Compound Interest

Percentage Accurate: 28.1% → 95.7%

Time: 10.8s

Alternatives: 18

Speedup: 24.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i}{n} - -1\\ t_1 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({t\_0}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{n}{2} \cdot \log \left({t\_0}^{2}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (/ i n) -1.0))
        (t_1 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_1 -2e-28)
     (* (- (pow t_0 n) 1.0) (/ (* 100.0 n) i))
     (if (<= t_1 2e-262)
       (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ i n))
       (if (<= t_1 INFINITY)
         (/ (* (expm1 (* (/ n 2.0) (log (pow t_0 2.0)))) 100.0) (/ i n))
         (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = (i / n) - -1.0;
	double t_1 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_1 <= -2e-28) {
		tmp = (pow(t_0, n) - 1.0) * ((100.0 * n) / i);
	} else if (t_1 <= 2e-262) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (expm1(((n / 2.0) * log(pow(t_0, 2.0)))) * 100.0) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (i / n) - -1.0;
	double t_1 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_1 <= -2e-28) {
		tmp = (Math.pow(t_0, n) - 1.0) * ((100.0 * n) / i);
	} else if (t_1 <= 2e-262) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.expm1(((n / 2.0) * Math.log(Math.pow(t_0, 2.0)))) * 100.0) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i / n) - -1.0
	t_1 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_1 <= -2e-28:
		tmp = (math.pow(t_0, n) - 1.0) * ((100.0 * n) / i)
	elif t_1 <= 2e-262:
		tmp = (math.expm1((math.log1p((i / n)) * n)) * 100.0) / (i / n)
	elif t_1 <= math.inf:
		tmp = (math.expm1(((n / 2.0) * math.log(math.pow(t_0, 2.0)))) * 100.0) / (i / n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) - -1.0)
	t_1 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_1 <= -2e-28)
		tmp = Float64(Float64((t_0 ^ n) - 1.0) * Float64(Float64(100.0 * n) / i));
	elseif (t_1 <= 2e-262)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(expm1(Float64(Float64(n / 2.0) * log((t_0 ^ 2.0)))) * 100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-28], N[(N[(N[Power[t$95$0, n], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-262], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(Exp[N[(N[(n / 2.0), $MachinePrecision] * N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -1.99999999999999994e-28

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 49.5

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

   4. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 49.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lift-/.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.99999999999999994e-28 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 2.00000000000000002e-262

   1. Initial program 23.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 23.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 2.00000000000000002e-262 < (*.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 96.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 61.9

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

   4. Applied rewrites 61.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log1p.f64 N/A

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

      4. log-pow-rev N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

      7. log-pow N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

   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 80.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\frac{n}{2} \cdot \log \left({\left(\frac{i}{n} - -1\right)}^{2}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -2e-28)
     (* (- (pow (- (/ i n) -1.0) n) 1.0) (/ (* 100.0 n) i))
     (if (<= t_0 2e-262)
       (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ i n))
       (if (<= t_0 INFINITY) t_0 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e-28) {
		tmp = (pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i);
	} else if (t_0 <= 2e-262) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e-28) {
		tmp = (Math.pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i);
	} else if (t_0 <= 2e-262) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= -2e-28:
		tmp = (math.pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i)
	elif t_0 <= 2e-262:
		tmp = (math.expm1((math.log1p((i / n)) * n)) * 100.0) / (i / n)
	elif t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -2e-28)
		tmp = Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) * Float64(Float64(100.0 * n) / i));
	elseif (t_0 <= 2e-262)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-28], N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-262], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -1.99999999999999994e-28

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 49.5

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

   4. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 49.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lift-/.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.99999999999999994e-28 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 2.00000000000000002e-262

   1. Initial program 23.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 23.3

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lower-log1p.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 2.00000000000000002e-262 < (*.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 96.7%

