Compound Interest

Percentage Accurate: 28.3% → 93.7%

Time: 10.3s

Alternatives: 16

Speedup: 24.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.3% 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: 93.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 29.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-*.f64 96.8

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

   4. Applied rewrites 96.8%

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

   if 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 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 2: 91.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[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], Infinity], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]

Derivation

1. Split input into 2 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 34.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 34.6

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-log1p.f64 96.6

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

   4. Applied rewrites 96.6%

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

   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

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

      1. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 3: 90.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 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 34.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 34.3

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 95.2

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

   4. Applied rewrites 95.2%

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

   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

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

      1. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 4: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -440:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 2.85 \cdot 10^{-16}:\\ \;\;\;\;\left(100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -440.0)
     (* (* t_0 100.0) n)
     (if (<= n 2.85e-16)
       (* (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ n i))
       (* t_0 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -440.0) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= 2.85e-16) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) * (n / i);
	} else {
		tmp = t_0 * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -440.0) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= 2.85e-16) {
		tmp = (100.0 * Math.expm1((Math.log1p((i / n)) * n))) * (n / i);
	} else {
		tmp = t_0 * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -440.0:
		tmp = (t_0 * 100.0) * n
	elif n <= 2.85e-16:
		tmp = (100.0 * math.expm1((math.log1p((i / n)) * n))) * (n / i)
	else:
		tmp = t_0 * (100.0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -440.0)
		tmp = Float64(Float64(t_0 * 100.0) * n);
	elseif (n <= 2.85e-16)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) * Float64(n / i));
	else
		tmp = Float64(t_0 * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -440.0], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.85e-16], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -440

   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

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -440 < n < 2.85e-16

   1. Initial program 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 33.0

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 93.6

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

   6. Applied rewrites 93.6%

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

   if 2.85e-16 < n

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.1

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

   7. Applied rewrites 97.1%

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

Alternative 5: 81.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\frac{\log \left(-i\right) - \log \left(-n\right)}{i} \cdot n\right) \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log i - \log n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -4.2e-79)
     (* (* t_0 100.0) n)
     (if (<= n -4e-310)
       (* (* (* (/ (- (log (- i)) (log (- n))) i) n) n) 100.0)
       (if (<= n 2.55e-52)
         (* 100.0 (* (* n n) (/ (- (log i) (log n)) i)))
         (* t_0 (* 100.0 n)))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -4.2e-79) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= -4e-310) {
		tmp = ((((log(-i) - log(-n)) / i) * n) * n) * 100.0;
	} else if (n <= 2.55e-52) {
		tmp = 100.0 * ((n * n) * ((log(i) - log(n)) / i));
	} else {
		tmp = t_0 * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -4.2e-79) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= -4e-310) {
		tmp = ((((Math.log(-i) - Math.log(-n)) / i) * n) * n) * 100.0;
	} else if (n <= 2.55e-52) {
		tmp = 100.0 * ((n * n) * ((Math.log(i) - Math.log(n)) / i));
	} else {
		tmp = t_0 * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -4.2e-79:
		tmp = (t_0 * 100.0) * n
	elif n <= -4e-310:
		tmp = ((((math.log(-i) - math.log(-n)) / i) * n) * n) * 100.0
	elif n <= 2.55e-52:
		tmp = 100.0 * ((n * n) * ((math.log(i) - math.log(n)) / i))
	else:
		tmp = t_0 * (100.0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -4.2e-79)
		tmp = Float64(Float64(t_0 * 100.0) * n);
	elseif (n <= -4e-310)
		tmp = Float64(Float64(Float64(Float64(Float64(log(Float64(-i)) - log(Float64(-n))) / i) * n) * n) * 100.0);
	elseif (n <= 2.55e-52)
		tmp = Float64(100.0 * Float64(Float64(n * n) * Float64(Float64(log(i) - log(n)) / i)));
	else
		tmp = Float64(t_0 * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -4.2e-79], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -4e-310], N[(N[(N[(N[(N[(N[Log[(-i)], $MachinePrecision] - N[Log[(-n)], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 2.55e-52], N[(100.0 * N[(N[(n * n), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.1999999999999999e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -4.1999999999999999e-79 < n < -3.999999999999988e-310

