Compound Interest

Percentage Accurate: 27.9% → 94.3%

Time: 9.6s

Alternatives: 19

Speedup: 24.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} - 1\\ t_1 := 100 \cdot \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ 1.0 (/ i n)) n) 1.0)) (t_1 (* 100.0 (/ t_0 (/ i n)))))
   (if (<= t_1 0.0)
     (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
     (if (<= t_1 INFINITY) (* (/ (* t_0 100.0) i) n) (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) / i) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * Math.expm1((Math.log1p((i / n)) * n))) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) / i) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) - 1.0
	t_1 = 100.0 * (t_0 / (i / n))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (100.0 * math.expm1((math.log1p((i / n)) * n))) / (i / n)
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) / i) * n
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0)
	t_1 = Float64(100.0 * Float64(t_0 / Float64(i / n)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) / i) * n);
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

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

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 73.6%

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

Alternative 2: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} - 1\\ t_1 := 100 \cdot \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ 1.0 (/ i n)) n) 1.0)) (t_1 (* 100.0 (/ t_0 (/ i n)))))
   (if (<= t_1 0.0)
     (* (* 100.0 (/ (expm1 (* (log1p (/ i n)) n)) i)) n)
     (if (<= t_1 INFINITY) (* (/ (* t_0 100.0) i) n) (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * (expm1((log1p((i / n)) * n)) / i)) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) / i) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (100.0 * (Math.expm1((Math.log1p((i / n)) * n)) / i)) * n;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) / i) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) - 1.0
	t_1 = 100.0 * (t_0 / (i / n))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (100.0 * (math.expm1((math.log1p((i / n)) * n)) / i)) * n
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) / i) * n
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0)
	t_1 = Float64(100.0 * Float64(t_0 / Float64(i / n)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i)) * n);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) / i) * n);
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 98.5%

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

   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 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 73.6%

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

Alternative 3: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} - 1\\ t_1 := 100 \cdot \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ 1.0 (/ i n)) n) 1.0)) (t_1 (* 100.0 (/ t_0 (/ i n)))))
   (if (<= t_1 0.0)
     (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
     (if (<= t_1 INFINITY) (* (/ (* t_0 100.0) i) n) (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) / i) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) - 1.0;
	double t_1 = 100.0 * (t_0 / (i / n));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) / i) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) - 1.0
	t_1 = 100.0 * (t_0 / (i / n))
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * ((math.expm1((math.log1p((i / n)) * n)) / i) * n)
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) / i) * n
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0)
	t_1 = Float64(100.0 * Float64(t_0 / Float64(i / n)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) / i) * n);
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

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

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift--.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 73.6%

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

Alternative 4: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(n \cdot \left(\log i - \log n\right)\right) \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5e-310)
     (* (* 100.0 t_0) n)
     (if (<= n 1.1e-78)
       (* (/ (* (* n (- (log i) (log n))) 100.0) i) n)
       (* 100.0 (* t_0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5e-310) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 1.1e-78) {
		tmp = (((n * (log(i) - log(n))) * 100.0) / i) * n;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5e-310) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 1.1e-78) {
		tmp = (((n * (Math.log(i) - Math.log(n))) * 100.0) / i) * n;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5e-310:
		tmp = (100.0 * t_0) * n
	elif n <= 1.1e-78:
		tmp = (((n * (math.log(i) - math.log(n))) * 100.0) / i) * n
	else:
		tmp = 100.0 * (t_0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5e-310)
		tmp = Float64(Float64(100.0 * t_0) * n);
	elseif (n <= 1.1e-78)
		tmp = Float64(Float64(Float64(Float64(n * Float64(log(i) - log(n))) * 100.0) / i) * n);
	else
		tmp = Float64(100.0 * Float64(t_0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5e-310], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.1e-78], N[(N[(N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.999999999999985e-310

