Compound Interest

Percentage Accurate: 28.7% → 94.2%

Time: 8.1s

Alternatives: 14

Speedup: 24.3×

• could not determine a ground truth

  1. i = 2.80643759469396e-191
  2. n = 1.3156781859645709e+28

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: 28.7% 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: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-266}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 1e-266)
     (* (/ (expm1 (* (log1p (/ i n)) n)) (/ i n)) 100.0)
     (if (<= t_0 INFINITY)
       (* 100.0 (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) n))
       (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-266) {
		tmp = (expm1((log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-266) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 1e-266:
		tmp = (math.expm1((math.log1p((i / n)) * n)) / (i / n)) * 100.0
	elif t_0 <= math.inf:
		tmp = 100.0 * (((math.pow(((i / n) + 1.0), n) - 1.0) / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 1e-266)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / Float64(i / n)) * 100.0);
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / 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, 1e-266], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $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))) < 9.9999999999999998e-267

   1. Initial program 27.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 97.5

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

   3. Applied rewrites 97.5%

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

   if 9.9999999999999998e-267 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 55.2

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

   3. Applied rewrites 55.2%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 79.4%

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

Alternative 2: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-266}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 1e-266)
     (* (* 100.0 (/ (expm1 (* (log1p (/ i n)) n)) i)) n)
     (if (<= t_0 INFINITY)
       (* 100.0 (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) n))
       (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-266) {
		tmp = (100.0 * (expm1((log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-266) {
		tmp = (100.0 * (Math.expm1((Math.log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 1e-266:
		tmp = (100.0 * (math.expm1((math.log1p((i / n)) * n)) / i)) * n
	elif t_0 <= math.inf:
		tmp = 100.0 * (((math.pow(((i / n) + 1.0), n) - 1.0) / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 1e-266)
		tmp = Float64(Float64(100.0 * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / 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, 1e-266], N[(N[(100.0 * N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $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))) < 9.9999999999999998e-267

   1. Initial program 27.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 96.4

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

   3. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   if 9.9999999999999998e-267 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 55.2

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

   3. Applied rewrites 55.2%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 79.4%

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

Alternative 3: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-266}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 1e-266)
     (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
     (if (<= t_0 INFINITY)
       (* 100.0 (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) n))
       (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-266) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-266) {
		tmp = 100.0 * ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 1e-266:
		tmp = 100.0 * ((math.expm1((math.log1p((i / n)) * n)) / i) * n)
	elif t_0 <= math.inf:
		tmp = 100.0 * (((math.pow(((i / n) + 1.0), n) - 1.0) / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 1e-266)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / 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, 1e-266], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $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))) < 9.9999999999999998e-267

   1. Initial program 27.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 96.4

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

   3. Applied rewrites 96.4%

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

   if 9.9999999999999998e-267 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 55.2

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

   3. Applied rewrites 55.2%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 79.4%

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

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-211}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 0.0135:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -4.2e-50)
     t_0
     (if (<= n -7.5e-242)
       t_1
       (if (<= n 1.5e-211)
         (* (* (/ (- 1.0 1.0) i) n) 100.0)
         (if (<= n 0.0135) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -4.2e-50) {
		tmp = t_0;
	} else if (n <= -7.5e-242) {
		tmp = t_1;
	} else if (n <= 1.5e-211) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else if (n <= 0.0135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.expm1(i) * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -4.2e-50) {
		tmp = t_0;
	} else if (n <= -7.5e-242) {
		tmp = t_1;
	} else if (n <= 1.5e-211) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else if (n <= 0.0135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) * n) / i)
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -4.2e-50:
		tmp = t_0
	elif n <= -7.5e-242:
		tmp = t_1
	elif n <= 1.5e-211:
		tmp = (((1.0 - 1.0) / i) * n) * 100.0
	elif n <= 0.0135:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -4.2e-50)
		tmp = t_0;
	elseif (n <= -7.5e-242)
		tmp = t_1;
	elseif (n <= 1.5e-211)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	elseif (n <= 0.0135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.2e-50], t$95$0, If[LessEqual[n, -7.5e-242], t$95$1, If[LessEqual[n, 1.5e-211], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 0.0135], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.2000000000000002e-50 or 0.0134999999999999998 < n

   1. Initial program 25.8%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-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 89.1

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

   4. Applied rewrites 89.1%

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

   if -4.2000000000000002e-50 < n < -7.4999999999999998e-242 or 1.50000000000000002e-211 < n < 0.0134999999999999998

