Compound Interest

Percentage Accurate: 29.4% → 81.9%

Time: 17.0s

Alternatives: 21

Speedup: 8.9×

• could not determine a ground truth

  1. i = 2.0595570987171685e-281
  2. n = 2.9957072087602534e+29

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: 29.4% 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: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\log \left(-\frac{1}{n}\right) + -1 \cdot \log \left(\frac{-1}{i}\right)\right)\\ t_1 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-35}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(-50, \frac{i \cdot e^{i}}{n}, t\_1\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-1 \cdot \frac{\mathsf{fma}\left(-100, \mathsf{expm1}\left(t\_0\right), -100 \cdot \frac{{n}^{2} \cdot e^{t\_0}}{i}\right)}{i}\right) \cdot n\\ \mathbf{elif}\;n \leq 1.36 \cdot 10^{-132}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i + \mathsf{fma}\left(-1, \log n, n \cdot \mathsf{fma}\left(0.5, {\left(\log i + -1 \cdot \log n\right)}^{2}, \frac{1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ (log (- (/ 1.0 n))) (* -1.0 (log (/ -1.0 i))))))
        (t_1 (* 100.0 (/ (expm1 i) i))))
   (if (<= n -9.5e-35)
     (* n (fma -50.0 (/ (* i (exp i)) n) t_1))
     (if (<= n -5e-310)
       (*
        (*
         -1.0
         (/
          (fma -100.0 (expm1 t_0) (* -100.0 (/ (* (pow n 2.0) (exp t_0)) i)))
          i))
        n)
       (if (<= n 1.36e-132)
         (*
          100.0
          (/
           (*
            n
            (+
             (log i)
             (fma
              -1.0
              (log n)
              (*
               n
               (fma 0.5 (pow (+ (log i) (* -1.0 (log n))) 2.0) (/ 1.0 i))))))
           (/ i n)))
         (* t_1 n))))))

C

double code(double i, double n) {
	double t_0 = n * (log(-(1.0 / n)) + (-1.0 * log((-1.0 / i))));
	double t_1 = 100.0 * (expm1(i) / i);
	double tmp;
	if (n <= -9.5e-35) {
		tmp = n * fma(-50.0, ((i * exp(i)) / n), t_1);
	} else if (n <= -5e-310) {
		tmp = (-1.0 * (fma(-100.0, expm1(t_0), (-100.0 * ((pow(n, 2.0) * exp(t_0)) / i))) / i)) * n;
	} else if (n <= 1.36e-132) {
		tmp = 100.0 * ((n * (log(i) + fma(-1.0, log(n), (n * fma(0.5, pow((log(i) + (-1.0 * log(n))), 2.0), (1.0 / i)))))) / (i / n));
	} else {
		tmp = t_1 * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * Float64(log(Float64(-Float64(1.0 / n))) + Float64(-1.0 * log(Float64(-1.0 / i)))))
	t_1 = Float64(100.0 * Float64(expm1(i) / i))
	tmp = 0.0
	if (n <= -9.5e-35)
		tmp = Float64(n * fma(-50.0, Float64(Float64(i * exp(i)) / n), t_1));
	elseif (n <= -5e-310)
		tmp = Float64(Float64(-1.0 * Float64(fma(-100.0, expm1(t_0), Float64(-100.0 * Float64(Float64((n ^ 2.0) * exp(t_0)) / i))) / i)) * n);
	elseif (n <= 1.36e-132)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(log(i) + fma(-1.0, log(n), Float64(n * fma(0.5, (Float64(log(i) + Float64(-1.0 * log(n))) ^ 2.0), Float64(1.0 / i)))))) / Float64(i / n)));
	else
		tmp = Float64(t_1 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[Log[(-N[(1.0 / n), $MachinePrecision])], $MachinePrecision] + N[(-1.0 * N[Log[N[(-1.0 / i), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e-35], N[(n * N[(-50.0 * N[(N[(i * N[Exp[i], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5e-310], N[(N[(-1.0 * N[(N[(-100.0 * N[(Exp[t$95$0] - 1), $MachinePrecision] + N[(-100.0 * N[(N[(N[Power[n, 2.0], $MachinePrecision] * N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.36e-132], N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] + N[(-1.0 * N[Log[n], $MachinePrecision] + N[(n * N[(0.5 * N[Power[N[(N[Log[i], $MachinePrecision] + N[(-1.0 * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.5000000000000003e-35

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(-50, \frac{i \cdot e^{i}}{n}, 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   if -9.5000000000000003e-35 < n < -4.999999999999985e-310

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in i around -inf

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

   8. Applied rewrites 14.7%

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

   if -4.999999999999985e-310 < n < 1.36000000000000002e-132

   1. Initial program 29.4%

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

   2. Taylor expanded in n around 0

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

   4. Applied rewrites 17.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(\log i + \mathsf{fma}\left(-1, \log n, n \cdot \mathsf{fma}\left(0.5, {\left(\log i + -1 \cdot \log n\right)}^{2}, \frac{1}{i}\right)\right)\right)}}{\frac{i}{n}} \]

   if 1.36000000000000002e-132 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