   1. Initial program 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 76.9%

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

   if -4.999999999999985e-310 < n < 1.0999999999999999e-78

   1. Initial program 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 72.0

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in n around 0

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

      1. pow-to-exp N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 80.2

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

   9. Applied rewrites 80.2%

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

   if 1.0999999999999999e-78 < n

   1. Initial program 15.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. 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 95.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(n \cdot \left(\log i - \log n\right)\right) \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(100 \cdot n\right) \cdot \left(\log i - \log n\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5e-310)
     (* (* 100.0 t_0) n)
     (if (<= n 1.1e-78)
       (* (/ (* (* 100.0 n) (- (log i) (log n))) i) n)
       (* 100.0 (* t_0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5e-310) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 1.1e-78) {
		tmp = (((100.0 * n) * (log(i) - log(n))) / i) * n;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5e-310) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 1.1e-78) {
		tmp = (((100.0 * n) * (Math.log(i) - Math.log(n))) / i) * n;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5e-310:
		tmp = (100.0 * t_0) * n
	elif n <= 1.1e-78:
		tmp = (((100.0 * n) * (math.log(i) - math.log(n))) / i) * n
	else:
		tmp = 100.0 * (t_0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5e-310)
		tmp = Float64(Float64(100.0 * t_0) * n);
	elseif (n <= 1.1e-78)
		tmp = Float64(Float64(Float64(Float64(100.0 * n) * Float64(log(i) - log(n))) / i) * n);
	else
		tmp = Float64(100.0 * Float64(t_0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5e-310], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.1e-78], N[(N[(N[(N[(100.0 * n), $MachinePrecision] * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.999999999999985e-310

   1. Initial program 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 76.9%

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

   if -4.999999999999985e-310 < n < 1.0999999999999999e-78

   1. Initial program 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 72.0

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 72.3%

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

   7. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if 1.0999999999999999e-78 < n

   1. Initial program 15.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. 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 95.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(100 \cdot n\right) \cdot \left(\log i - \log n\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \left(\frac{n \cdot \left(\log i - \log n\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5e-310)
     (* (* 100.0 t_0) n)
     (if (<= n 1.1e-78)
       (* 100.0 (* (/ (* n (- (log i) (log n))) i) n))
       (* 100.0 (* t_0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5e-310) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 1.1e-78) {
		tmp = 100.0 * (((n * (log(i) - log(n))) / i) * n);
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5e-310) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 1.1e-78) {
		tmp = 100.0 * (((n * (Math.log(i) - Math.log(n))) / i) * n);
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5e-310:
		tmp = (100.0 * t_0) * n
	elif n <= 1.1e-78:
		tmp = 100.0 * (((n * (math.log(i) - math.log(n))) / i) * n)
	else:
		tmp = 100.0 * (t_0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5e-310)
		tmp = Float64(Float64(100.0 * t_0) * n);
	elseif (n <= 1.1e-78)
		tmp = Float64(100.0 * Float64(Float64(Float64(n * Float64(log(i) - log(n))) / i) * n));
	else
		tmp = Float64(100.0 * Float64(t_0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5e-310], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.1e-78], N[(100.0 * N[(N[(N[(n * N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.999999999999985e-310

   1. Initial program 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 75.1%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 76.9%

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

   if -4.999999999999985e-310 < n < 1.0999999999999999e-78

   1. Initial program 36.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 72.2

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 20.3

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

   7. Applied rewrites 20.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 21.2%

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

   11. Taylor expanded in n around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-log.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 N/A

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

       7. lower-log.f64 80.0

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

   13. Applied rewrites 80.0%

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

   if 1.0999999999999999e-78 < n

   1. Initial program 15.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 79.8

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

   4. Applied rewrites 79.8%

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

   5. 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 95.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \left(\frac{n \cdot \left(\log i - \log n\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{if}\;n \leq -3 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.25 \cdot 10^{-273}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 800000000:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))))
   (if (<= n -3e-127)
     t_0
     (if (<= n -2.25e-273)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 800000000.0) (* 100.0 (/ (* n n) n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double tmp;
	if (n <= -3e-127) {
		tmp = t_0;
	} else if (n <= -2.25e-273) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 800000000.0) {
		tmp = 100.0 * ((n * n) / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.expm1(i) * n) / i);
	double tmp;
	if (n <= -3e-127) {
		tmp = t_0;
	} else if (n <= -2.25e-273) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 800000000.0) {
		tmp = 100.0 * ((n * n) / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) * n) / i)
	tmp = 0
	if n <= -3e-127:
		tmp = t_0
	elif n <= -2.25e-273:
		tmp = 100.0 * (i / (i / n))
	elif n <= 800000000.0:
		tmp = 100.0 * ((n * n) / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	tmp = 0.0
	if (n <= -3e-127)
		tmp = t_0;
	elseif (n <= -2.25e-273)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 800000000.0)
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3e-127], t$95$0, If[LessEqual[n, -2.25e-273], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 800000000.0], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.00000000000000009e-127 or 8e8 < n