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

   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 80.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -2e-28)
     (* (- (pow (- (/ i n) -1.0) n) 1.0) (/ (* 100.0 n) i))
     (if (<= t_0 2e-262)
       (* (/ (expm1 (* (log1p (/ i n)) n)) i) (* n 100.0))
       (if (<= t_0 INFINITY) t_0 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e-28) {
		tmp = (pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i);
	} else if (t_0 <= 2e-262) {
		tmp = (expm1((log1p((i / n)) * n)) / i) * (n * 100.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e-28) {
		tmp = (Math.pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i);
	} else if (t_0 <= 2e-262) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) / i) * (n * 100.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= -2e-28:
		tmp = (math.pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i)
	elif t_0 <= 2e-262:
		tmp = (math.expm1((math.log1p((i / n)) * n)) / i) * (n * 100.0)
	elif t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -2e-28)
		tmp = Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) * Float64(Float64(100.0 * n) / i));
	elseif (t_0 <= 2e-262)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * Float64(n * 100.0));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-28], N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-262], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -1.99999999999999994e-28

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 49.5

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

   4. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 49.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lift-/.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.99999999999999994e-28 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 2.00000000000000002e-262

   1. Initial program 23.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 2.00000000000000002e-262 < (*.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 96.7%

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

   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 80.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-262}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -2e-28)
     (* (- (pow (- (/ i n) -1.0) n) 1.0) (/ (* 100.0 n) i))
     (if (<= t_0 2e-262)
       (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
       (if (<= t_0 INFINITY) t_0 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e-28) {
		tmp = (pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i);
	} else if (t_0 <= 2e-262) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e-28) {
		tmp = (Math.pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i);
	} else if (t_0 <= 2e-262) {
		tmp = 100.0 * ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= -2e-28:
		tmp = (math.pow(((i / n) - -1.0), n) - 1.0) * ((100.0 * n) / i)
	elif t_0 <= 2e-262:
		tmp = 100.0 * ((math.expm1((math.log1p((i / n)) * n)) / i) * n)
	elif t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -2e-28)
		tmp = Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) * Float64(Float64(100.0 * n) / i));
	elseif (t_0 <= 2e-262)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-28], N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-262], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -1.99999999999999994e-28

   1. Initial program 99.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 49.5

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

   4. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 49.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lift-/.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.99999999999999994e-28 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 2.00000000000000002e-262

   1. Initial program 23.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 23.3

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 98.5

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

   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 2.00000000000000002e-262 < (*.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 96.7%

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

   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 80.4%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq -2 \cdot 10^{-28}:\\ \;\;\;\;\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100 \cdot n}{i}\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 2 \cdot 10^{-262}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-230} \lor \neg \left(n \leq 2.6 \cdot 10^{-205}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -3.4e-230) (not (<= n 2.6e-205)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -3.4e-230) || !(n <= 2.6e-205)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -3.4e-230) || !(n <= 2.6e-205)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -3.4e-230) or not (n <= 2.6e-205):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -3.4e-230) || !(n <= 2.6e-205))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -3.4e-230], N[Not[LessEqual[n, 2.6e-205]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.4e-230 or 2.5999999999999998e-205 < n

   1. Initial program 23.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 81.2%

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

   if -3.4e-230 < n < 2.5999999999999998e-205

   1. Initial program 64.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 85.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-230} \lor \neg \left(n \leq 2.6 \cdot 10^{-205}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, n \cdot i, 16.666666666666668 \cdot n\right), i, 50 \cdot n\right), i, 100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.5e-142)
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   (if (<= n 2.6e-205)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (fma
      (fma
       (fma 4.166666666666667 (* n i) (* 16.666666666666668 n))
       i
       (* 50.0 n))
      i
      (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.5e-142) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else if (n <= 2.6e-205) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = fma(fma(fma(4.166666666666667, (n * i), (16.666666666666668 * n)), i, (50.0 * n)), i, (100.0 * n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.5e-142)
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	elseif (n <= 2.6e-205)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = fma(fma(fma(4.166666666666667, Float64(n * i), Float64(16.666666666666668 * n)), i, Float64(50.0 * n)), i, Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.5e-142], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.6e-205], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.166666666666667 * N[(n * i), $MachinePrecision] + N[(16.666666666666668 * n), $MachinePrecision]), $MachinePrecision] * i + N[(50.0 * n), $MachinePrecision]), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.50000000000000019e-142

   1. Initial program 20.4%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 66.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   if -4.50000000000000019e-142 < n < 2.5999999999999998e-205

