Compound Interest

Percentage Accurate: 28.6% → 97.6%

Time: 7.8s

Alternatives: 10

Speedup: 24.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 0.0)
     (* (/ (expm1 (* (log1p (/ i n)) n)) (/ i n)) 100.0)
     (if (<= t_0 INFINITY)
       (* (/ (* 100.0 (- (pow (/ i n) n) 1.0)) i) n)
       (* 100.0 (* (/ (expm1 (fma (/ (* i i) n) -0.5 i)) i) n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((100.0 * (pow((i / n), n) - 1.0)) / i) * n;
	} else {
		tmp = 100.0 * ((expm1(fma(((i * i) / n), -0.5, i)) / i) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / Float64(i / n)) * 100.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * Float64((Float64(i / n) ^ n) - 1.0)) / i) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(fma(Float64(Float64(i * i) / n), -0.5, i)) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(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[(N[(N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * -0.5 + i), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.1%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 98.9

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

   9. Applied rewrites 98.9%

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-/.f64 99.2

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

   11. Applied rewrites 99.2%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + \color{blue}{i}\right)}{i} \cdot n\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{{i}^{2}}{n} \cdot \frac{-1}{2} + i\right)}{i} \cdot n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \color{blue}{\frac{-1}{2}}, i\right)\right)}{i} \cdot n\right) \]

      4. lower-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      5. unpow2 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      6. lower-*.f64 97.5

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right) \]

   7. Applied rewrites 97.5%

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot n\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 0.0)
     (* (* 100.0 (/ (expm1 (* (log1p (/ i n)) n)) i)) n)
     (if (<= t_0 INFINITY)
       (* (/ (* 100.0 (- (pow (/ i n) n) 1.0)) i) n)
       (* 100.0 (* (/ (expm1 (fma (/ (* i i) n) -0.5 i)) i) n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (100.0 * (expm1((log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((100.0 * (pow((i / n), n) - 1.0)) / i) * n;
	} else {
		tmp = 100.0 * ((expm1(fma(((i * i) / n), -0.5, i)) / i) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * Float64((Float64(i / n) ^ n) - 1.0)) / i) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(fma(Float64(Float64(i * i) / n), -0.5, i)) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * -0.5 + i), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.1%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. *-commutative N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. 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. lower-*.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-/.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-/.f64 97.4

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

   6. Applied rewrites 97.4%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 98.9

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

   9. Applied rewrites 98.9%

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-/.f64 99.2

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

   11. Applied rewrites 99.2%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + \color{blue}{i}\right)}{i} \cdot n\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{{i}^{2}}{n} \cdot \frac{-1}{2} + i\right)}{i} \cdot n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \color{blue}{\frac{-1}{2}}, i\right)\right)}{i} \cdot n\right) \]

      4. lower-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      5. unpow2 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      6. lower-*.f64 97.5

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right) \]

   7. Applied rewrites 97.5%

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot n\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 0.0)
     (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
     (if (<= t_0 INFINITY)
       (* (/ (* 100.0 (- (pow (/ i n) n) 1.0)) i) n)
       (* 100.0 (* (/ (expm1 (fma (/ (* i i) n) -0.5 i)) i) n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((100.0 * (pow((i / n), n) - 1.0)) / i) * n;
	} else {
		tmp = 100.0 * ((expm1(fma(((i * i) / n), -0.5, i)) / i) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * Float64((Float64(i / n) ^ n) - 1.0)) / i) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(fma(Float64(Float64(i * i) / n), -0.5, i)) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * -0.5 + i), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.1%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 97.3

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

   4. Applied rewrites 97.3%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 98.9

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

   9. Applied rewrites 98.9%

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-/.f64 99.2

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

   11. Applied rewrites 99.2%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + \color{blue}{i}\right)}{i} \cdot n\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{{i}^{2}}{n} \cdot \frac{-1}{2} + i\right)}{i} \cdot n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \color{blue}{\frac{-1}{2}}, i\right)\right)}{i} \cdot n\right) \]

      4. lower-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      5. unpow2 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      6. lower-*.f64 97.5