   1. Initial program 24.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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 84.6

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

   5. Applied rewrites 84.6%

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

   if -3.00000000000000009e-127 < n < -2.2499999999999998e-273

   1. Initial program 41.5%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 92.0%

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

   if -2.2499999999999998e-273 < n < 8e8

   1. Initial program 32.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in n around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 34.7%

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

   9. Taylor expanded in i around 0

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

       1. unpow2 N/A

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

       2. lower-*.f64 70.0

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

   11. Applied rewrites 70.0%

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

Alternative 8: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.25 \cdot 10^{-273} \lor \neg \left(n \leq 2.85 \cdot 10^{-151}\right):\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{1 - 1}{i}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.25e-273) (not (<= n 2.85e-151)))
   (* 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 <= -2.25e-273) || !(n <= 2.85e-151)) {
		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 <= -2.25e-273) || !(n <= 2.85e-151)) {
		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 <= -2.25e-273) or not (n <= 2.85e-151):
		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 <= -2.25e-273) || !(n <= 2.85e-151))
		tmp = Float64(100.0 * Float64(Float64(expm1(i) / i) * n));
	else
		tmp = Float64(Float64(100.0 * Float64(Float64(1.0 - 1.0) / i)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.25e-273], N[Not[LessEqual[n, 2.85e-151]], $MachinePrecision]], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.2499999999999998e-273 or 2.84999999999999994e-151 < n

   1. Initial program 24.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 78.1

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

   4. Applied rewrites 78.1%

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

   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 82.6%

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

   if -2.2499999999999998e-273 < n < 2.84999999999999994e-151

   1. Initial program 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 62.4

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

   4. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      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-pow.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 84.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.25 \cdot 10^{-273} \lor \neg \left(n \leq 2.85 \cdot 10^{-151}\right):\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{1 - 1}{i}\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.25 \cdot 10^{-273}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \mathbf{elif}\;n \leq 2.85 \cdot 10^{-151}:\\ \;\;\;\;\left(100 \cdot \frac{1 - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.25e-273)
     (* (* 100.0 t_0) n)
     (if (<= n 2.85e-151)
       (* (* 100.0 (/ (- 1.0 1.0) i)) n)
       (* 100.0 (* t_0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.25e-273) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 2.85e-151) {
		tmp = (100.0 * ((1.0 - 1.0) / i)) * n;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.25e-273) {
		tmp = (100.0 * t_0) * n;
	} else if (n <= 2.85e-151) {
		tmp = (100.0 * ((1.0 - 1.0) / i)) * n;
	} else {
		tmp = 100.0 * (t_0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.25e-273:
		tmp = (100.0 * t_0) * n
	elif n <= 2.85e-151:
		tmp = (100.0 * ((1.0 - 1.0) / i)) * n
	else:
		tmp = 100.0 * (t_0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.25e-273)
		tmp = Float64(Float64(100.0 * t_0) * n);
	elseif (n <= 2.85e-151)
		tmp = Float64(Float64(100.0 * Float64(Float64(1.0 - 1.0) / i)) * n);
	else
		tmp = Float64(100.0 * Float64(t_0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.25e-273], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.85e-151], N[(N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.2499999999999998e-273

   1. Initial program 30.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 74.7

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

   4. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 74.5%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 77.8%

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

   if -2.2499999999999998e-273 < n < 2.84999999999999994e-151

   1. Initial program 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 62.4

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

   4. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      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-pow.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 84.9%