   1. Initial program 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 50.6

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

      4. lift--.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-log1p.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-log.f64 0.0

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

   7. Applied rewrites 0.0%

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

   9. Applied rewrites 49.5%

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

   11. Applied rewrites 71.5%

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

   if -3.999999999999988e-310 < n < 2.54999999999999995e-52

   1. Initial program 32.9%

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

   3. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-log.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if 2.54999999999999995e-52 < n

   1. Initial program 20.6%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 94.2

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

   7. Applied rewrites 94.2%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\frac{\log \left(-i\right) - \log \left(-n\right)}{i} \cdot n\right) \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log i - \log n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right)}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-168} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.1e-168) (not (<= n 1.65e-93)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* (/ (- 1.0 1.0) i) (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.1e-168) || !(n <= 1.65e-93)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.1e-168) || !(n <= 1.65e-93)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.1e-168) or not (n <= 1.65e-93):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = ((1.0 - 1.0) / i) * (n * 100.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.1e-168) || !(n <= 1.65e-93))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.1e-168], N[Not[LessEqual[n, 1.65e-93]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.0999999999999998e-168 or 1.6500000000000001e-93 < n

   1. Initial program 22.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 85.1

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

   5. Applied rewrites 85.1%

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

   if -4.0999999999999998e-168 < n < 1.6500000000000001e-93

   1. Initial program 55.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 67.5

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

   7. Applied rewrites 67.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.1 \cdot 10^{-168} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -4.1 \cdot 10^{-168}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{-93}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -4.1e-168)
     (* (* t_0 100.0) n)
     (if (<= n 1.65e-93)
       (* (/ (- 1.0 1.0) i) (* n 100.0))
       (* t_0 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -4.1e-168) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= 1.65e-93) {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	} else {
		tmp = t_0 * (100.0 * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -4.1e-168) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= 1.65e-93) {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	} else {
		tmp = t_0 * (100.0 * n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -4.1e-168:
		tmp = (t_0 * 100.0) * n
	elif n <= 1.65e-93:
		tmp = ((1.0 - 1.0) / i) * (n * 100.0)
	else:
		tmp = t_0 * (100.0 * n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -4.1e-168)
		tmp = Float64(Float64(t_0 * 100.0) * n);
	elseif (n <= 1.65e-93)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	else
		tmp = Float64(t_0 * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -4.1e-168], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.65e-93], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.0999999999999998e-168

   1. Initial program 25.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -4.0999999999999998e-168 < n < 1.6500000000000001e-93

   1. Initial program 55.9%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 67.5

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

   7. Applied rewrites 67.5%

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

   if 1.6500000000000001e-93 < n

   1. Initial program 19.5%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 70.6

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

   5. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 8: 65.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.3e-148) (not (<= n 1.65e-93)))
   (*
    100.0
    (fma
     (fma
      (* n i)
      (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
      (* (- 0.5 (/ 0.5 n)) n))
     i
     n))
   (* (/ (- 1.0 1.0) i) (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.3e-148) || !(n <= 1.65e-93)) {
		tmp = 100.0 * fma(fma((n * i), (((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), ((0.5 - (0.5 / n)) * n)), i, n);
	} else {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.3e-148) || !(n <= 1.65e-93))
		tmp = Float64(100.0 * fma(fma(Float64(n * i), Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), Float64(Float64(0.5 - Float64(0.5 / n)) * n)), i, n));
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.3e-148], N[Not[LessEqual[n, 1.65e-93]], $MachinePrecision]], N[(100.0 * N[(N[(N[(n * i), $MachinePrecision] * N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.29999999999999995e-148 or 1.6500000000000001e-93 < n