Alternative 2: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-35}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(-50, \frac{i \cdot e^{i}}{n}, t\_0\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(n \cdot \left(-\log \left(\frac{n}{i}\right)\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 1.36 \cdot 10^{-132}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(\log i + \mathsf{fma}\left(-1, \log n, n \cdot \mathsf{fma}\left(0.5, {\left(\log i + -1 \cdot \log n\right)}^{2}, \frac{1}{i}\right)\right)\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) i))))
   (if (<= n -9.5e-35)
     (* n (fma -50.0 (/ (* i (exp i)) n) t_0))
     (if (<= n -5e-310)
       (* 100.0 (* (/ (expm1 (* n (- (log (/ n i))))) i) n))
       (if (<= n 1.36e-132)
         (*
          100.0
          (/
           (*
            n
            (+
             (log i)
             (fma
              -1.0
              (log n)
              (*
               n
               (fma 0.5 (pow (+ (log i) (* -1.0 (log n))) 2.0) (/ 1.0 i))))))
           (/ i n)))
         (* t_0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / i);
	double tmp;
	if (n <= -9.5e-35) {
		tmp = n * fma(-50.0, ((i * exp(i)) / n), t_0);
	} else if (n <= -5e-310) {
		tmp = 100.0 * ((expm1((n * -log((n / i)))) / i) * n);
	} else if (n <= 1.36e-132) {
		tmp = 100.0 * ((n * (log(i) + fma(-1.0, log(n), (n * fma(0.5, pow((log(i) + (-1.0 * log(n))), 2.0), (1.0 / i)))))) / (i / n));
	} else {
		tmp = t_0 * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / i))
	tmp = 0.0
	if (n <= -9.5e-35)
		tmp = Float64(n * fma(-50.0, Float64(Float64(i * exp(i)) / n), t_0));
	elseif (n <= -5e-310)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(n * Float64(-log(Float64(n / i))))) / i) * n));
	elseif (n <= 1.36e-132)
		tmp = Float64(100.0 * Float64(Float64(n * Float64(log(i) + fma(-1.0, log(n), Float64(n * fma(0.5, (Float64(log(i) + Float64(-1.0 * log(n))) ^ 2.0), Float64(1.0 / i)))))) / Float64(i / n)));
	else
		tmp = Float64(t_0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e-35], N[(n * N[(-50.0 * N[(N[(i * N[Exp[i], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5e-310], N[(100.0 * N[(N[(N[(Exp[N[(n * (-N[Log[N[(n / i), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.36e-132], N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] + N[(-1.0 * N[Log[n], $MachinePrecision] + N[(n * N[(0.5 * N[Power[N[(N[Log[i], $MachinePrecision] + N[(-1.0 * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.5000000000000003e-35

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{n \cdot \mathsf{fma}\left(-50, \frac{i \cdot e^{i}}{n}, 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)} \]

   if -9.5000000000000003e-35 < n < -4.999999999999985e-310

   1. Initial program 29.4%

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

   2. Taylor expanded in i around -inf

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

   4. Applied rewrites 14.7%

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

   6. Applied rewrites 28.6%

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

   if -4.999999999999985e-310 < n < 1.36000000000000002e-132

   1. Initial program 29.4%

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

   2. Taylor expanded in n around 0

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

   4. Applied rewrites 17.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(\log i + \mathsf{fma}\left(-1, \log n, n \cdot \mathsf{fma}\left(0.5, {\left(\log i + -1 \cdot \log n\right)}^{2}, \frac{1}{i}\right)\right)\right)}}{\frac{i}{n}} \]

   if 1.36000000000000002e-132 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

Alternative 3: 81.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-283}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{n + i}{n}\right)}^{n}}{i}, n, -\frac{n}{i}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -2e-283)
     (* 100.0 (fma (/ (pow (/ (+ n i) n) n) i) n (- (/ n i))))
     (if (<= t_0 0.0)
       (* (* 100.0 (/ (expm1 i) i)) n)
       (if (<= t_0 INFINITY)
         (* (* (- (pow (- (/ i n) -1.0) n) 1.0) (/ 100.0 i)) n)
         (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e-283) {
		tmp = 100.0 * fma((pow(((n + i) / n), n) / i), n, -(n / i));
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((pow(((i / n) - -1.0), n) - 1.0) * (100.0 / i)) * n;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -2e-283)
		tmp = Float64(100.0 * fma(Float64((Float64(Float64(n + i) / n) ^ n) / i), n, Float64(-Float64(n / i))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) * Float64(100.0 / i)) * n);
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-283], N[(100.0 * N[(N[(N[Power[N[(N[(n + i), $MachinePrecision] / n), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision] * n + (-N[(n / i), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -1.99999999999999989e-283

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   if -1.99999999999999989e-283 < (*.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 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

