Compound Interest

Percentage Accurate: 28.0% → 94.6%

Time: 8.5s

Alternatives: 13

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: 28.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 0.3× speedup

Math

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

FPCore

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 99.0

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

   4. Applied rewrites 99.0%

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

   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 99.8%

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

      1. lift-*.f64 N/A

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

      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 57.1

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 57.1%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. +-commutative N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 83.2%

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

Alternative 2: 94.0% accurate, 0.3× speedup

Math

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

FPCore

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 98.2%

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

   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 99.8%

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

      1. lift-*.f64 N/A

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

      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 57.1

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 57.1%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. +-commutative N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 83.2%

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

Alternative 3: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 98.1

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

   4. Applied rewrites 98.1%

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

   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 99.8%

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

      1. lift-*.f64 N/A

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

      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 57.1

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

   4. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 57.1%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. +-commutative N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 83.2%

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

Alternative 4: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.2 \cdot 10^{-303}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-133}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot \frac{\mathsf{fma}\left(-1, \log n, \log i\right)}{i}\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -3.4e-132)
     t_0
     (if (<= n -1.2e-303)
       (* (* 100.0 n) (/ (expm1 (* (log (/ i n)) n)) i))
       (if (<= n 2.7e-133)
         (* 100.0 (* (* n (/ (fma -1.0 (log n) (log i)) i)) n))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -3.4e-132) {
		tmp = t_0;
	} else if (n <= -1.2e-303) {
		tmp = (100.0 * n) * (expm1((log((i / n)) * n)) / i);
	} else if (n <= 2.7e-133) {
		tmp = 100.0 * ((n * (fma(-1.0, log(n), log(i)) / i)) * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -3.4e-132)
		tmp = t_0;
	elseif (n <= -1.2e-303)
		tmp = Float64(Float64(100.0 * n) * Float64(expm1(Float64(log(Float64(i / n)) * n)) / i));
	elseif (n <= 2.7e-133)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(fma(-1.0, log(n), log(i)) / i)) * n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -3.4e-132], t$95$0, If[LessEqual[n, -1.2e-303], N[(N[(100.0 * n), $MachinePrecision] * N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.7e-133], N[(100.0 * N[(N[(n * N[(N[(-1.0 * N[Log[n], $MachinePrecision] + N[Log[i], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.39999999999999983e-132 or 2.6999999999999999e-133 < n

   1. Initial program 21.5%

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

      1. lift-*.f64 N/A

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

      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 71.5

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

   4. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 71.2%

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

   7. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 84.5

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

   9. Applied rewrites 84.5%

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

   if -3.39999999999999983e-132 < n < -1.2e-303

   1. Initial program 69.7%

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

   3. Taylor expanded in i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in n around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. diff-log N/A

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

      7. lower-log.f64 N/A

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

      8. lift-/.f64 57.4

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

   8. Applied rewrites 57.4%

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

   9. Taylor expanded in i around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

   11. Applied rewrites 85.5%

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

   if -1.2e-303 < n < 2.6999999999999999e-133

   1. Initial program 36.3%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 30.5

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

   5. Applied rewrites 30.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 10.2

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 10.2

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

   7. Applied rewrites 10.2%

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

   8. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 74.0

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

   10. Applied rewrites 74.0%

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

Alternative 5: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{if}\;n \leq -6.8 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-177}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))))
   (if (<= n -6.8e-132)
     t_0
     (if (<= n 5.8e-177)
       (* 100.0 (* (/ (- 1.0 1.0) i) n))
       (if (<= n 1.2e-11) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double tmp;
	if (n <= -6.8e-132) {
		tmp = t_0;
	} else if (n <= 5.8e-177) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.2e-11) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.expm1(i) * n) / i);
	double tmp;
	if (n <= -6.8e-132) {
		tmp = t_0;
	} else if (n <= 5.8e-177) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.2e-11) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) * n) / i)
	tmp = 0
	if n <= -6.8e-132:
		tmp = t_0
	elif n <= 5.8e-177:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	elif n <= 1.2e-11:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	tmp = 0.0
	if (n <= -6.8e-132)
		tmp = t_0;
	elseif (n <= 5.8e-177)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	elseif (n <= 1.2e-11)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.8e-132], t$95$0, If[LessEqual[n, 5.8e-177], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-11], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.79999999999999965e-132 or 1.2000000000000001e-11 < 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 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 86.6