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

   if 2.84999999999999994e-151 < n

   1. Initial program 17.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 82.6

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

   4. Applied rewrites 82.6%

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

   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 88.5%

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

Alternative 10: 65.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+105}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{elif}\;n \leq -2.25 \cdot 10^{-273}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.85 \cdot 10^{-151}:\\ \;\;\;\;\left(100 \cdot \frac{1 - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left(1 + i \cdot \left(\left(0.5 + i \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) - \frac{0.5}{n}\right)\right) - \frac{0.5}{n}\right)\right)\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.6e+105)
   (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n))
   (if (<= n -2.25e-273)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 2.85e-151)
       (* (* 100.0 (/ (- 1.0 1.0) i)) n)
       (*
        (*
         100.0
         (+
          1.0
          (*
           i
           (-
            (+
             0.5
             (*
              i
              (-
               (+ 0.16666666666666666 (/ 0.3333333333333333 (* n n)))
               (/ 0.5 n))))
            (/ 0.5 n)))))
        n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.6e+105) {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	} else if (n <= -2.25e-273) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.85e-151) {
		tmp = (100.0 * ((1.0 - 1.0) / i)) * n;
	} else {
		tmp = (100.0 * (1.0 + (i * ((0.5 + (i * ((0.16666666666666666 + (0.3333333333333333 / (n * n))) - (0.5 / n)))) - (0.5 / n))))) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.6e+105)
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	elseif (n <= -2.25e-273)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.85e-151)
		tmp = Float64(Float64(100.0 * Float64(Float64(1.0 - 1.0) / i)) * n);
	else
		tmp = Float64(Float64(100.0 * Float64(1.0 + Float64(i * Float64(Float64(0.5 + Float64(i * Float64(Float64(0.16666666666666666 + Float64(0.3333333333333333 / Float64(n * n))) - Float64(0.5 / n)))) - Float64(0.5 / n))))) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.6e+105], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.25e-273], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.85e-151], N[(N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(100.0 * N[(1.0 + N[(i * N[(N[(0.5 + N[(i * N[(N[(0.16666666666666666 + N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.59999999999999995e105

   1. Initial program 17.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -6.59999999999999995e105 < n < -2.2499999999999998e-273

   1. Initial program 41.6%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 56.9%

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

   if -2.2499999999999998e-273 < n < 2.84999999999999994e-151

   1. Initial program 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 62.4

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

   4. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 63.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      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-pow.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 48.0

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

   8. Applied rewrites 48.0%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 84.9%

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

   if 2.84999999999999994e-151 < n

   1. Initial program 17.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 82.8

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

   4. Applied rewrites 82.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-/r/ N/A

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

      11. pow-to-exp N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 82.5%

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

   7. Taylor expanded in i around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   9. Applied rewrites 72.5%

      \[\leadsto \left(100 \cdot \color{blue}{\left(1 + i \cdot \left(\left(0.5 + i \cdot \left(\left(0.16666666666666666 + \frac{0.3333333333333333}{n \cdot n}\right) - \frac{0.5}{n}\right)\right) - \frac{0.5}{n}\right)\right)}\right) \cdot n \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.6 \cdot 10^{+105}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{elif}\;n \leq -2.25 \cdot 10^{-273}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.85 \cdot 10^{-151}:\\ \;\;\;\;\left(100 \cdot \frac{1 - 1}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.6e+105)
   (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n))
   (if (<= n -2.25e-273)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 2.85e-151)
       (* (* 100.0 (/ (- 1.0 1.0) i)) n)
       (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.6e+105) {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	} else if (n <= -2.25e-273) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.85e-151) {
		tmp = (100.0 * ((1.0 - 1.0) / i)) * n;
	} else {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.6e+105)
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	elseif (n <= -2.25e-273)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.85e-151)
		tmp = Float64(Float64(100.0 * Float64(Float64(1.0 - 1.0) / i)) * n);
	else
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.6e+105], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.25e-273], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.85e-151], N[(N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -6.59999999999999995e105

   1. Initial program 17.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -6.59999999999999995e105 < n < -2.2499999999999998e-273

   1. Initial program 41.6%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 56.9%

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

   if -2.2499999999999998e-273 < n < 2.84999999999999994e-151

   1. Initial program 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 62.4

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

   4. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1\right)}{\frac{i}{n}} \]