   1. Initial program 22.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 67.5%

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

   if -5.29999999999999995e-148 < n < 1.6500000000000001e-93

   1. Initial program 55.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 66.4

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

   7. Applied rewrites 66.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 + \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, \frac{-0.5}{n}\right), i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.3e-148) (not (<= n 1.65e-93)))
   (*
    (fma
     (+
      0.5
      (fma
       (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
       i
       (/ -0.5 n)))
     i
     1.0)
    (* n 100.0))
   (* (/ (- 1.0 1.0) i) (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.3e-148) || !(n <= 1.65e-93)) {
		tmp = fma((0.5 + fma((((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), i, (-0.5 / n))), i, 1.0) * (n * 100.0);
	} else {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.3e-148) || !(n <= 1.65e-93))
		tmp = Float64(fma(Float64(0.5 + fma(Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), i, Float64(-0.5 / n))), i, 1.0) * Float64(n * 100.0));
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.3e-148], N[Not[LessEqual[n, 1.65e-93]], $MachinePrecision]], N[(N[(N[(0.5 + N[(N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + 1.0), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.29999999999999995e-148 or 1.6500000000000001e-93 < n

   1. Initial program 22.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 4.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 4.5

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

   7. Applied rewrites 4.5%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 67.5%

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

   if -5.29999999999999995e-148 < n < 1.6500000000000001e-93

   1. Initial program 55.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 66.4

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

   7. Applied rewrites 66.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 + \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, \frac{-0.5}{n}\right), i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;i \leq 5 \cdot 10^{-10}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.1e+39)
   (* (/ (- 1.0 1.0) i) (* n 100.0))
   (if (<= i 5e-10)
     (* 100.0 n)
     (*
      100.0
      (/
       (*
        (fma
         (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
         i
         1.0)
        i)
       (/ i n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.1e+39) {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	} else if (i <= 5e-10) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.1e+39)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	elseif (i <= 5e-10)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.1e+39], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5e-10], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.10000000000000004e39

   1. Initial program 69.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 32.8

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

   7. Applied rewrites 32.8%

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

   if -4.10000000000000004e39 < i < 5.00000000000000031e-10

   1. Initial program 10.6%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if 5.00000000000000031e-10 < i

   1. Initial program 41.3%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 47.3%

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

Alternative 11: 63.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.1e+39)
   (* (/ (- 1.0 1.0) i) (* n 100.0))
   (if (<= i 2.8e-16)
     (* 100.0 n)
     (* 100.0 (/ (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) i) (/ i n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.1e+39) {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	} else if (i <= 2.8e-16) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * ((fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * i) / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.1e+39)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	elseif (i <= 2.8e-16)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * i) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.1e+39], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.8e-16], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.10000000000000004e39

   1. Initial program 69.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 32.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 32.8

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

   7. Applied rewrites 32.8%

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

   if -4.10000000000000004e39 < i < 2.8000000000000001e-16

   1. Initial program 10.6%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if 2.8000000000000001e-16 < i

   1. Initial program 40.7%

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

   3. Taylor expanded in n around inf

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

      1. lower-expm1.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 43.5%

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

Alternative 12: 63.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.68:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;i \leq 7.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, i, 1\right) \cdot i}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{fma}\left(i \cdot i, 0.16666666666666666 \cdot i, i\right)}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.68)
   (* (/ (- 1.0 1.0) i) (* n 100.0))
   (if (<= i 7.6e+15)
     (* (/ (* (fma 0.5 i 1.0) i) i) (* 100.0 n))
     (* 100.0 (/ (fma (* i i) (* 0.16666666666666666 i) i) (/ i n))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.68) {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	} else if (i <= 7.6e+15) {
		tmp = ((fma(0.5, i, 1.0) * i) / i) * (100.0 * n);
	} else {
		tmp = 100.0 * (fma((i * i), (0.16666666666666666 * i), i) / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.68)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	elseif (i <= 7.6e+15)
		tmp = Float64(Float64(Float64(fma(0.5, i, 1.0) * i) / i) * Float64(100.0 * n));
	else
		tmp = Float64(100.0 * Float64(fma(Float64(i * i), Float64(0.16666666666666666 * i), i) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.68], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 7.6e+15], N[(N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(i * i), $MachinePrecision] * N[(0.16666666666666666 * i), $MachinePrecision] + i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -0.680000000000000049

   1. Initial program 66.6%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 31.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 31.6

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

   7. Applied rewrites 31.6%

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

   if -0.680000000000000049 < i < 7.6e15

   1. Initial program 11.2%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 83.4%

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

   if 7.6e15 < i

   1. Initial program 42.3%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 16.8%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 48.5%