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

   5. Applied rewrites 86.6%

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

   if -6.79999999999999965e-132 < n < 5.79999999999999994e-177

   1. Initial program 59.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 43.2

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

   5. Applied rewrites 43.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 20.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 20.3

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

   7. Applied rewrites 20.3%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative 71.0

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

   10. Applied rewrites 71.0%

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

   if 5.79999999999999994e-177 < n < 1.2000000000000001e-11

   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 i around 0

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

   5. Applied rewrites 66.2%

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

Alternative 6: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-132} \lor \neg \left(n \leq 6.5 \cdot 10^{-165}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3e-132) (not (<= n 6.5e-165)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (* (/ (- 1.0 1.0) i) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3e-132) || !(n <= 6.5e-165)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -3e-132) || !(n <= 6.5e-165)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -3e-132) or not (n <= 6.5e-165):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3e-132) || !(n <= 6.5e-165))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * 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, -3e-132], N[Not[LessEqual[n, 6.5e-165]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $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 < -3e-132 or 6.5000000000000004e-165 < n

   1. Initial program 21.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 72.7

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

   4. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 72.4%

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

   7. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-expm1.f64 82.9

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

   9. Applied rewrites 82.9%

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

   if -3e-132 < n < 6.5000000000000004e-165

   1. Initial program 58.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 19.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative 71.7

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

   10. Applied rewrites 71.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-132} \lor \neg \left(n \leq 6.5 \cdot 10^{-165}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(i \cdot 100\right) \cdot n}{i}\\ \mathbf{if}\;n \leq -1.36 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-177}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* (* i 100.0) n) i)))
   (if (<= n -1.36e-104)
     t_0
     (if (<= n 5.8e-177)
       (* 100.0 (* (/ (- 1.0 1.0) i) n))
       (if (<= n 1.2e-11) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = ((i * 100.0) * n) / i;
	double tmp;
	if (n <= -1.36e-104) {
		tmp = t_0;
	} else if (n <= 5.8e-177) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.2e-11) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((i * 100.0d0) * n) / i
    if (n <= (-1.36d-104)) then
        tmp = t_0
    else if (n <= 5.8d-177) then
        tmp = 100.0d0 * (((1.0d0 - 1.0d0) / i) * n)
    else if (n <= 1.2d-11) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = ((i * 100.0) * n) / i;
	double tmp;
	if (n <= -1.36e-104) {
		tmp = t_0;
	} else if (n <= 5.8e-177) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.2e-11) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((i * 100.0) * n) / i
	tmp = 0
	if n <= -1.36e-104:
		tmp = t_0
	elif n <= 5.8e-177:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	elif n <= 1.2e-11:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(i * 100.0) * n) / i)
	tmp = 0.0
	if (n <= -1.36e-104)
		tmp = t_0;
	elseif (n <= 5.8e-177)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	elseif (n <= 1.2e-11)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = ((i * 100.0) * n) / i;
	tmp = 0.0;
	if (n <= -1.36e-104)
		tmp = t_0;
	elseif (n <= 5.8e-177)
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	elseif (n <= 1.2e-11)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(i * 100.0), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.36e-104], t$95$0, If[LessEqual[n, 5.8e-177], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.2e-11], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.35999999999999997e-104 or 1.2000000000000001e-11 < n

   1. Initial program 22.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 69.4

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

   4. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 69.0%

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

   7. Taylor expanded in i around 0

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

      1. pow-to-exp 55.4

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

      2. +-commutative 55.4

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

   9. Applied rewrites 55.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 63.6

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

       6. +-commutative 63.6

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

   11. Applied rewrites 63.6%

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

   if -1.35999999999999997e-104 < n < 5.79999999999999994e-177

   1. Initial program 56.9%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 40.4

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

   5. Applied rewrites 40.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 18.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 18.3

