Compound Interest

Percentage Accurate: 28.6% → 95.6%

Time: 9.2s

Alternatives: 16

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: 28.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (* 100.0 (* (/ (- (pow (/ i n) n) 1.0) i) n))
     (if (<= t_0 0.0)
       (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
       (if (<= t_0 INFINITY)
         (* 100.0 (- (/ (pow (+ (/ i n) 1.0) n) (/ i n)) (/ 1.0 (/ i 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 <= -((double) INFINITY)) {
		tmp = 100.0 * (((pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * ((pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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 <= -Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * Math.expm1((Math.log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((Math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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 <= -math.inf:
		tmp = 100.0 * (((math.pow((i / n), n) - 1.0) / i) * n)
	elif t_0 <= 0.0:
		tmp = (100.0 * math.expm1((math.log1p((i / n)) * n))) / (i / n)
	elif t_0 <= math.inf:
		tmp = 100.0 * ((math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(i / n) ^ n) - 1.0) / i) * n));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) / Float64(i / n)) - Float64(1.0 / Float64(i / 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, (-Infinity)], N[(100.0 * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in i around inf

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

      1. lift-/.f64 100.0

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

   9. Applied rewrites 100.0%

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

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

   1. Initial program 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

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

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/r/ N/A

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

      9. pow-to-exp N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 98.5%

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

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

   1. Initial program 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 83.9%

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

Alternative 2: 94.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-28}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

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-28)
     (* 100.0 (* (/ (- (pow (/ i n) n) 1.0) i) n))
     (if (<= t_0 0.0)
       (* (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) i) n)
       (if (<= t_0 INFINITY)
         (* 100.0 (- (/ (pow (+ (/ i n) 1.0) n) (/ i n)) (/ 1.0 (/ i 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-28) {
		tmp = 100.0 * (((pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = ((100.0 * expm1((log1p((i / n)) * n))) / i) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * ((pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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-28) {
		tmp = 100.0 * (((Math.pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = ((100.0 * Math.expm1((Math.log1p((i / n)) * n))) / i) * n;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((Math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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-28:
		tmp = 100.0 * (((math.pow((i / n), n) - 1.0) / i) * n)
	elif t_0 <= 0.0:
		tmp = ((100.0 * math.expm1((math.log1p((i / n)) * n))) / i) * n
	elif t_0 <= math.inf:
		tmp = 100.0 * ((math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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-28)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(i / n) ^ n) - 1.0) / i) * n));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / i) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) / Float64(i / n)) - Float64(1.0 / Float64(i / 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-28], N[(100.0 * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -5.0000000000000002e-28

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in i around inf

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

      1. lift-/.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -5.0000000000000002e-28 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 26.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

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

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/r/ N/A

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

      9. pow-to-exp N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 98.5%

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

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

   1. Initial program 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 83.9%

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

Alternative 3: 94.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-20}:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

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-20)
     (* 100.0 (* (/ (- (pow (/ i n) n) 1.0) i) n))
     (if (<= t_0 0.0)
       (* (* 100.0 (/ (expm1 (* (log1p (/ i n)) n)) i)) n)
       (if (<= t_0 INFINITY)
         (* 100.0 (- (/ (pow (+ (/ i n) 1.0) n) (/ i n)) (/ 1.0 (/ i 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-20) {
		tmp = 100.0 * (((pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * (expm1((log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * ((pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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-20) {
		tmp = 100.0 * (((Math.pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * (Math.expm1((Math.log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((Math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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-20:
		tmp = 100.0 * (((math.pow((i / n), n) - 1.0) / i) * n)
	elif t_0 <= 0.0:
		tmp = (100.0 * (math.expm1((math.log1p((i / n)) * n)) / i)) * n
	elif t_0 <= math.inf:
		tmp = 100.0 * ((math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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-20)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(i / n) ^ n) - 1.0) / i) * n));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) / Float64(i / n)) - Float64(1.0 / Float64(i / 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-20], N[(100.0 * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], 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[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -4.9999999999999999e-20

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 33.3

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

   4. Applied rewrites 33.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in i around inf

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

      1. lift-/.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -4.9999999999999999e-20 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 27.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