   1. Initial program 61.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 75.2%

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

   if 2.5999999999999998e-205 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot n + \color{blue}{i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, n \cdot i, 16.666666666666668 \cdot n\right), i, 50 \cdot n\right), \color{blue}{i}, 100 \cdot n\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.5e-142)
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   (if (<= n 2.6e-205)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (*
      (*
       (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
       100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.5e-142) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else if (n <= 2.6e-205) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.5e-142)
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	elseif (n <= 2.6e-205)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.5e-142], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.6e-205], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.50000000000000019e-142

   1. Initial program 20.4%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 66.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   if -4.50000000000000019e-142 < n < 2.5999999999999998e-205

   1. Initial program 61.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 75.2%

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

   if 2.5999999999999998e-205 < n

   1. Initial program 21.5%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 81.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 71.6%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.9e+27)
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   (if (<= n 6.5e-8)
     (* 100.0 (/ i (/ i n)))
     (*
      (*
       (fma (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5) i 1.0)
       100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.9e+27) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else if (n <= 6.5e-8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.9e+27)
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	elseif (n <= 6.5e-8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.9e+27], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 6.5e-8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.90000000000000015e27

   1. Initial program 21.8%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 63.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   if -4.90000000000000015e27 < n < 6.49999999999999997e-8

   1. Initial program 34.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 61.6%

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

   if 6.49999999999999997e-8 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 78.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27} \lor \neg \left(n \leq 6.5 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.9e+27) (not (<= n 6.5e-8)))
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.9e+27) || !(n <= 6.5e-8)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.9e+27) || !(n <= 6.5e-8))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.9e+27], N[Not[LessEqual[n, 6.5e-8]], $MachinePrecision]], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.90000000000000015e27 or 6.49999999999999997e-8 < n

   1. Initial program 23.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 70.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   if -4.90000000000000015e27 < n < 6.49999999999999997e-8

   1. Initial program 34.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 61.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27} \lor \neg \left(n \leq 6.5 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27} \lor \neg \left(n \leq 6.5 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{100 \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.9e+27) (not (<= n 6.5e-8)))
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   (* i (/ (* 100.0 n) i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.9e+27) || !(n <= 6.5e-8)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else {
		tmp = i * ((100.0 * n) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.9e+27) || !(n <= 6.5e-8))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	else
		tmp = Float64(i * Float64(Float64(100.0 * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.9e+27], N[Not[LessEqual[n, 6.5e-8]], $MachinePrecision]], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], N[(i * N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.90000000000000015e27 or 6.49999999999999997e-8 < n

   1. Initial program 23.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 70.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   if -4.90000000000000015e27 < n < 6.49999999999999997e-8

   1. Initial program 34.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.2

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 60.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27} \lor \neg \left(n \leq 6.5 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{100 \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27} \lor \neg \left(n \leq 6.5 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot i\right) \cdot 4.166666666666667, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{100 \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.9e+27) (not (<= n 6.5e-8)))
   (* (fma (* (* i i) 4.166666666666667) i 100.0) n)
   (* i (/ (* 100.0 n) i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.9e+27) || !(n <= 6.5e-8)) {
		tmp = fma(((i * i) * 4.166666666666667), i, 100.0) * n;
	} else {
		tmp = i * ((100.0 * n) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.9e+27) || !(n <= 6.5e-8))
		tmp = Float64(fma(Float64(Float64(i * i) * 4.166666666666667), i, 100.0) * n);
	else
		tmp = Float64(i * Float64(Float64(100.0 * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.9e+27], N[Not[LessEqual[n, 6.5e-8]], $MachinePrecision]], N[(N[(N[(N[(i * i), $MachinePrecision] * 4.166666666666667), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], N[(i * N[(N[(100.0 * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.90000000000000015e27 or 6.49999999999999997e-8 < n

   1. Initial program 23.6%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 70.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   9. Taylor expanded in i around inf

      \[\leadsto \mathsf{fma}\left(\frac{25}{6} \cdot {i}^{2}, i, 100\right) \cdot n \]
   10. Step-by-step derivation

   11. Applied rewrites 69.4%

       \[\leadsto \mathsf{fma}\left(\left(i \cdot i\right) \cdot 4.166666666666667, i, 100\right) \cdot n \]

   if -4.90000000000000015e27 < n < 6.49999999999999997e-8

   1. Initial program 34.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.2

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 60.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.9 \cdot 10^{+27} \lor \neg \left(n \leq 6.5 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot i\right) \cdot 4.166666666666667, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{100 \cdot n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(i \cdot i\right) \cdot 4.166666666666667, i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* (fma (* (* i i) 4.166666666666667) i 100.0) n))