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

   7. Applied rewrites 18.3%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative 66.9

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

   10. Applied rewrites 66.9%

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

   if 5.79999999999999994e-177 < n < 1.2000000000000001e-11

   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 i around 0

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

   5. Applied rewrites 66.2%

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

Alternative 8: 63.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot i, 100, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(0.5, i, 1\right) \cdot i}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3e-132)
   (* (fma (* (- 0.5 (/ 0.5 n)) i) 100.0 100.0) n)
   (if (<= n 6.5e-165)
     (* 100.0 (* (/ (- 1.0 1.0) i) n))
     (* 100.0 (* (/ (* (fma 0.5 i 1.0) i) i) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3e-132) {
		tmp = fma(((0.5 - (0.5 / n)) * i), 100.0, 100.0) * n;
	} else if (n <= 6.5e-165) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * (((fma(0.5, i, 1.0) * i) / i) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3e-132)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(0.5 / n)) * i), 100.0, 100.0) * n);
	elseif (n <= 6.5e-165)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(fma(0.5, i, 1.0) * i) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3e-132], N[(N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 6.5e-165], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3e-132

   1. Initial program 21.5%

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

      1. lift-*.f64 N/A

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

      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 66.9

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

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 66.4%

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

   7. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 58.4

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

   9. Applied rewrites 58.4%

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

   if -3e-132 < n < 6.5000000000000004e-165

   1. Initial program 58.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 19.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative 71.7

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

   10. Applied rewrites 71.7%

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

   if 6.5000000000000004e-165 < n

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 53.5%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(0.5, i, 1\right) \cdot i}{\frac{i}{n}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.8

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

   10. Applied rewrites 68.8%

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

Alternative 9: 62.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-132} \lor \neg \left(n \leq 6.5 \cdot 10^{-165}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3e-132) (not (<= n 6.5e-165)))
   (* 100.0 (fma (* (- 0.5 (/ 0.5 n)) n) i n))
   (* 100.0 (* (/ (- 1.0 1.0) i) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3e-132) || !(n <= 6.5e-165)) {
		tmp = 100.0 * fma(((0.5 - (0.5 / n)) * 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 <= -3e-132) || !(n <= 6.5e-165))
		tmp = Float64(100.0 * fma(Float64(Float64(0.5 - Float64(0.5 / n)) * 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, -3e-132], N[Not[LessEqual[n, 6.5e-165]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $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 < -3e-132 or 6.5000000000000004e-165 < n

   1. Initial program 21.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if -3e-132 < n < 6.5000000000000004e-165

   1. Initial program 58.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 19.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative 71.7

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

   10. Applied rewrites 71.7%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-132} \lor \neg \left(n \leq 6.5 \cdot 10^{-165}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{0.5}{n}\\ \mathbf{if}\;n \leq -3 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot i, 100, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(t\_0 \cdot n, i, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ 0.5 n))))
   (if (<= n -3e-132)
     (* (fma (* t_0 i) 100.0 100.0) n)
     (if (<= n 6.5e-165)
       (* 100.0 (* (/ (- 1.0 1.0) i) n))
       (* 100.0 (fma (* t_0 n) i n))))))

C

double code(double i, double n) {
	double t_0 = 0.5 - (0.5 / n);
	double tmp;
	if (n <= -3e-132) {
		tmp = fma((t_0 * i), 100.0, 100.0) * n;
	} else if (n <= 6.5e-165) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * fma((t_0 * n), i, n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(0.5 - Float64(0.5 / n))
	tmp = 0.0
	if (n <= -3e-132)
		tmp = Float64(fma(Float64(t_0 * i), 100.0, 100.0) * n);
	elseif (n <= 6.5e-165)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * fma(Float64(t_0 * n), i, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3e-132], N[(N[(N[(t$95$0 * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 6.5e-165], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(t$95$0 * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3e-132