   1. Initial program 27.7%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 62.1%

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

   if -7.4999999999999998e-242 < n < 1.50000000000000002e-211

   1. Initial program 54.8%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 81.5

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 81.5

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

   6. Applied rewrites 81.5%

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

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-211}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 0.0135:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* 100.0 (* (expm1 i) n)) i)) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -3.9e-50)
     t_0
     (if (<= n -7.5e-242)
       t_1
       (if (<= n 1.5e-211)
         (* (* (/ (- 1.0 1.0) i) n) 100.0)
         (if (<= n 0.0135) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * (expm1(i) * n)) / i;
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -3.9e-50) {
		tmp = t_0;
	} else if (n <= -7.5e-242) {
		tmp = t_1;
	} else if (n <= 1.5e-211) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else if (n <= 0.0135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (100.0 * (Math.expm1(i) * n)) / i;
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -3.9e-50) {
		tmp = t_0;
	} else if (n <= -7.5e-242) {
		tmp = t_1;
	} else if (n <= 1.5e-211) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else if (n <= 0.0135) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * (math.expm1(i) * n)) / i
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -3.9e-50:
		tmp = t_0
	elif n <= -7.5e-242:
		tmp = t_1
	elif n <= 1.5e-211:
		tmp = (((1.0 - 1.0) / i) * n) * 100.0
	elif n <= 0.0135:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * Float64(expm1(i) * n)) / i)
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -3.9e-50)
		tmp = t_0;
	elseif (n <= -7.5e-242)
		tmp = t_1;
	elseif (n <= 1.5e-211)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	elseif (n <= 0.0135)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.9e-50], t$95$0, If[LessEqual[n, -7.5e-242], t$95$1, If[LessEqual[n, 1.5e-211], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 0.0135], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.90000000000000021e-50 or 0.0134999999999999998 < n

   1. Initial program 25.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 68.2

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

   3. Applied rewrites 68.2%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 89.3%

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

   7. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 88.7

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

   9. Applied rewrites 88.7%

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

   if -3.90000000000000021e-50 < n < -7.4999999999999998e-242 or 1.50000000000000002e-211 < n < 0.0134999999999999998

   1. Initial program 27.7%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 62.1%

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

   if -7.4999999999999998e-242 < n < 1.50000000000000002e-211

   1. Initial program 54.8%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 81.5

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 81.5

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

   6. Applied rewrites 81.5%

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

Alternative 6: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -7.5 \cdot 10^{-242}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -7.5e-242)
     (* (* 100.0 t_0) n)
     (if (<= n 9e-199)
       (* (* (/ (- 1.0 1.0) i) n) 100.0)
       (* 100.0 (* t_0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -7.5e-242) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 9e-199) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -7.5e-242) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 9e-199) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -7.5e-242:
		tmp = (100.0 * t_0) * n
	elif n <= 9e-199:
		tmp = (((1.0 - 1.0) / i) * n) * 100.0
	else:
		tmp = 100.0 * (t_0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -7.5e-242)
		tmp = Float64(Float64(100.0 * t_0) * n);
	elseif (n <= 9e-199)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	else
		tmp = Float64(100.0 * Float64(t_0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -7.5e-242], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 9e-199], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.4999999999999998e-242

   1. Initial program 31.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 75.6

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

   3. Applied rewrites 75.6%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 78.1

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

   8. Applied rewrites 78.1%

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

   if -7.4999999999999998e-242 < n < 8.99999999999999995e-199

   1. Initial program 52.7%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.8

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 78.8

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

   6. Applied rewrites 78.8%

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

   if 8.99999999999999995e-199 < n

   1. Initial program 20.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 75.4

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

   3. Applied rewrites 75.4%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 80.9%

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

Alternative 7: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{if}\;n \leq -7.5 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* (/ (expm1 i) i) n))))
   (if (<= n -7.5e-242)
     t_0
     (if (<= n 9e-199) (* (* (/ (- 1.0 1.0) i) n) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) / i) * n);
	double tmp;
	if (n <= -7.5e-242) {
		tmp = t_0;
	} else if (n <= 9e-199) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.expm1(i) / i) * n);
	double tmp;
	if (n <= -7.5e-242) {
		tmp = t_0;
	} else if (n <= 9e-199) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) / i) * n)
	tmp = 0
	if n <= -7.5e-242:
		tmp = t_0
	elif n <= 9e-199:
		tmp = (((1.0 - 1.0) / i) * n) * 100.0
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) / i) * n))
	tmp = 0.0
	if (n <= -7.5e-242)
		tmp = t_0;
	elseif (n <= 9e-199)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	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] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7.5e-242], t$95$0, If[LessEqual[n, 9e-199], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7.4999999999999998e-242 or 8.99999999999999995e-199 < n

   1. Initial program 26.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 75.5

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 79.4%

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

   if -7.4999999999999998e-242 < n < 8.99999999999999995e-199

   1. Initial program 52.7%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.8

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 78.8

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

   6. Applied rewrites 78.8%

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

Alternative 8: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, 0.041666666666666664, 0.16666666666666666 \cdot n\right), i, 0.5 \cdot n\right), i, n\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          100.0
          (fma
           (fma
            (fma (* n i) 0.041666666666666664 (* 0.16666666666666666 n))
            i
            (* 0.5 n))
           i
           n))))
   (if (<= n -2e-88)
     t_0
     (if (<= n 9e-199) (* (* (/ (- 1.0 1.0) i) n) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * fma(fma(fma((n * i), 0.041666666666666664, (0.16666666666666666 * n)), i, (0.5 * n)), i, n);
	double tmp;
	if (n <= -2e-88) {
		tmp = t_0;
	} else if (n <= 9e-199) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * fma(fma(fma(Float64(n * i), 0.041666666666666664, Float64(0.16666666666666666 * n)), i, Float64(0.5 * n)), i, n))
	tmp = 0.0
	if (n <= -2e-88)
		tmp = t_0;
	elseif (n <= 9e-199)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(N[(n * i), $MachinePrecision] * 0.041666666666666664 + N[(0.16666666666666666 * n), $MachinePrecision]), $MachinePrecision] * i + N[(0.5 * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2e-88], t$95$0, If[LessEqual[n, 9e-199], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.99999999999999987e-88 or 8.99999999999999995e-199 < n