      6. lift-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1\right)}{\frac{i}{n}} \]

      7. pow-to-exp N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}{\frac{i}{n}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      10. associate-/r/ N/A

          \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      11. pow-to-exp N/A

          \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{i} \cdot n\right) \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1}{i}\right) \cdot n} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1}{i}\right) \cdot n} \]

   6. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n} \]
   7. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \left(100 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1}}{i}\right) \cdot n \]

      2. lift-*.f64 N/A

         \[\leadsto \left(100 \cdot \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1}{i}\right) \cdot n \]

      3. lift-/.f64 N/A

         \[\leadsto \left(100 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1}{i}\right) \cdot n \]

      4. lift-log1p.f64 N/A

         \[\leadsto \left(100 \cdot \frac{e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1}{i}\right) \cdot n \]

      5. lower--.f64 N/A

         \[\leadsto \left(100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1}}{i}\right) \cdot n \]

      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-pow.f64 N/A

         \[\leadsto \left(100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{i}\right) \cdot n \]

      8. lower-+.f64 N/A

         \[\leadsto \left(100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{i}\right) \cdot n \]

      9. lift-/.f64 48.0

         \[\leadsto \left(100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{i}\right) \cdot n \]

   8. Applied rewrites 48.0%

      \[\leadsto \left(100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot n \]

   9. Taylor expanded in i around 0

      \[\leadsto \left(100 \cdot \frac{\color{blue}{1} - 1}{i}\right) \cdot n \]
   10. Step-by-step derivation

   11. Applied rewrites 84.9%

       \[\leadsto \left(100 \cdot \frac{\color{blue}{1} - 1}{i}\right) \cdot n \]

   if 2.84999999999999994e-151 < n

   1. Initial program 17.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 72.5%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(n \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right), i, n\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot n, i, n\right) \]

      4. lower-fma.f64 71.8

         \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   8. Applied rewrites 71.8%

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 64.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-203} \lor \neg \left(n \leq 2.85 \cdot 10^{-151}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{1 - 1}{i}\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9e-203) (not (<= n 2.85e-151)))
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (* (* 100.0 (/ (- 1.0 1.0) i)) n)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9e-203) || !(n <= 2.85e-151)) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else {
		tmp = (100.0 * ((1.0 - 1.0) / i)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9e-203) || !(n <= 2.85e-151))
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	else
		tmp = Float64(Float64(100.0 * Float64(Float64(1.0 - 1.0) / i)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9e-203], N[Not[LessEqual[n, 2.85e-151]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.0000000000000003e-203 or 2.84999999999999994e-151 < n

   1. Initial program 23.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 58.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(n \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right), i, n\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot n, i, n\right) \]

      4. lower-fma.f64 61.8

         \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   8. Applied rewrites 61.8%

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   if -9.0000000000000003e-203 < n < 2.84999999999999994e-151

   1. Initial program 49.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      4. lift-pow.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \]

      5. lift-+.f64 N/A

         \[\leadsto 100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

      6. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{\frac{i}{n}} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{\frac{i}{n}}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}}{\frac{i}{n}} \]

      11. pow-to-exp N/A

          \[\leadsto \frac{100 \cdot \left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right)}{\frac{i}{n}} \]

      12. lower-expm1.f64 N/A

          \[\leadsto \frac{100 \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}{\frac{i}{n}} \]

      14. lower-log1p.f64 N/A

          \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

      16. lift-/.f64 69.1

          \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\color{blue}{\frac{i}{n}}} \]

   4. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}{\frac{i}{n}} \]

      3. lift-expm1.f64 N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1\right)}}{\frac{i}{n}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1\right)}{\frac{i}{n}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1\right)}{\frac{i}{n}} \]

      6. lift-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \left(e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1\right)}{\frac{i}{n}} \]

      7. pow-to-exp N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right)}{\frac{i}{n}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      10. associate-/r/ N/A

          \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

      11. pow-to-exp N/A

          \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{i} \cdot n\right) \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1}{i}\right) \cdot n} \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1}{i}\right) \cdot n} \]