   1. Initial program 21.5%

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

      1. lift-*.f64 N/A

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

      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 66.9

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

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 66.4%

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

   7. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 58.4

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

   9. Applied rewrites 58.4%

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

   if -3e-132 < n < 6.5000000000000004e-165

   1. Initial program 58.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 19.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative 71.7

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

   10. Applied rewrites 71.7%

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

   if 6.5000000000000004e-165 < n

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

Alternative 11: 60.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.36 \cdot 10^{-104} \lor \neg \left(n \leq 5.6 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{\left(i \cdot 100\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.36e-104) (not (<= n 5.6e-95)))
   (/ (* (* i 100.0) n) i)
   (* 100.0 (* (/ (- 1.0 1.0) i) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.36e-104) || !(n <= 5.6e-95)) {
		tmp = ((i * 100.0) * n) / i;
	} else {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.36d-104)) .or. (.not. (n <= 5.6d-95))) then
        tmp = ((i * 100.0d0) * n) / i
    else
        tmp = 100.0d0 * (((1.0d0 - 1.0d0) / i) * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.36e-104) || !(n <= 5.6e-95)) {
		tmp = ((i * 100.0) * n) / i;
	} else {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.36e-104) or not (n <= 5.6e-95):
		tmp = ((i * 100.0) * n) / i
	else:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.36e-104) || !(n <= 5.6e-95))
		tmp = Float64(Float64(Float64(i * 100.0) * n) / i);
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.36e-104) || ~((n <= 5.6e-95)))
		tmp = ((i * 100.0) * n) / i;
	else
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.36e-104], N[Not[LessEqual[n, 5.6e-95]], $MachinePrecision]], N[(N[(N[(i * 100.0), $MachinePrecision] * n), $MachinePrecision] / i), $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 < -1.35999999999999997e-104 or 5.5999999999999998e-95 < n

   1. Initial program 21.3%

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

      1. lift-*.f64 N/A

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

      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 70.7

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

   4. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 70.4%

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

   7. Taylor expanded in i around 0

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

      1. pow-to-exp 55.4

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

      2. +-commutative 55.4

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

   9. Applied rewrites 55.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 62.9

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

       6. +-commutative 62.9

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

   11. Applied rewrites 62.9%

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

   if -1.35999999999999997e-104 < n < 5.5999999999999998e-95

   1. Initial program 49.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 36.1

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

   5. Applied rewrites 36.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 15.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 15.6

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

   7. Applied rewrites 15.6%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative 60.1

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

   10. Applied rewrites 60.1%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.36 \cdot 10^{-104} \lor \neg \left(n \leq 5.6 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{\left(i \cdot 100\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.85 \cdot 10^{-151}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i \cdot 100\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.85e-151) (* 100.0 n) (/ (* (* i 100.0) n) i)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.85e-151) {
		tmp = 100.0 * n;
	} else {
		tmp = ((i * 100.0) * n) / i;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 2.85d-151) then
        tmp = 100.0d0 * n
    else
        tmp = ((i * 100.0d0) * n) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.85e-151) {
		tmp = 100.0 * n;
	} else {
		tmp = ((i * 100.0) * n) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.85e-151:
		tmp = 100.0 * n
	else:
		tmp = ((i * 100.0) * n) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.85e-151)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(Float64(Float64(i * 100.0) * n) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.85e-151)
		tmp = 100.0 * n;
	else
		tmp = ((i * 100.0) * n) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.85e-151], N[(100.0 * n), $MachinePrecision], N[(N[(N[(i * 100.0), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2.84999999999999994e-151

   1. Initial program 22.6%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 58.8%

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

   if 2.84999999999999994e-151 < i

   1. Initial program 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      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 74.3

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

   4. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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 74.5%

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

   7. Taylor expanded in i around 0

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

      1. pow-to-exp 31.4

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

      2. +-commutative 31.4

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

   9. Applied rewrites 31.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 47.5

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

       6. +-commutative 47.5

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

   11. Applied rewrites 47.5%

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

Alternative 13: 50.1% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot n \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.1%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 48.3%

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

Developer Target 1: 33.4% 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 2025037 
(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))))