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

   12. Taylor expanded in n around inf

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

   14. Applied rewrites 45.4%

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

Alternative 13: 63.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, i, 1\right) \cdot i}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.3e-148) (not (<= n 1.65e-93)))
   (* (/ (* (fma 0.5 i 1.0) i) i) (* 100.0 n))
   (* (/ (- 1.0 1.0) i) (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.3e-148) || !(n <= 1.65e-93)) {
		tmp = ((fma(0.5, i, 1.0) * i) / i) * (100.0 * n);
	} else {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.3e-148) || !(n <= 1.65e-93))
		tmp = Float64(Float64(Float64(fma(0.5, i, 1.0) * i) / i) * Float64(100.0 * n));
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.3e-148], N[Not[LessEqual[n, 1.65e-93]], $MachinePrecision]], N[(N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.29999999999999995e-148 or 1.6500000000000001e-93 < n

   1. Initial program 22.0%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 47.5

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

   5. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 63.5

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 63.7%

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

   if -5.29999999999999995e-148 < n < 1.6500000000000001e-93

   1. Initial program 55.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 66.4

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

   7. Applied rewrites 66.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, i, 1\right) \cdot i}{i} \cdot \left(100 \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5.3e-148) (not (<= n 1.65e-93)))
   (* 100.0 (fma (* (- 0.5 (/ 0.5 n)) n) i n))
   (* (/ (- 1.0 1.0) i) (* n 100.0))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5.3e-148) || !(n <= 1.65e-93)) {
		tmp = 100.0 * fma(((0.5 - (0.5 / n)) * n), i, n);
	} else {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5.3e-148) || !(n <= 1.65e-93))
		tmp = Float64(100.0 * fma(Float64(Float64(0.5 - Float64(0.5 / n)) * n), i, n));
	else
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5.3e-148], N[Not[LessEqual[n, 1.65e-93]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.29999999999999995e-148 or 1.6500000000000001e-93 < n

   1. Initial program 22.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 63.1

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

   5. Applied rewrites 63.1%

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

   if -5.29999999999999995e-148 < n < 1.6500000000000001e-93

   1. Initial program 55.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 66.4

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

   7. Applied rewrites 66.4%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.3 \cdot 10^{-148} \lor \neg \left(n \leq 1.65 \cdot 10^{-93}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.1 \cdot 10^{+39} \lor \neg \left(i \leq 1650000\right):\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -4.1e+39) (not (<= i 1650000.0)))
   (* (/ (- 1.0 1.0) i) (* n 100.0))
   (* 100.0 n)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -4.1e+39) || !(i <= 1650000.0)) {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-4.1d+39)) .or. (.not. (i <= 1650000.0d0))) then
        tmp = ((1.0d0 - 1.0d0) / i) * (n * 100.0d0)
    else
        tmp = 100.0d0 * n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -4.1e+39) || !(i <= 1650000.0)) {
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -4.1e+39) or not (i <= 1650000.0):
		tmp = ((1.0 - 1.0) / i) * (n * 100.0)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -4.1e+39) || !(i <= 1650000.0))
		tmp = Float64(Float64(Float64(1.0 - 1.0) / i) * Float64(n * 100.0));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -4.1e+39) || ~((i <= 1650000.0)))
		tmp = ((1.0 - 1.0) / i) * (n * 100.0);
	else
		tmp = 100.0 * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -4.1e+39], N[Not[LessEqual[i, 1650000.0]], $MachinePrecision]], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -4.10000000000000004e39 or 1.65e6 < i

   1. Initial program 54.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 28.4

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

   7. Applied rewrites 28.4%

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

   if -4.10000000000000004e39 < i < 1.65e6

   1. Initial program 10.5%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.1 \cdot 10^{+39} \lor \neg \left(i \leq 1650000\right):\\ \;\;\;\;\frac{1 - 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.6% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.4%

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

3. Taylor expanded in i around 0

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

   1. lower-*.f64 50.5

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

5. Applied rewrites 50.5%

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

Developer Target 1: 34.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025017 
(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))))