Compound Interest

Percentage Accurate: 28.1% → 95.9%

Time: 10.2s

Alternatives: 17

Speedup: 24.3×

• could not determine a ground truth

  1. i = 3.5831789894651354e-41
  2. n = -5.528691157533801e+27

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: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ t_1 := \frac{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right) \cdot 100}{i} \cdot n\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
        (t_1 (* (/ (* (- (pow (+ (/ i n) 1.0) n) 1.0) 100.0) i) n)))
   (if (<= t_0 -200000.0)
     t_1
     (if (<= t_0 0.0)
       (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
       (if (<= t_0 INFINITY) t_1 (* 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 t_1 = (((pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} 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 t_1 = (((Math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * Math.expm1((Math.log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} 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))
	t_1 = (((math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n
	tmp = 0
	if t_0 <= -200000.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (100.0 * math.expm1((math.log1p((i / n)) * n))) / (i / n)
	elif t_0 <= math.inf:
		tmp = t_1
	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)))
	t_1 = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) * 100.0) / i) * n)
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(100.0 * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -2e5 or -0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 55.1

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

   4. Applied rewrites 55.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 55.1%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if -2e5 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -0.0

   1. Initial program 23.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 89.7%

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

Alternative 2: 95.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 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))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

      1. lift-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 24.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 98.4%

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

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

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 64.9%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 89.7%

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

Alternative 3: 95.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 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))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

      1. lift-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 24.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

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

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 64.9%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 89.7%

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

Alternative 4: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n)
     (if (<= t_0 INFINITY) (* 100.0 (* (expm1 i) (/ n i))) (* 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 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (expm1(i) * (n / i));
	} 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 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(expm1(i) * Float64(n / i)));
	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, (-Infinity)], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 25.0

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

   4. Applied rewrites 25.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 25.0%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-expm1.f64 25.7

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

   9. Applied rewrites 25.7%

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

   10. Taylor expanded in i around 0

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

       1. +-commutative N/A

          \[\leadsto \left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + 100\right) \cdot n \]

       2. *-commutative N/A

          \[\leadsto \left(\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) \cdot i + 100\right) \cdot n \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right), i, 100\right) \cdot n \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50, i, 100\right) \cdot n \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{50}{3} + \frac{25}{6} \cdot i\right) \cdot i + 50, i, 100\right) \cdot n \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{50}{3} + \frac{25}{6} \cdot i, i, 50\right), i, 100\right) \cdot n \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{6} \cdot i + \frac{50}{3}, i, 50\right), i, 100\right) \cdot n \]

       8. lower-fma.f64 88.2

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

   12. Applied rewrites 88.2%

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

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

   1. Initial program 31.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 72.0

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

   5. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-expm1.f64 N/A

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

      7. lower-/.f64 76.2

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

   7. Applied rewrites 76.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 89.7%

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

Alternative 5: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.08 \cdot 10^{+141}:\\ \;\;\;\;100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -1 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+167}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right) \cdot 100}{i} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.08e+141)
   (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))
   (if (<= i -1e-186)
     (* 100.0 (/ (* (expm1 i) n) i))
     (if (<= i 4e+167)
       (* (* 100.0 (/ (expm1 i) i)) n)
       (* (/ (* (- (pow (+ (/ i n) 1.0) n) 1.0) 100.0) i) n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.08e+141) {
		tmp = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	} else if (i <= -1e-186) {
		tmp = 100.0 * ((expm1(i) * n) / i);
	} else if (i <= 4e+167) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else {
		tmp = (((pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.08e+141) {
		tmp = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	} else if (i <= -1e-186) {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	} else if (i <= 4e+167) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else {
		tmp = (((Math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.08e+141:
		tmp = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	elif i <= -1e-186:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	elif i <= 4e+167:
		tmp = (100.0 * (math.expm1(i) / i)) * n
	else:
		tmp = (((math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.08e+141)
		tmp = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)));
	elseif (i <= -1e-186)
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	elseif (i <= 4e+167)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	else
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) * 100.0) / i) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.08e+141], 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[i, -1e-186], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4e+167], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.08000000000000007e141

