tmp = 100.0 * fma(fma(fma((n * i), 0.041666666666666664, ((((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)) * n)), i, ((0.5 - (0.5 / n)) * n)), i, n);
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.85e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e-100)
		tmp = Float64(100.0 * fma(fma(fma(Float64(n * i), 0.041666666666666664, Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)) * n)), i, Float64(Float64(0.5 - Float64(0.5 / n)) * n)), i, n));
	elseif (n <= 5.8e-204)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.85e-9)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e-100], N[(100.0 * N[(N[(N[(N[(n * i), $MachinePrecision] * 0.041666666666666664 + N[(N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-204], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-9], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.5999999999999999e-100

   1. Initial program 30.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \frac{1}{24}, \left(\left(\frac{\frac{1}{3}}{n \cdot n} + \frac{1}{6}\right) - \frac{\frac{1}{2}}{n}\right) \cdot n\right), i, \left(\frac{1}{2} - \frac{\frac{1}{2}}{n}\right) \cdot n\right), i, n\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.4%

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, 0.041666666666666664, \left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}\right) \cdot n\right), i, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right) \]

   if -3.5999999999999999e-100 < n < 5.80000000000000018e-204

   1. Initial program 50.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 69.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 5.80000000000000018e-204 < n < 1.85e-9

   1. Initial program 12.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 59.8%

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

   if 1.85e-9 < n

   1. Initial program 19.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 85.3%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-100}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right) \cdot n, i, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e-100)
   (*
    100.0
    (fma
     (fma
      (* (fma 0.041666666666666664 i 0.16666666666666666) n)
      i
      (* (- 0.5 (/ 0.5 n)) n))
     i
     n))
   (if (<= n 5.8e-204)
     (* (/ (- 1.0 1.0) i) (* 100.0 n))
     (if (<= n 1.85e-9)
       (* 100.0 (/ i (/ i n)))
       (*
        100.0
        (*
         (fma
          (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
          i
          1.0)
         n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e-100) {
		tmp = 100.0 * fma(fma((fma(0.041666666666666664, i, 0.16666666666666666) * n), i, ((0.5 - (0.5 / n)) * n)), i, n);
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.85e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e-100)
		tmp = Float64(100.0 * fma(fma(Float64(fma(0.041666666666666664, i, 0.16666666666666666) * n), i, Float64(Float64(0.5 - Float64(0.5 / n)) * n)), i, n));
	elseif (n <= 5.8e-204)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.85e-9)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e-100], N[(100.0 * N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * n), $MachinePrecision] * i + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-204], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-9], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.5999999999999999e-100

   1. Initial program 30.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.0%

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

   if -3.5999999999999999e-100 < n < 5.80000000000000018e-204

   1. Initial program 50.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 69.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 5.80000000000000018e-204 < n < 1.85e-9

   1. Initial program 12.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 59.8%

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

   if 1.85e-9 < n

   1. Initial program 19.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 85.3%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-100}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.041666666666666664, i, n\right)\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.6e-100)
   (* 100.0 (fma (* (* (* i i) n) 0.041666666666666664) i n))
   (if (<= n 5.8e-204)
     (* (/ (- 1.0 1.0) i) (* 100.0 n))
     (if (<= n 1.85e-9)
       (* 100.0 (/ i (/ i n)))
       (*
        100.0
        (*
         (fma
          (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
          i
          1.0)
         n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.6e-100) {
		tmp = 100.0 * fma((((i * i) * n) * 0.041666666666666664), i, n);
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.85e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.6e-100)
		tmp = Float64(100.0 * fma(Float64(Float64(Float64(i * i) * n) * 0.041666666666666664), i, n));
	elseif (n <= 5.8e-204)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.85e-9)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.6e-100], N[(100.0 * N[(N[(N[(N[(i * i), $MachinePrecision] * n), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-204], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-9], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.5999999999999999e-100

   1. Initial program 30.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left({i}^{2} \cdot n\right), i, n\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 62.3%

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

   if -3.5999999999999999e-100 < n < 5.80000000000000018e-204

   1. Initial program 50.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 69.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 5.80000000000000018e-204 < n < 1.85e-9

   1. Initial program 12.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 59.8%

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

   if 1.85e-9 < n

   1. Initial program 19.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 85.3%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 66.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \mathsf{fma}\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.041666666666666664, i, n\right)\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-204}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.85 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (fma (* (* (* i i) n) 0.041666666666666664) i n))))
   (if (<= n -3.6e-100)
     t_0
     (if (<= n 5.8e-204)
       (* (/ (- 1.0 1.0) i) (* 100.0 n))
       (if (<= n 1.85e-9) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * fma((((i * i) * n) * 0.041666666666666664), i, n);
	double tmp;
	if (n <= -3.6e-100) {
		tmp = t_0;
	} else if (n <= 5.8e-204) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else if (n <= 1.85e-9) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * fma(Float64(Float64(Float64(i * i) * n) * 0.041666666666666664), i, n))
	tmp = 0.0
	if (n <= -3.6e-100)
		tmp = t_0;
	elseif (n <= 5.8e-204)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	elseif (n <= 1.85e-9)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(N[(i * i), $MachinePrecision] * n), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.6e-100], t$95$0, If[LessEqual[n, 5.8e-204], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.85e-9], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.5999999999999999e-100 or 1.85e-9 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 72.7%