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

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/r/ N/A

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

      9. pow-to-exp N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 98.5%

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

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

   1. Initial program 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 83.9%

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

Alternative 4: 94.9% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (* 100.0 (* (/ (- (pow (/ i n) n) 1.0) i) n))
     (if (<= t_0 0.0)
       (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
       (if (<= t_0 INFINITY)
         (* 100.0 (- (/ (pow (+ (/ i n) 1.0) n) (/ i n)) (/ 1.0 (/ i 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 <= -((double) INFINITY)) {
		tmp = 100.0 * (((pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * ((pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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 <= -Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow((i / n), n) - 1.0) / i) * n);
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((Math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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 <= -math.inf:
		tmp = 100.0 * (((math.pow((i / n), n) - 1.0) / i) * n)
	elif t_0 <= 0.0:
		tmp = 100.0 * ((math.expm1((math.log1p((i / n)) * n)) / i) * n)
	elif t_0 <= math.inf:
		tmp = 100.0 * ((math.pow(((i / n) + 1.0), n) / (i / n)) - (1.0 / (i / 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 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(i / n) ^ n) - 1.0) / i) * n));
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) / Float64(i / n)) - Float64(1.0 / Float64(i / 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, (-Infinity)], N[(100.0 * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in i around inf

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

      1. lift-/.f64 100.0

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

   9. Applied rewrites 100.0%

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

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

   1. Initial program 28.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

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

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/r/ N/A

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

      9. pow-to-exp N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

   8. Applied rewrites 98.5%

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

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

   1. Initial program 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 83.9%

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

Alternative 5: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-225}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-271}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-21}:\\ \;\;\;\;\left(100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.32e-225)
   (* (* 100.0 (/ (expm1 i) i)) n)
   (if (<= n 6.8e-271)
     (* 100.0 (* (/ (- 1.0 1.0) i) n))
     (if (<= n 5.6e-21)
       (* (* 100.0 (/ (* n (- (log i) (log n))) i)) n)
       (* 100.0 (/ (* (expm1 i) n) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else if (n <= 6.8e-271) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 5.6e-21) {
		tmp = (100.0 * ((n * (log(i) - log(n))) / i)) * n;
	} else {
		tmp = 100.0 * ((expm1(i) * n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else if (n <= 6.8e-271) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 5.6e-21) {
		tmp = (100.0 * ((n * (Math.log(i) - Math.log(n))) / i)) * n;
	} else {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.32e-225:
		tmp = (100.0 * (math.expm1(i) / i)) * n
	elif n <= 6.8e-271:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	elif n <= 5.6e-21:
		tmp = (100.0 * ((n * (math.log(i) - math.log(n))) / i)) * n
	else:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.32e-225)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	elseif (n <= 6.8e-271)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	elseif (n <= 5.6e-21)
		tmp = Float64(Float64(100.0 * Float64(Float64(n * Float64(log(i) - log(n))) / i)) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.32e-225], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 6.8e-271], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.6e-21], N[(N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.32e-225

   1. Initial program 27.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 81.4%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 80.7%