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right) \]

   7. Applied rewrites 97.5%

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot n\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 0.0)
     (* 100.0 (/ (expm1 i) (/ i n)))
     (if (<= t_0 INFINITY)
       (* (/ (* 100.0 (- (pow (/ i n) n) 1.0)) i) n)
       (* 100.0 (* (/ (expm1 (fma (/ (* i i) n) -0.5 i)) i) n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((100.0 * (pow((i / n), n) - 1.0)) / i) * n;
	} else {
		tmp = 100.0 * ((expm1(fma(((i * i) / n), -0.5, i)) / i) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * Float64((Float64(i / n) ^ n) - 1.0)) / i) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(fma(Float64(Float64(i * i) / n), -0.5, i)) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * -0.5 + i), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 80.9

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

   5. Applied rewrites 80.9%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 98.9

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

   9. Applied rewrites 98.9%

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-/.f64 99.2

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

   11. Applied rewrites 99.2%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 1.9

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

   4. Applied rewrites 1.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{i + \frac{-1}{2} \cdot \frac{{i}^{2}}{n}}\right)}{i} \cdot n\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{-1}{2} \cdot \frac{{i}^{2}}{n} + \color{blue}{i}\right)}{i} \cdot n\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\frac{{i}^{2}}{n} \cdot \frac{-1}{2} + i\right)}{i} \cdot n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \color{blue}{\frac{-1}{2}}, i\right)\right)}{i} \cdot n\right) \]

      4. lower-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{{i}^{2}}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      5. unpow2 N/A

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, \frac{-1}{2}, i\right)\right)}{i} \cdot n\right) \]

      6. lower-*.f64 97.5

         \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)\right)}{i} \cdot n\right) \]

   7. Applied rewrites 97.5%

      \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\frac{i \cdot i}{n}, -0.5, i\right)}\right)}{i} \cdot n\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -9 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-235}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-223}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 1.8:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -9e-65)
     t_0
     (if (<= n -1.35e-235)
       t_1
       (if (<= n 4.5e-223)
         (* 100.0 (* (/ (- 1.0 1.0) i) n))
         (if (<= n 1.8) 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 <= -9e-65) {
		tmp = t_0;
	} else if (n <= -1.35e-235) {
		tmp = t_1;
	} else if (n <= 4.5e-223) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.8) {
		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 <= -9e-65) {
		tmp = t_0;
	} else if (n <= -1.35e-235) {
		tmp = t_1;
	} else if (n <= 4.5e-223) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.8) {
		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 <= -9e-65:
		tmp = t_0
	elif n <= -1.35e-235:
		tmp = t_1
	elif n <= 4.5e-223:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	elif n <= 1.8:
		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 <= -9e-65)
		tmp = t_0;
	elseif (n <= -1.35e-235)
		tmp = t_1;
	elseif (n <= 4.5e-223)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	elseif (n <= 1.8)
		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, -9e-65], t$95$0, If[LessEqual[n, -1.35e-235], t$95$1, If[LessEqual[n, 4.5e-223], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.9999999999999995e-65 or 1.80000000000000004 < n

   1. Initial program 20.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 91.4

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

   5. Applied rewrites 91.4%

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

   if -8.9999999999999995e-65 < n < -1.3500000000000001e-235 or 4.49999999999999968e-223 < n < 1.80000000000000004

   1. Initial program 28.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 70.0%

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

   if -1.3500000000000001e-235 < n < 4.49999999999999968e-223

   1. Initial program 68.3%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 66.7

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 88.3%

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

Alternative 6: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-235} \lor \neg \left(n \leq 1.12 \cdot 10^{-116}\right):\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.35e-235) (not (<= n 1.12e-116)))
   (* 100.0 (* (/ (expm1 i) i) n))
   (* 100.0 (* (/ (- 1.0 1.0) i) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.35e-235) || !(n <= 1.12e-116)) {
		tmp = 100.0 * ((expm1(i) / i) * n);
	} else {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.35e-235) || !(n <= 1.12e-116)) {
		tmp = 100.0 * ((Math.expm1(i) / i) * n);
	} else {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.35e-235) or not (n <= 1.12e-116):
		tmp = 100.0 * ((math.expm1(i) / i) * n)
	else:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.35e-235) || !(n <= 1.12e-116))
		tmp = Float64(100.0 * Float64(Float64(expm1(i) / i) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.35e-235], N[Not[LessEqual[n, 1.12e-116]], $MachinePrecision]], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.3500000000000001e-235 or 1.12e-116 < n