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

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left({i}^{2} \cdot n\right), i, n\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 72.2%

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

   if -3.5999999999999999e-100 < n < 5.80000000000000018e-204

   1. Initial program 50.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 69.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 5.80000000000000018e-204 < n < 1.85e-9

   1. Initial program 12.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 59.8%

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

Alternative 16: 65.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-100} \lor \neg \left(n \leq 2.7 \cdot 10^{-137}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.041666666666666664, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.6e-100) (not (<= n 2.7e-137)))
   (* 100.0 (fma (* (* (* i i) n) 0.041666666666666664) i n))
   (* (/ (- 1.0 1.0) i) (* 100.0 n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.6e-100) || !(n <= 2.7e-137)) {
		tmp = 100.0 * fma((((i * i) * n) * 0.041666666666666664), i, n);
	} else {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.6e-100) || !(n <= 2.7e-137))
		tmp = Float64(100.0 * fma(Float64(Float64(Float64(i * i) * n) * 0.041666666666666664), i, n));
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.6e-100], N[Not[LessEqual[n, 2.7e-137]], $MachinePrecision]], N[(100.0 * N[(N[(N[(N[(i * i), $MachinePrecision] * n), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.5999999999999999e-100 or 2.69999999999999993e-137 < n

   1. Initial program 23.4%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left({i}^{2} \cdot n\right), i, n\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.6%

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

   if -3.5999999999999999e-100 < n < 2.69999999999999993e-137

   1. Initial program 43.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 64.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 64.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.7

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

   7. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{-100} \lor \neg \left(n \leq 2.7 \cdot 10^{-137}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.041666666666666664, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{+21} \lor \neg \left(i \leq 6 \cdot 10^{+225}\right):\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -3.5e+21) (not (<= i 6e+225)))
   (* (/ (- 1.0 1.0) i) (* 100.0 n))
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -3.5e+21) || !(i <= 6e+225)) {
		tmp = ((1.0 - 1.0) / i) * (100.0 * n);
	} else {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -3.5e+21) || !(i <= 6e+225))
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(100.0 * n));
	else
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -3.5e+21], N[Not[LessEqual[i, 6e+225]], $MachinePrecision]], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -3.5e21 or 6.000000000000001e225 < i

   1. Initial program 67.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 49.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 49.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 49.6

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

   7. Applied rewrites 49.6%

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

   if -3.5e21 < i < 6.000000000000001e225

   1. Initial program 15.1%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 71.6%

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{+21} \lor \neg \left(i \leq 6 \cdot 10^{+225}\right):\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 7.4e+22)
   (* 100.0 (* (fma 0.5 i 1.0) n))
   (* 100.0 (* (* (* i i) n) 0.16666666666666666))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 7.4e+22) {
		tmp = 100.0 * (fma(0.5, i, 1.0) * n);
	} else {
		tmp = 100.0 * (((i * i) * n) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 7.4e+22)
		tmp = Float64(100.0 * Float64(fma(0.5, i, 1.0) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(i * i) * n) * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 7.4e+22], N[(100.0 * N[(N[(0.5 * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(i * i), $MachinePrecision] * n), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 7.3999999999999996e22

   1. Initial program 24.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, i, 1\right) \cdot n\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 63.8%

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

   if 7.3999999999999996e22 < i

   1. Initial program 38.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 35.3%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 35.3%

       \[\leadsto 100 \cdot \left(\left(\left(i \cdot i\right) \cdot n\right) \cdot 0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 57.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n)))

C

double code(double i, double n) {
	return 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
}

Julia

function code(i, n)
	return Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 51.9%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 56.0%

   \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]
9. Add Preprocessing

Alternative 20: 57.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 51.9%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 56.0%

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
9. Add Preprocessing

Alternative 21: 55.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 (* (fma 0.5 i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma(0.5, i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(0.5, i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(0.5 * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 51.9%

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

6. Taylor expanded in n around inf

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

8. Applied rewrites 56.0%

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

9. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, i, 1\right) \cdot n\right) \]
10. Step-by-step derivation

11. Applied rewrites 52.5%

    \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right) \]
12. Add Preprocessing

Alternative 22: 50.1% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot n \end{array} \]

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 28.3%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 48.3%

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

Developer Target 1: 33.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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 2025018 
(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))))