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

   if -1.32e-225 < n < 6.8000000000000001e-271

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 93.9

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

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 93.9%

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

   if 6.8000000000000001e-271 < n < 5.60000000000000008e-21

   1. Initial program 27.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 80.0

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

   4. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 80.0%

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

   7. Taylor expanded in n around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 75.2

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

   9. Applied rewrites 75.2%

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

   if 5.60000000000000008e-21 < n

   1. Initial program 23.8%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-225}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{-271}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-21}:\\ \;\;\;\;\left(100 \cdot \frac{n \cdot \left(\log i - \log n\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-225}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-116}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log i - \log n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.32e-225)
   (* (* 100.0 (/ (expm1 i) i)) n)
   (if (<= n 1.35e-116)
     (* 100.0 (/ (* n n) n))
     (if (<= n 5.6e-21)
       (* 100.0 (* (* n n) (/ (- (log i) (log n)) i)))
       (* 100.0 (/ (* (expm1 i) n) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else if (n <= 1.35e-116) {
		tmp = 100.0 * ((n * n) / n);
	} else if (n <= 5.6e-21) {
		tmp = 100.0 * ((n * n) * ((log(i) - log(n)) / i));
	} else {
		tmp = 100.0 * ((expm1(i) * n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else if (n <= 1.35e-116) {
		tmp = 100.0 * ((n * n) / n);
	} else if (n <= 5.6e-21) {
		tmp = 100.0 * ((n * n) * ((Math.log(i) - Math.log(n)) / i));
	} else {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.32e-225:
		tmp = (100.0 * (math.expm1(i) / i)) * n
	elif n <= 1.35e-116:
		tmp = 100.0 * ((n * n) / n)
	elif n <= 5.6e-21:
		tmp = 100.0 * ((n * n) * ((math.log(i) - math.log(n)) / i))
	else:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.32e-225)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	elseif (n <= 1.35e-116)
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	elseif (n <= 5.6e-21)
		tmp = Float64(100.0 * Float64(Float64(n * n) * Float64(Float64(log(i) - log(n)) / i)));
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.32e-225], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.35e-116], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.6e-21], N[(100.0 * N[(N[(n * n), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.32e-225

   1. Initial program 27.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 81.4%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 80.7%

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

   if -1.32e-225 < n < 1.35e-116

   1. Initial program 59.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 8.5%

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

   6. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 17.5%

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

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

   if 1.35e-116 < n < 5.60000000000000008e-21

   1. Initial program 10.5%

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. log-pow-rev N/A

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

      9. unpow1 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 5.60000000000000008e-21 < n

   1. Initial program 23.8%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 7: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-225}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-198}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \frac{\log \left(\frac{i}{n}\right) \cdot n}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.32e-225)
   (* (* 100.0 (/ (expm1 i) i)) n)
   (if (<= n 3.7e-198)
     (* 100.0 (/ (* n n) n))
     (if (<= n 5.6e-21)
       (* 100.0 (/ (* (log (/ i n)) n) (/ i n)))
       (* 100.0 (/ (* (expm1 i) n) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else if (n <= 3.7e-198) {
		tmp = 100.0 * ((n * n) / n);
	} else if (n <= 5.6e-21) {
		tmp = 100.0 * ((log((i / n)) * n) / (i / n));
	} else {
		tmp = 100.0 * ((expm1(i) * n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else if (n <= 3.7e-198) {
		tmp = 100.0 * ((n * n) / n);
	} else if (n <= 5.6e-21) {
		tmp = 100.0 * ((Math.log((i / n)) * n) / (i / n));
	} else {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.32e-225:
		tmp = (100.0 * (math.expm1(i) / i)) * n
	elif n <= 3.7e-198:
		tmp = 100.0 * ((n * n) / n)
	elif n <= 5.6e-21:
		tmp = 100.0 * ((math.log((i / n)) * n) / (i / n))
	else:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.32e-225)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	elseif (n <= 3.7e-198)
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	elseif (n <= 5.6e-21)
		tmp = Float64(100.0 * Float64(Float64(log(Float64(i / n)) * n) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.32e-225], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 3.7e-198], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.6e-21], N[(100.0 * N[(N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.32e-225

   1. Initial program 27.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 81.4%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 80.7%

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

   if -1.32e-225 < n < 3.69999999999999971e-198

   1. Initial program 72.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 0.8%

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

   6. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 12.8%

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

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f64 90.3

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

   11. Applied rewrites 90.3%

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

   if 3.69999999999999971e-198 < n < 5.60000000000000008e-21

   1. Initial program 17.6%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. log-pow-rev N/A

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

      6. unpow1 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 66.6

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

   5. Applied rewrites 66.6%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lift-/.f64 55.7

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

   7. Applied rewrites 55.7%

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

   if 5.60000000000000008e-21 < n

   1. Initial program 23.8%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 91.6

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

   5. Applied rewrites 91.6%

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

Alternative 8: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-45} \lor \neg \left(n \leq 4.2 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9.8e-45) (not (<= n 4.2e-6)))
   (* 100.0 (/ (* (expm1 i) n) i))
   (* 100.0 (/ (* n n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9.8e-45) || !(n <= 4.2e-6)) {
		tmp = 100.0 * ((expm1(i) * n) / i);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9.8e-45) || !(n <= 4.2e-6)) {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9.8e-45) or not (n <= 4.2e-6):
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	else:
		tmp = 100.0 * ((n * n) / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9.8e-45) || !(n <= 4.2e-6))
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9.8e-45], N[Not[LessEqual[n, 4.2e-6]], $MachinePrecision]], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.7999999999999996e-45 or 4.1999999999999996e-6 < n