   1. Initial program 89.5%

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

   if -1.08000000000000007e141 < i < -9.9999999999999991e-187

   1. Initial program 21.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 42.6

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

   8. Applied rewrites 42.6%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-expm1.f64 78.7

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

   11. Applied rewrites 78.7%

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

   if -9.9999999999999991e-187 < i < 4.0000000000000002e167

   1. Initial program 12.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 78.7

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

   4. Applied rewrites 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-expm1.f64 88.0

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

   9. Applied rewrites 88.0%

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

   if 4.0000000000000002e167 < i

   1. Initial program 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 47.6

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

   4. Applied rewrites 47.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 47.9%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 71.8

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

   8. Applied rewrites 71.8%

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

Alternative 6: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right) \cdot 100}{i} \cdot n\\ \mathbf{if}\;i \leq -1.08 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq -1 \cdot 10^{-186}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+167}:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (/ (* (- (pow (+ (/ i n) 1.0) n) 1.0) 100.0) i) n)))
   (if (<= i -1.08e+141)
     t_0
     (if (<= i -1e-186)
       (* 100.0 (/ (* (expm1 i) n) i))
       (if (<= i 4e+167) (* (* 100.0 (/ (expm1 i) i)) n) t_0)))))

C

double code(double i, double n) {
	double t_0 = (((pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n;
	double tmp;
	if (i <= -1.08e+141) {
		tmp = t_0;
	} else if (i <= -1e-186) {
		tmp = 100.0 * ((expm1(i) * n) / i);
	} else if (i <= 4e+167) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (((Math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n;
	double tmp;
	if (i <= -1.08e+141) {
		tmp = t_0;
	} else if (i <= -1e-186) {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	} else if (i <= 4e+167) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (((math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / i) * n
	tmp = 0
	if i <= -1.08e+141:
		tmp = t_0
	elif i <= -1e-186:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	elif i <= 4e+167:
		tmp = (100.0 * (math.expm1(i) / i)) * n
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) * 100.0) / i) * n)
	tmp = 0.0
	if (i <= -1.08e+141)
		tmp = t_0;
	elseif (i <= -1e-186)
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	elseif (i <= 4e+167)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[i, -1.08e+141], t$95$0, If[LessEqual[i, -1e-186], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4e+167], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.08000000000000007e141 or 4.0000000000000002e167 < i

   1. Initial program 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 68.6

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

   4. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 67.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 80.9

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

   8. Applied rewrites 80.9%

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

   if -1.08000000000000007e141 < i < -9.9999999999999991e-187

   1. Initial program 21.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 42.6

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

   8. Applied rewrites 42.6%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-expm1.f64 78.7

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

   11. Applied rewrites 78.7%

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

   if -9.9999999999999991e-187 < i < 4.0000000000000002e167

   1. Initial program 12.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 78.7

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

   4. Applied rewrites 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-expm1.f64 88.0

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

   9. Applied rewrites 88.0%

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

Alternative 7: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{+33}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right), n\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))))
   (if (<= n -2.9e-34)
     t_0
     (if (<= n -5e-201)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 2.5e-141)
         (* 100.0 (/ (- 1.0 1.0) (/ i n)))
         (if (<= n 1.6e+33)
           (*
            100.0
            (fma
             i
             (fma
              0.5
              n
              (*
               i
               (fma 0.041666666666666664 (* i n) (* 0.16666666666666666 n))))
             n))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double tmp;
	if (n <= -2.9e-34) {
		tmp = t_0;
	} else if (n <= -5e-201) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.5e-141) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 1.6e+33) {
		tmp = 100.0 * fma(i, fma(0.5, n, (i * fma(0.041666666666666664, (i * n), (0.16666666666666666 * n)))), n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	tmp = 0.0
	if (n <= -2.9e-34)
		tmp = t_0;
	elseif (n <= -5e-201)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.5e-141)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 1.6e+33)
		tmp = Float64(100.0 * fma(i, fma(0.5, n, Float64(i * fma(0.041666666666666664, Float64(i * n), Float64(0.16666666666666666 * n)))), n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.9e-34], t$95$0, If[LessEqual[n, -5e-201], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-141], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.6e+33], N[(100.0 * N[(i * N[(0.5 * n + N[(i * N[(0.041666666666666664 * N[(i * n), $MachinePrecision] + N[(0.16666666666666666 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.9000000000000002e-34 or 1.60000000000000009e33 < n