   6. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n} \]
   7. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \left(100 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1}}{i}\right) \cdot n \]

      2. lift-*.f64 N/A

         \[\leadsto \left(100 \cdot \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1}{i}\right) \cdot n \]

      3. lift-/.f64 N/A

         \[\leadsto \left(100 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1}{i}\right) \cdot n \]

      4. lift-log1p.f64 N/A

         \[\leadsto \left(100 \cdot \frac{e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1}{i}\right) \cdot n \]

      5. lower--.f64 N/A

         \[\leadsto \left(100 \cdot \frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1}}{i}\right) \cdot n \]

      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-pow.f64 N/A

         \[\leadsto \left(100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{i}\right) \cdot n \]

      8. lower-+.f64 N/A

         \[\leadsto \left(100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{i}\right) \cdot n \]

      9. lift-/.f64 50.2

         \[\leadsto \left(100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{i}\right) \cdot n \]

   8. Applied rewrites 50.2%

      \[\leadsto \left(100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{i}\right) \cdot n \]

   9. Taylor expanded in i around 0

      \[\leadsto \left(100 \cdot \frac{\color{blue}{1} - 1}{i}\right) \cdot n \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto \left(100 \cdot \frac{\color{blue}{1} - 1}{i}\right) \cdot n \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-203} \lor \neg \left(n \leq 2.85 \cdot 10^{-151}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \frac{1 - 1}{i}\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-203} \lor \neg \left(n \leq 3.6 \cdot 10^{-37}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9e-203) (not (<= n 3.6e-37)))
   (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n))
   (* 100.0 (/ (* n n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9e-203) || !(n <= 3.6e-37)) {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9e-203) || !(n <= 3.6e-37))
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9e-203], N[Not[LessEqual[n, 3.6e-37]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.0000000000000003e-203 or 3.60000000000000007e-37 < n

   1. Initial program 23.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 59.4%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right) + 1\right) \cdot n\right) \]

      4. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot i + 1\right) \cdot n\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]

      6. +-commutative N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i + \frac{1}{2}, i, 1\right) \cdot n\right) \]

      7. lower-fma.f64 63.2

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   8. Applied rewrites 63.2%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]

   if -9.0000000000000003e-203 < n < 3.60000000000000007e-37

   1. Initial program 39.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 18.1%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{n} \]

   8. Applied rewrites 22.2%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

       2. lower-*.f64 64.8

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   11. Applied rewrites 64.8%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-203} \lor \neg \left(n \leq 3.6 \cdot 10^{-37}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-203}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9e-203)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 3.6e-37)
     (* 100.0 (/ (* n n) n))
     (* 100.0 (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9e-203) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 3.6e-37) {
		tmp = 100.0 * ((n * n) / n);
	} else {
		tmp = 100.0 * (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9e-203)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 3.6e-37)
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	else
		tmp = Float64(100.0 * Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9e-203], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.6e-37], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.0000000000000003e-203

   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

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 46.9%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(n \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right), i, n\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot n, i, n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(\left(\frac{1}{6} \cdot i + \frac{1}{2}\right) \cdot n, i, n\right) \]

      4. lower-fma.f64 53.3

         \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   8. Applied rewrites 53.3%

      \[\leadsto 100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right) \]

   if -9.0000000000000003e-203 < n < 3.60000000000000007e-37

   1. Initial program 39.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 18.1%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{n} \]

   8. Applied rewrites 22.2%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

       2. lower-*.f64 64.8

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   11. Applied rewrites 64.8%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   if 3.60000000000000007e-37 < n

   1. Initial program 16.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 77.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right) + 1\right) \cdot n\right) \]

      4. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot i + 1\right) \cdot n\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]

      6. +-commutative N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i + \frac{1}{2}, i, 1\right) \cdot n\right) \]