   1. Initial program 22.4%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-/.f64 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 86.6%

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

   if -1.3500000000000001e-235 < n < 1.12e-116

   1. Initial program 51.5%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 51.3

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

   5. Applied rewrites 51.3%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 50.5

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 74.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-235} \lor \neg \left(n \leq 1.12 \cdot 10^{-116}\right):\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.82:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{elif}\;i \leq 2.2 \cdot 10^{-34}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(0.5, i, 1\right) \cdot i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.82)
   (* 100.0 (* (/ (- 1.0 1.0) i) n))
   (if (<= i 2.2e-34)
     (* 100.0 (fma (* (- 0.5 (/ 0.5 n)) n) i n))
     (* 100.0 (/ (* (fma 0.5 i 1.0) i) (/ i n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.82) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (i <= 2.2e-34) {
		tmp = 100.0 * fma(((0.5 - (0.5 / n)) * n), i, n);
	} else {
		tmp = 100.0 * ((fma(0.5, i, 1.0) * i) / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.82)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	elseif (i <= 2.2e-34)
		tmp = Float64(100.0 * fma(Float64(Float64(0.5 - Float64(0.5 / n)) * n), i, n));
	else
		tmp = Float64(100.0 * Float64(Float64(fma(0.5, i, 1.0) * i) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.82], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.2e-34], N[(100.0 * N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.82000000000000006

   1. Initial program 53.0%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 73.7

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

   7. Applied rewrites 73.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 28.1%

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

   if -1.82000000000000006 < i < 2.1999999999999999e-34

   1. Initial program 7.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if 2.1999999999999999e-34 < i

   1. Initial program 44.1%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 41.2%

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

Alternative 8: 59.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6400000000 \lor \neg \left(i \leq 9.8 \cdot 10^{+29}\right):\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -6400000000.0) (not (<= i 9.8e+29)))
   (* 100.0 (* (/ (- 1.0 1.0) i) n))
   (* 100.0 n)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -6400000000.0) || !(i <= 9.8e+29)) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-6400000000.0d0)) .or. (.not. (i <= 9.8d+29))) then
        tmp = 100.0d0 * (((1.0d0 - 1.0d0) / i) * n)
    else
        tmp = 100.0d0 * n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -6400000000.0) || !(i <= 9.8e+29)) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -6400000000.0) or not (i <= 9.8e+29):
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -6400000000.0) || !(i <= 9.8e+29))
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -6400000000.0) || ~((i <= 9.8e+29)))
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	else
		tmp = 100.0 * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -6400000000.0], N[Not[LessEqual[i, 9.8e+29]], $MachinePrecision]], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -6.4e9 or 9.8000000000000003e29 < i

   1. Initial program 52.6%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 70.1

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 29.2%

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

   if -6.4e9 < i < 9.8000000000000003e29

   1. Initial program 8.6%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 83.8%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6400000000 \lor \neg \left(i \leq 9.8 \cdot 10^{+29}\right):\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.82:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.82)
   (* 100.0 (* (/ (- 1.0 1.0) i) n))
   (* 100.0 (fma (* (- 0.5 (/ 0.5 n)) n) i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.82) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = 100.0 * fma(((0.5 - (0.5 / n)) * n), i, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.82)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * fma(Float64(Float64(0.5 - Float64(0.5 / n)) * n), i, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.82], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.82000000000000006

   1. Initial program 53.0%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. associate-/r/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 73.7

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

   7. Applied rewrites 73.7%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 28.1%

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

   if -1.82000000000000006 < i

   1. Initial program 17.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.6% 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 26.0%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 52.7%

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

Developer Target 1: 34.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025044 
(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))))