   1. Initial program 23.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 63.9

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

   8. Applied rewrites 63.9%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-expm1.f64 92.0

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

   11. Applied rewrites 92.0%

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

   if -2.9000000000000002e-34 < n < -4.9999999999999999e-201

   1. Initial program 27.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 54.2%

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

   if -4.9999999999999999e-201 < n < 2.5e-141

   1. Initial program 64.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.5%

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

   if 2.5e-141 < n < 1.60000000000000009e33

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 71.7

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

   8. Applied rewrites 71.7%

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

Alternative 8: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-201} \lor \neg \left(n \leq 2.5 \cdot 10^{-141}\right):\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-201) (not (<= n 2.5e-141)))
   (* (* 100.0 (/ (expm1 i) i)) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-201) || !(n <= 2.5e-141)) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-201) || !(n <= 2.5e-141)) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-201) or not (n <= 2.5e-141):
		tmp = (100.0 * (math.expm1(i) / i)) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-201) || !(n <= 2.5e-141))
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * 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, -5e-201], N[Not[LessEqual[n, 2.5e-141]], $MachinePrecision]], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $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 < -4.9999999999999999e-201 or 2.5e-141 < n

   1. Initial program 24.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-expm1.f64 83.0

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

   9. Applied rewrites 83.0%

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

   if -4.9999999999999999e-201 < n < 2.5e-141

   1. Initial program 64.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-201} \lor \neg \left(n \leq 2.5 \cdot 10^{-141}\right):\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+17}:\\ \;\;\;\;100 \cdot \frac{\left(i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), 0.5\right), 1\right)\right) \cdot n}{i}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right), n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.3e+17)
   (*
    100.0
    (/
     (*
      (*
       i
       (fma
        i
        (fma i (fma 0.041666666666666664 i 0.16666666666666666) 0.5)
        1.0))
      n)
     i))
   (if (<= n -5e-201)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 2.5e-141)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (*
        100.0
        (fma
         i
         (fma
          0.5
          n
          (* i (fma 0.041666666666666664 (* i n) (* 0.16666666666666666 n))))
         n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.3e+17) {
		tmp = 100.0 * (((i * fma(i, fma(i, fma(0.041666666666666664, i, 0.16666666666666666), 0.5), 1.0)) * n) / i);
	} else if (n <= -5e-201) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 2.5e-141) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = 100.0 * fma(i, fma(0.5, n, (i * fma(0.041666666666666664, (i * n), (0.16666666666666666 * n)))), n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.3e+17)
		tmp = Float64(100.0 * Float64(Float64(Float64(i * fma(i, fma(i, fma(0.041666666666666664, i, 0.16666666666666666), 0.5), 1.0)) * n) / i));
	elseif (n <= -5e-201)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 2.5e-141)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(100.0 * fma(i, fma(0.5, n, Float64(i * fma(0.041666666666666664, Float64(i * n), Float64(0.16666666666666666 * n)))), n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.3e+17], N[(100.0 * N[(N[(N[(i * N[(i * N[(i * N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5e-201], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-141], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i * N[(0.5 * n + N[(i * N[(0.041666666666666664 * N[(i * n), $MachinePrecision] + N[(0.16666666666666666 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.3e17

   1. Initial program 22.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 64.5

         \[\leadsto 100 \cdot \frac{\left(i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), 0.5\right), 1\right)\right) \cdot n}{i} \]