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-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 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 65.7

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

   7. Applied rewrites 65.7%

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

   if -1.99999999999999987e-88 < n < 8.99999999999999995e-199

   1. Initial program 48.8%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.2

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 61.2

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

   6. Applied rewrites 61.2%

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

Alternative 9: 63.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, 0.16666666666666666, 0.5 \cdot n\right), i, n\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (fma (fma (* n i) 0.16666666666666666 (* 0.5 n)) i n))))
   (if (<= n -2e-88)
     t_0
     (if (<= n 9e-199) (* (* (/ (- 1.0 1.0) i) n) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * fma(fma((n * i), 0.16666666666666666, (0.5 * n)), i, n);
	double tmp;
	if (n <= -2e-88) {
		tmp = t_0;
	} else if (n <= 9e-199) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * fma(fma(Float64(n * i), 0.16666666666666666, Float64(0.5 * n)), i, n))
	tmp = 0.0
	if (n <= -2e-88)
		tmp = t_0;
	elseif (n <= 9e-199)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(n * i), $MachinePrecision] * 0.16666666666666666 + N[(0.5 * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2e-88], t$95$0, If[LessEqual[n, 9e-199], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.99999999999999987e-88 or 8.99999999999999995e-199 < n

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-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 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 63.5

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

   7. Applied rewrites 63.5%

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

   if -1.99999999999999987e-88 < n < 8.99999999999999995e-199

   1. Initial program 48.8%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.2

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 61.2

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

   6. Applied rewrites 61.2%

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

Alternative 10: 60.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right)\\ \mathbf{if}\;n \leq -2 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (fma (* n i) 0.5 n))))
   (if (<= n -2e-88)
     t_0
     (if (<= n 9e-199) (* (* (/ (- 1.0 1.0) i) n) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * fma((n * i), 0.5, n);
	double tmp;
	if (n <= -2e-88) {
		tmp = t_0;
	} else if (n <= 9e-199) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * fma(Float64(n * i), 0.5, n))
	tmp = 0.0
	if (n <= -2e-88)
		tmp = t_0;
	elseif (n <= 9e-199)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5 + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2e-88], t$95$0, If[LessEqual[n, 9e-199], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.99999999999999987e-88 or 8.99999999999999995e-199 < n

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-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 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

   if -1.99999999999999987e-88 < n < 8.99999999999999995e-199

   1. Initial program 48.8%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.2

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 61.2

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

   6. Applied rewrites 61.2%

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

Alternative 11: 55.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.85 \cdot 10^{-195}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.85e-195) (* 100.0 n) (* 100.0 (/ (* i n) i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.85e-195) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.85d-195) then
        tmp = 100.0d0 * n
    else
        tmp = 100.0d0 * ((i * n) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.85e-195) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((i * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.85e-195:
		tmp = 100.0 * n
	else:
		tmp = 100.0 * ((i * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.85e-195)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.85e-195)
		tmp = 100.0 * n;
	else
		tmp = 100.0 * ((i * n) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.85e-195], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.84999999999999981e-195

   1. Initial program 26.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 58.1%

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

   if 1.84999999999999981e-195 < i

   1. Initial program 32.8%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-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 64.2

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 50.5%

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

Alternative 12: 54.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot i\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.0) (* 100.0 n) (* 100.0 (* (* n i) 0.5))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((n * i) * 0.5);
	}
	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 <= 2.0d0) then
        tmp = 100.0d0 * n
    else
        tmp = 100.0d0 * ((n * i) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.0) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((n * i) * 0.5);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.0:
		tmp = 100.0 * n
	else:
		tmp = 100.0 * ((n * i) * 0.5)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.0)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(Float64(n * i) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.0)
		tmp = 100.0 * n;
	else
		tmp = 100.0 * ((n * i) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.0], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2

   1. Initial program 22.9%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 62.4%

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

   if 2 < i

   1. Initial program 47.6%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-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 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 28.6

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

   7. Applied rewrites 28.6%

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

   8. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 28.6

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

   10. Applied rewrites 28.6%

       \[\leadsto 100 \cdot \left(\left(n \cdot i\right) \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.7% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. *-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 70.2

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

4. Applied rewrites 70.2%

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

5. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 54.7

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

7. Applied rewrites 54.7%

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

Alternative 14: 48.9% 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 28.7%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 48.9%

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

Developer Target 1: 33.8% 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 2025093 
(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))))