      7. lower-fma.f64 77.8

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   8. Applied rewrites 77.8%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 63.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-203} \lor \neg \left(n \leq 3.6 \cdot 10^{-37}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9e-203) (not (<= n 3.6e-37)))
   (* 100.0 (* (fma (* 0.16666666666666666 i) i 1.0) n))
   (* 100.0 (/ (* n n) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9e-203) || !(n <= 3.6e-37)) {
		tmp = 100.0 * (fma((0.16666666666666666 * i), i, 1.0) * n);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9e-203) || !(n <= 3.6e-37))
		tmp = Float64(100.0 * Float64(fma(Float64(0.16666666666666666 * i), i, 1.0) * n));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9e-203], N[Not[LessEqual[n, 3.6e-37]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.0000000000000003e-203 or 3.60000000000000007e-37 < n

   1. Initial program 23.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 59.4%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right) + 1\right) \cdot n\right) \]

      4. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot i + 1\right) \cdot n\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]

      6. +-commutative N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i + \frac{1}{2}, i, 1\right) \cdot n\right) \]

      7. lower-fma.f64 63.2

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   8. Applied rewrites 63.2%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 62.9

          \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \]

   11. Applied rewrites 62.9%

       \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \]

   if -9.0000000000000003e-203 < n < 3.60000000000000007e-37

   1. Initial program 39.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 18.1%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{n} \]

   8. Applied rewrites 22.2%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

       2. lower-*.f64 64.8

          \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   11. Applied rewrites 64.8%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-203} \lor \neg \left(n \leq 3.6 \cdot 10^{-37}\right):\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.4:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(0.16666666666666666 \cdot \left(\left(i \cdot i\right) \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.4) (* 100.0 n) (* 100.0 (* 0.16666666666666666 (* (* i i) n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.4) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * (0.16666666666666666 * ((i * i) * n));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 2.4d0) then
        tmp = 100.0d0 * n
    else
        tmp = 100.0d0 * (0.16666666666666666d0 * ((i * i) * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.4) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * (0.16666666666666666 * ((i * i) * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.4:
		tmp = 100.0 * n
	else:
		tmp = 100.0 * (0.16666666666666666 * ((i * i) * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.4)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(0.16666666666666666 * Float64(Float64(i * i) * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.4)
		tmp = 100.0 * n;
	else
		tmp = 100.0 * (0.16666666666666666 * ((i * i) * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.4], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(0.16666666666666666 * N[(N[(i * i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2.39999999999999991

   1. Initial program 20.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{n} \]
   4. Step-by-step derivation

   5. Applied rewrites 61.8%

      \[\leadsto 100 \cdot \color{blue}{n} \]

   if 2.39999999999999991 < i

   1. Initial program 47.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

   5. Applied rewrites 35.0%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

      3. +-commutative N/A

         \[\leadsto 100 \cdot \left(\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right) + 1\right) \cdot n\right) \]

      4. *-commutative N/A

         \[\leadsto 100 \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot i + 1\right) \cdot n\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]

      6. +-commutative N/A

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i + \frac{1}{2}, i, 1\right) \cdot n\right) \]

      7. lower-fma.f64 35.3

         \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

   8. Applied rewrites 35.3%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]

   9. Taylor expanded in i around inf

      \[\leadsto 100 \cdot \left(\frac{1}{6} \cdot \left({i}^{2} \cdot \color{blue}{n}\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 100 \cdot \left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 100 \cdot \left(\frac{1}{6} \cdot \left({i}^{2} \cdot n\right)\right) \]

       3. pow2 N/A

          \[\leadsto 100 \cdot \left(\frac{1}{6} \cdot \left(\left(i \cdot i\right) \cdot n\right)\right) \]

       4. lift-*.f64 35.3

          \[\leadsto 100 \cdot \left(0.16666666666666666 \cdot \left(\left(i \cdot i\right) \cdot n\right)\right) \]

   11. Applied rewrites 35.3%

       \[\leadsto 100 \cdot \left(0.16666666666666666 \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{n}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 56.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (* (fma (* 0.16666666666666666 i) i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma((0.16666666666666666 * i), i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(Float64(0.16666666666666666 * i), i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[(0.16666666666666666 * i), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.4%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 100 \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \color{blue}{n}\right) \]

   2. *-commutative N/A

      \[\leadsto 100 \cdot \left(\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + n\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \mathsf{fma}\left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right), \color{blue}{i}, n\right) \]

5. Applied rewrites 49.9%

   \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]