   8. Applied rewrites 64.5%

      \[\leadsto 100 \cdot \frac{\left(i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), 0.5\right), 1\right)\right) \cdot n}{i} \]

   if -2.3e17 < n < -4.9999999999999999e-201

   1. Initial program 26.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 59.2%

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

   if -4.9999999999999999e-201 < n < 2.5e-141

   1. Initial program 64.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.5%

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

   if 2.5e-141 < n

   1. Initial program 24.4%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 77.0

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

   8. Applied rewrites 77.0%

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

Alternative 10: 65.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-199} \lor \neg \left(n \leq 2.5 \cdot 10^{-141}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right), n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.05e-199) (not (<= n 2.5e-141)))
   (*
    100.0
    (fma
     i
     (fma
      0.5
      n
      (* i (fma 0.041666666666666664 (* i n) (* 0.16666666666666666 n))))
     n))
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.05e-199) || !(n <= 2.5e-141)) {
		tmp = 100.0 * fma(i, fma(0.5, n, (i * fma(0.041666666666666664, (i * n), (0.16666666666666666 * n)))), n);
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.05e-199) || !(n <= 2.5e-141))
		tmp = Float64(100.0 * fma(i, fma(0.5, n, Float64(i * fma(0.041666666666666664, Float64(i * n), Float64(0.16666666666666666 * n)))), 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, -1.05e-199], N[Not[LessEqual[n, 2.5e-141]], $MachinePrecision]], N[(100.0 * N[(i * N[(0.5 * n + N[(i * N[(0.041666666666666664 * N[(i * n), $MachinePrecision] + N[(0.16666666666666666 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $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 < -1.05000000000000001e-199 or 2.5e-141 < n

   1. Initial program 24.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 66.4

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

   8. Applied rewrites 66.4%

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

   if -1.05000000000000001e-199 < n < 2.5e-141

   1. Initial program 64.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-199} \lor \neg \left(n \leq 2.5 \cdot 10^{-141}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(0.5, n, i \cdot \mathsf{fma}\left(0.041666666666666664, i \cdot n, 0.16666666666666666 \cdot n\right)\right), n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-199} \lor \neg \left(n \leq 2.5 \cdot 10^{-141}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.05e-199) (not (<= n 2.5e-141)))
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.05e-199) || !(n <= 2.5e-141)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), 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 <= -1.05e-199) || !(n <= 2.5e-141))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), 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, -1.05e-199], N[Not[LessEqual[n, 2.5e-141]], $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[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.05000000000000001e-199 or 2.5e-141 < n

   1. Initial program 24.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-expm1.f64 83.0

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

   9. Applied rewrites 83.0%

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

   10. Taylor expanded in i around 0

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

       1. +-commutative N/A

          \[\leadsto \left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + 100\right) \cdot n \]

       2. *-commutative N/A

          \[\leadsto \left(\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) \cdot i + 100\right) \cdot n \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right), i, 100\right) \cdot n \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50, i, 100\right) \cdot n \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{50}{3} + \frac{25}{6} \cdot i\right) \cdot i + 50, i, 100\right) \cdot n \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{50}{3} + \frac{25}{6} \cdot i, i, 50\right), i, 100\right) \cdot n \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{6} \cdot i + \frac{50}{3}, i, 50\right), i, 100\right) \cdot n \]

       8. lower-fma.f64 66.4

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

   12. Applied rewrites 66.4%

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

   if -1.05000000000000001e-199 < n < 2.5e-141

   1. Initial program 64.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-199} \lor \neg \left(n \leq 2.5 \cdot 10^{-141}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+17} \lor \neg \left(n \leq 3.7 \cdot 10^{-40}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.3e+17) (not (<= n 3.7e-40)))
   (*
    (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 <= -2.3e+17) || !(n <= 3.7e-40)) {
		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 <= -2.3e+17) || !(n <= 3.7e-40))
		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, -2.3e+17], N[Not[LessEqual[n, 3.7e-40]], $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 < -2.3e17 or 3.69999999999999998e-40 < n

   1. Initial program 24.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 71.8

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

   4. Applied rewrites 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 71.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-expm1.f64 91.8

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

   9. Applied rewrites 91.8%

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

   10. Taylor expanded in i around 0

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

       1. +-commutative N/A

          \[\leadsto \left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + 100\right) \cdot n \]