C

double code(double i, double n) {
	return fma(((i * i) * 4.166666666666667), i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(Float64(Float64(i * i) * 4.166666666666667), i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(N[(N[(i * i), $MachinePrecision] * 4.166666666666667), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

3. Taylor expanded in n around inf

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

5. Applied rewrites 75.0%

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

6. Taylor expanded in i around 0

   \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
7. Step-by-step derivation

8. Applied rewrites 59.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

9. Taylor expanded in i around inf

   \[\leadsto \mathsf{fma}\left(\frac{25}{6} \cdot {i}^{2}, i, 100\right) \cdot n \]
10. Step-by-step derivation

11. Applied rewrites 58.3%

    \[\leadsto \mathsf{fma}\left(\left(i \cdot i\right) \cdot 4.166666666666667, i, 100\right) \cdot n \]
12. Add Preprocessing

Alternative 13: 57.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n))

C

double code(double i, double n) {
	return fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

3. Taylor expanded in n around inf

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

5. Applied rewrites 75.0%

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

6. Taylor expanded in i around 0

   \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
7. Step-by-step derivation

8. Applied rewrites 56.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]
9. Add Preprocessing

Alternative 14: 54.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(50 \cdot i\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.1e+30) (* 100.0 n) (* (* 50.0 i) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.1e+30) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.1d+30) then
        tmp = 100.0d0 * n
    else
        tmp = (50.0d0 * i) * n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.1e+30) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.1e+30:
		tmp = 100.0 * n
	else:
		tmp = (50.0 * i) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.1e+30)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(Float64(50.0 * i) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.1e+30)
		tmp = 100.0 * n;
	else
		tmp = (50.0 * i) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.1e+30], N[(100.0 * n), $MachinePrecision], N[(N[(50.0 * i), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.1e30

   1. Initial program 24.3%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 59.3%

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

   if 1.1e30 < i

   1. Initial program 46.1%

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

   3. Taylor expanded in n around inf

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 33.2%

      \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]

   9. Taylor expanded in i around inf

      \[\leadsto \left(50 \cdot i\right) \cdot n \]
   10. Step-by-step derivation

   11. Applied rewrites 33.2%

       \[\leadsto \left(50 \cdot i\right) \cdot n \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(50 \cdot i\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.8% 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 28.8%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 12.2%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

7. Applied rewrites 12.7%

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

8. Taylor expanded in i around 0

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

10. Applied rewrites 42.5%

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

11. Taylor expanded in n around inf

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

13. Applied rewrites 53.9%

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

Alternative 16: 54.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(n \cdot i, 50, 100 \cdot n\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (fma (* n i) 50.0 (* 100.0 n)))

C

double code(double i, double n) {
	return fma((n * i), 50.0, (100.0 * n));
}

Julia

function code(i, n)
	return fma(Float64(n * i), 50.0, Float64(100.0 * n))
end

Wolfram

code[i_, n_] := N[(N[(n * i), $MachinePrecision] * 50.0 + N[(100.0 * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

3. Taylor expanded in n around inf

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

5. Applied rewrites 75.0%

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

6. Taylor expanded in i around 0

   \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
7. Step-by-step derivation

8. Applied rewrites 53.9%

   \[\leadsto \mathsf{fma}\left(n \cdot i, \color{blue}{50}, 100 \cdot n\right) \]
9. Add Preprocessing

Alternative 17: 54.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(50, i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* (fma 50.0 i 100.0) n))

C

double code(double i, double n) {
	return fma(50.0, i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(50.0, i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

3. Taylor expanded in n around inf

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

5. Applied rewrites 75.0%

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

6. Taylor expanded in i around 0

   \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
7. Step-by-step derivation

8. Applied rewrites 53.9%

   \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]
9. Add Preprocessing

Alternative 18: 49.4% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.8%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 48.0%

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

6. Final simplification 48.0%

   \[\leadsto 100 \cdot n \]
7. Add Preprocessing

Developer Target 1: 33.7% 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 2025019 
(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))))