6. Taylor expanded in n around inf

   \[\leadsto 100 \cdot \left(n \cdot \color{blue}{\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

   2. lower-*.f64 N/A

      \[\leadsto 100 \cdot \left(\left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right) \cdot n\right) \]

   3. +-commutative N/A

      \[\leadsto 100 \cdot \left(\left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right) + 1\right) \cdot n\right) \]

   4. *-commutative N/A

      \[\leadsto 100 \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{6} \cdot i\right) \cdot i + 1\right) \cdot n\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]

   6. +-commutative N/A

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i + \frac{1}{2}, i, 1\right) \cdot n\right) \]

   7. lower-fma.f64 54.6

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \]

8. Applied rewrites 54.6%

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]

9. Taylor expanded in i around inf

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot i, i, 1\right) \cdot n\right) \]
10. Step-by-step derivation

    1. lower-*.f64 54.3

       \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \]

11. Applied rewrites 54.3%

    \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot i, i, 1\right) \cdot n\right) \]
12. Add Preprocessing

Alternative 18: 54.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 (* (fma 0.5 i 1.0) n)))

C

double code(double i, double n) {
	return 100.0 * (fma(0.5, i, 1.0) * n);
}

Julia

function code(i, n)
	return Float64(100.0 * Float64(fma(0.5, i, 1.0) * n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(0.5 * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.4%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

   2. lift--.f64 N/A

      \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

   3. lift-pow.f64 N/A

      \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1}{\frac{i}{n}} \]

   4. lift-+.f64 N/A

      \[\leadsto 100 \cdot \frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n} - 1}{\frac{i}{n}} \]

   5. lift-/.f64 N/A

      \[\leadsto 100 \cdot \frac{{\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1}{\frac{i}{n}} \]

   6. lift-/.f64 N/A

      \[\leadsto 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\color{blue}{\frac{i}{n}}} \]

   7. associate-/r/ N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot n\right)} \]

   9. lower-/.f64 N/A

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}} \cdot n\right) \]

   10. pow-to-exp N/A

       \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1}{i} \cdot n\right) \]

   11. lower-expm1.f64 N/A

       \[\leadsto 100 \cdot \left(\frac{\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)}}{i} \cdot n\right) \]

   12. lower-*.f64 N/A

       \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{n}\right) \cdot n}\right)}{i} \cdot n\right) \]

   13. lower-log1p.f64 N/A

       \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{i} \cdot n\right) \]

   14. lift-/.f64 76.2

       \[\leadsto 100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{i} \cdot n\right) \]

4. Applied rewrites 76.2%

   \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)} \]

5. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \left(\color{blue}{\left(1 + i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)} \cdot n\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 100 \cdot \left(\left(i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) + \color{blue}{1}\right) \cdot n\right) \]

   2. *-commutative N/A

      \[\leadsto 100 \cdot \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot i + 1\right) \cdot n\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}, \color{blue}{i}, 1\right) \cdot n\right) \]

   4. lower--.f64 N/A

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}, i, 1\right) \cdot n\right) \]

   5. associate-*r/ N/A

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{n}, i, 1\right) \cdot n\right) \]

   6. metadata-eval N/A

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2} - \frac{\frac{1}{2}}{n}, i, 1\right) \cdot n\right) \]

   7. lift-/.f64 51.2

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.5 - \frac{0.5}{n}, i, 1\right) \cdot n\right) \]

7. Applied rewrites 51.2%

   \[\leadsto 100 \cdot \left(\color{blue}{\mathsf{fma}\left(0.5 - \frac{0.5}{n}, i, 1\right)} \cdot n\right) \]

8. Taylor expanded in n around inf

   \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, i, 1\right) \cdot n\right) \]
9. Step-by-step derivation

10. Applied rewrites 51.3%

    \[\leadsto 100 \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right) \]
11. Add Preprocessing

Alternative 19: 49.2% 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 27.4%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \color{blue}{n} \]
4. Step-by-step derivation

5. Applied rewrites 46.4%

   \[\leadsto 100 \cdot \color{blue}{n} \]
6. Add Preprocessing

Developer Target 1: 33.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025064 
(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))))