   1. Initial program 23.7%

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -9.7999999999999996e-45 < n < 4.1999999999999996e-6

   1. Initial program 42.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 20.9%

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

   6. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 26.1%

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

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f64 60.8

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

   11. Applied rewrites 60.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.8 \cdot 10^{-45} \lor \neg \left(n \leq 4.2 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-225}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-92}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.32e-225)
   (* (* 100.0 (/ (expm1 i) i)) n)
   (if (<= n 2.6e-92)
     (* 100.0 (* (/ (- 1.0 1.0) i) n))
     (* 100.0 (/ (* (expm1 i) n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else if (n <= 2.6e-92) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * ((expm1(i) * n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else if (n <= 2.6e-92) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.32e-225:
		tmp = (100.0 * (math.expm1(i) / i)) * n
	elif n <= 2.6e-92:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	else:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.32e-225)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	elseif (n <= 2.6e-92)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.32e-225], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.6e-92], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.32e-225

   1. Initial program 27.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 81.4%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 80.7%

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

   if -1.32e-225 < n < 2.6e-92

   1. Initial program 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 84.2

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

   4. Applied rewrites 84.2%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 56.1

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

   6. Applied rewrites 56.1%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 71.7%

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

   if 2.6e-92 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 10: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.32 \cdot 10^{-225}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-92}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.32e-225)
   (* 100.0 (* (/ (expm1 i) i) n))
   (if (<= n 2.6e-92)
     (* 100.0 (* (/ (- 1.0 1.0) i) n))
     (* 100.0 (/ (* (expm1 i) n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = 100.0 * ((expm1(i) / i) * n);
	} else if (n <= 2.6e-92) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * ((expm1(i) * n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.32e-225) {
		tmp = 100.0 * ((Math.expm1(i) / i) * n);
	} else if (n <= 2.6e-92) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.32e-225:
		tmp = 100.0 * ((math.expm1(i) / i) * n)
	elif n <= 2.6e-92:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	else:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.32e-225)
		tmp = Float64(100.0 * Float64(Float64(expm1(i) / i) * n));
	elseif (n <= 2.6e-92)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.32e-225], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e-92], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.32e-225

   1. Initial program 27.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 80.7%

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

   if -1.32e-225 < n < 2.6e-92

   1. Initial program 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 84.2

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

   4. Applied rewrites 84.2%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 56.1

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

   6. Applied rewrites 56.1%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 71.7%

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

   if 2.6e-92 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 11: 63.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-163} \lor \neg \left(n \leq 8.5 \cdot 10^{-93}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.7e-163) (not (<= n 8.5e-93)))
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (* 100.0 (* (/ (- 1.0 1.0) i) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.7e-163) || !(n <= 8.5e-93)) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.7e-163) || !(n <= 8.5e-93))
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.7e-163], N[Not[LessEqual[n, 8.5e-93]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.6999999999999999e-163 or 8.5000000000000007e-93 < n

   1. Initial program 22.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 60.9%

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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 61.0

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

   8. Applied rewrites 61.0%

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

   if -3.6999999999999999e-163 < n < 8.5000000000000007e-93

   1. Initial program 56.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 56.5

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

   6. Applied rewrites 56.5%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 68.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-163} \lor \neg \left(n \leq 8.5 \cdot 10^{-93}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-163} \lor \neg \left(n \leq 3 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.7e-163) (not (<= n 3e-6)))
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (* 100.0 (/ (* n n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.7e-163) || !(n <= 3e-6)) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.7e-163) || !(n <= 3e-6))
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.7e-163], N[Not[LessEqual[n, 3e-6]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.6999999999999999e-163 or 3.0000000000000001e-6 < n

   1. Initial program 23.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 62.0%

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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if -3.6999999999999999e-163 < n < 3.0000000000000001e-6

   1. Initial program 47.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 13.1%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{n} \]