       2. *-commutative N/A

          \[\leadsto \left(\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) \cdot i + 100\right) \cdot n \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right), i, 100\right) \cdot n \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50, i, 100\right) \cdot n \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{50}{3} + \frac{25}{6} \cdot i\right) \cdot i + 50, i, 100\right) \cdot n \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{50}{3} + \frac{25}{6} \cdot i, i, 50\right), i, 100\right) \cdot n \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{6} \cdot i + \frac{50}{3}, i, 50\right), i, 100\right) \cdot n \]

       8. lower-fma.f64 72.3

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

   12. Applied rewrites 72.3%

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

   if -2.3e17 < n < 3.69999999999999998e-40

   1. Initial program 36.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 55.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+17} \lor \neg \left(n \leq 3.7 \cdot 10^{-40}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0) n))

C

double code(double i, double n) {
	return fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. pow-to-exp N/A

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

   12. lower-expm1.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log1p.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 80.6

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

4. Applied rewrites 80.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-expm1.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log1p.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 79.9%

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

7. Taylor expanded in n around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-expm1.f64 77.6

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

9. Applied rewrites 77.6%

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

10. Taylor expanded in i around 0

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

    1. +-commutative N/A

       \[\leadsto \left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + 100\right) \cdot n \]

    2. *-commutative N/A

       \[\leadsto \left(\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) \cdot i + 100\right) \cdot n \]

    3. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right), i, 100\right) \cdot n \]

    4. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50, i, 100\right) \cdot n \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{50}{3} + \frac{25}{6} \cdot i\right) \cdot i + 50, i, 100\right) \cdot n \]

    6. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{50}{3} + \frac{25}{6} \cdot i, i, 50\right), i, 100\right) \cdot n \]

    7. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{6} \cdot i + \frac{50}{3}, i, 50\right), i, 100\right) \cdot n \]

    8. lower-fma.f64 61.0

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

12. Applied rewrites 61.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), \color{blue}{i}, 100\right) \cdot n \]
13. Add Preprocessing

Alternative 14: 57.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (fma i (* n (fma 0.16666666666666666 i 0.5)) n)))

C

double code(double i, double n) {
	return 100.0 * fma(i, (n * fma(0.16666666666666666, i, 0.5)), n);
}

Julia

function code(i, n)
	return Float64(100.0 * fma(i, Float64(n * fma(0.16666666666666666, i, 0.5)), n))
end

Wolfram

code[i_, n_] := N[(100.0 * N[(i * N[(n * N[(0.16666666666666666 * i + 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-expm1.f64 72.9

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

5. Applied rewrites 72.9%

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

6. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 58.4

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

8. Applied rewrites 58.4%

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

Alternative 15: 57.5% 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 29.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. pow-to-exp N/A

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

   12. lower-expm1.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log1p.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 80.6

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

4. Applied rewrites 80.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-expm1.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log1p.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 79.9%

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

7. Taylor expanded in n around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-expm1.f64 77.6

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

9. Applied rewrites 77.6%

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

10. Taylor expanded in i around 0

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

    1. +-commutative N/A

       \[\leadsto \left(i \cdot \left(50 + \frac{50}{3} \cdot i\right) + 100\right) \cdot n \]

    2. *-commutative N/A

       \[\leadsto \left(\left(50 + \frac{50}{3} \cdot i\right) \cdot i + 100\right) \cdot n \]

    3. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(50 + \frac{50}{3} \cdot i, i, 100\right) \cdot n \]

    4. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{50}{3} \cdot i + 50, i, 100\right) \cdot n \]

    5. lower-fma.f64 58.4

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

12. Applied rewrites 58.4%

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

Alternative 16: 54.9% 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 29.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. pow-to-exp N/A

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

   12. lower-expm1.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log1p.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 80.6

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

4. Applied rewrites 80.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-expm1.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log1p.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 79.9%

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

7. Taylor expanded in n around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-expm1.f64 77.6

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

9. Applied rewrites 77.6%

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

10. Taylor expanded in i around 0

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

    1. +-commutative N/A

       \[\leadsto \left(50 \cdot i + 100\right) \cdot n \]

    2. lower-fma.f64 55.7

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

12. Applied rewrites 55.7%

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

Alternative 17: 49.8% 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 29.0%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 49.6%

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

Reproduce

herbie shell --seed 2025085 
(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))))