   8. Applied rewrites 18.5%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

       2. lift-*.f64 61.7

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   11. Applied rewrites 61.7%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-163} \lor \neg \left(n \leq 3 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.46 \cdot 10^{+143} \lor \neg \left(n \leq 3 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot i\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.46e+143) (not (<= n 3e-6)))
   (* 100.0 (fma (* (* 0.16666666666666666 i) n) i n))
   (* 100.0 (/ (* n n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.46e+143) || !(n <= 3e-6)) {
		tmp = 100.0 * fma(((0.16666666666666666 * i) * n), i, n);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.46e+143) || !(n <= 3e-6))
		tmp = Float64(100.0 * fma(Float64(Float64(0.16666666666666666 * i) * n), i, n));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.46e+143], N[Not[LessEqual[n, 3e-6]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.45999999999999997e143 or 3.0000000000000001e-6 < n

   1. Initial program 18.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 68.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   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

      1. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot n, i, n\right) \]

      4. lower-fma.f64 68.8

         \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   8. Applied rewrites 68.8%

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 68.0

          \[\leadsto 100 \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot i\right) \cdot n, i, n\right) \]

   11. Applied rewrites 68.0%

       \[\leadsto 100 \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot i\right) \cdot n, i, n\right) \]

   if -1.45999999999999997e143 < n < 3.0000000000000001e-6

   1. Initial program 42.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 27.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{n} \]

   8. Applied rewrites 31.5%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

       2. lift-*.f64 55.6

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   11. Applied rewrites 55.6%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.46 \cdot 10^{+143} \lor \neg \left(n \leq 3 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.16666666666666666 \cdot i\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.46 \cdot 10^{+153} \lor \neg \left(n \leq 3 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(0.5 \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.46e+153) (not (<= n 3e-6)))
   (* 100.0 (fma (* 0.5 n) i n))
   (* 100.0 (/ (* n n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.46e+153) || !(n <= 3e-6)) {
		tmp = 100.0 * fma((0.5 * n), i, n);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.46e+153) || !(n <= 3e-6))
		tmp = Float64(100.0 * fma(Float64(0.5 * n), i, n));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.46e+153], N[Not[LessEqual[n, 3e-6]], $MachinePrecision]], N[(100.0 * N[(N[(0.5 * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.4600000000000001e153 or 3.0000000000000001e-6 < n

   1. Initial program 18.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 68.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   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

      1. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot n, i, n\right) \]

      4. lower-fma.f64 68.8

         \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   8. Applied rewrites 68.8%

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot n, i, n\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 63.5%

       \[\leadsto 100 \cdot \mathsf{fma}\left(0.5 \cdot n, i, n\right) \]

   if -1.4600000000000001e153 < n < 3.0000000000000001e-6

   1. Initial program 42.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 27.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{n} \]

   8. Applied rewrites 31.5%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

       2. lift-*.f64 55.6

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   11. Applied rewrites 55.6%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.46 \cdot 10^{+153} \lor \neg \left(n \leq 3 \cdot 10^{-6}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(0.5 \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \mathsf{fma}\left(0.5 \cdot n, i, n\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 (fma (* 0.5 n) i n)))

C

double code(double i, double n) {
	return 100.0 * fma((0.5 * n), i, n);
}

Julia

function code(i, n)
	return Float64(100.0 * fma(Float64(0.5 * n), i, n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(0.5 * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
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

   1. +-commutative N/A

      \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

   2. *-commutative N/A

      \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

5. Applied rewrites 48.5%

   \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

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

   1. *-commutative N/A

      \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

   2. lower-*.f64 N/A

      \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

   3. +-commutative N/A

      \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot n, i, n\right) \]

   4. lower-fma.f64 52.2

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

8. Applied rewrites 52.2%

   \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

9. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot n, i, n\right) \]
10. Step-by-step derivation

11. Applied rewrites 49.5%

    \[\leadsto 100 \cdot \mathsf{fma}\left(0.5 \cdot n, i, n\right) \]
12. Add Preprocessing

Alternative 16: 48.1% 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 30.1%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \color{blue}{n} \]
4. Step-by-step derivation

5. Applied rewrites 42.7%

   \[\leadsto 100 \cdot \color{blue}{n} \]
6. Add Preprocessing

Developer Target 1: 35.0% 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 2025038 
(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))))