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

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

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

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.0%

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

Alternative 4: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ t_1 := \left(\left({\left(\frac{i}{n} - -1\right)}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-283}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
        (t_1 (* (* (- (pow (- (/ i n) -1.0) n) 1.0) (/ 100.0 i)) n)))
   (if (<= t_0 -2e-283)
     t_1
     (if (<= t_0 0.0)
       (* (* 100.0 (/ (expm1 i) i)) n)
       (if (<= t_0 INFINITY) t_1 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double t_1 = ((pow(((i / n) - -1.0), n) - 1.0) * (100.0 / i)) * n;
	double tmp;
	if (t_0 <= -2e-283) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * (expm1(i) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double t_1 = ((Math.pow(((i / n) - -1.0), n) - 1.0) * (100.0 / i)) * n;
	double tmp;
	if (t_0 <= -2e-283) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * (Math.expm1(i) / i)) * n;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	t_1 = ((math.pow(((i / n) - -1.0), n) - 1.0) * (100.0 / i)) * n
	tmp = 0
	if t_0 <= -2e-283:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (100.0 * (math.expm1(i) / i)) * n
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	t_1 = Float64(Float64(Float64((Float64(Float64(i / n) - -1.0) ^ n) - 1.0) * Float64(100.0 / i)) * n)
	tmp = 0.0
	if (t_0 <= -2e-283)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * Float64(expm1(i) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] - -1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-283], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(100.0 * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -1.99999999999999989e-283 or -0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   if -1.99999999999999989e-283 < (*.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 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

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

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.0%

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

Alternative 5: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-35}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(n \cdot \left(-\log \left(\frac{n}{i}\right)\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 1.36 \cdot 10^{-132}:\\ \;\;\;\;\left(100 \cdot \frac{n \cdot \left(\log i + -1 \cdot \log n\right)}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -9.5e-35)
     (* 100.0 (* t_0 n))
     (if (<= n -5e-310)
       (* 100.0 (* (/ (expm1 (* n (- (log (/ n i))))) i) n))
       (if (<= n 1.36e-132)
         (* (* 100.0 (/ (* n (+ (log i) (* -1.0 (log n)))) i)) n)
         (* (* 100.0 t_0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -9.5e-35) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -5e-310) {
		tmp = 100.0 * ((expm1((n * -log((n / i)))) / i) * n);
	} else if (n <= 1.36e-132) {
		tmp = (100.0 * ((n * (log(i) + (-1.0 * log(n)))) / i)) * n;
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -9.5e-35) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -5e-310) {
		tmp = 100.0 * ((Math.expm1((n * -Math.log((n / i)))) / i) * n);
	} else if (n <= 1.36e-132) {
		tmp = (100.0 * ((n * (Math.log(i) + (-1.0 * Math.log(n)))) / i)) * n;
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -9.5e-35:
		tmp = 100.0 * (t_0 * n)
	elif n <= -5e-310:
		tmp = 100.0 * ((math.expm1((n * -math.log((n / i)))) / i) * n)
	elif n <= 1.36e-132:
		tmp = (100.0 * ((n * (math.log(i) + (-1.0 * math.log(n)))) / i)) * n
	else:
		tmp = (100.0 * t_0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -9.5e-35)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= -5e-310)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(n * Float64(-log(Float64(n / i))))) / i) * n));
	elseif (n <= 1.36e-132)
		tmp = Float64(Float64(100.0 * Float64(Float64(n * Float64(log(i) + Float64(-1.0 * log(n)))) / i)) * n);
	else
		tmp = Float64(Float64(100.0 * t_0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -9.5e-35], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5e-310], N[(100.0 * N[(N[(N[(Exp[N[(n * (-N[Log[N[(n / i), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.36e-132], N[(N[(100.0 * N[(N[(n * N[(N[Log[i], $MachinePrecision] + N[(-1.0 * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.5000000000000003e-35

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 75.4%

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

   if -9.5000000000000003e-35 < n < -4.999999999999985e-310

   1. Initial program 29.4%

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

   2. Taylor expanded in i around -inf

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

   4. Applied rewrites 14.7%

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

   6. Applied rewrites 28.6%

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

   if -4.999999999999985e-310 < n < 1.36000000000000002e-132

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 11.9%

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

   if 1.36000000000000002e-132 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

Alternative 6: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-35}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq -7.6 \cdot 10^{-267}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(n \cdot \left(-\log \left(\frac{n}{i}\right)\right)\right)}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -9.5e-35)
     (* 100.0 (* t_0 n))
     (if (<= n -7.6e-267)
       (* 100.0 (* (/ (expm1 (* n (- (log (/ n i))))) i) n))
       (if (<= n 1.55e-142)
         (* 100.0 (/ (- 1.0 1.0) (/ i n)))
         (* (* 100.0 t_0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -9.5e-35) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -7.6e-267) {
		tmp = 100.0 * ((expm1((n * -log((n / i)))) / i) * n);
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -9.5e-35) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -7.6e-267) {
		tmp = 100.0 * ((Math.expm1((n * -Math.log((n / i)))) / i) * n);
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -9.5e-35:
		tmp = 100.0 * (t_0 * n)
	elif n <= -7.6e-267:
		tmp = 100.0 * ((math.expm1((n * -math.log((n / i)))) / i) * n)
	elif n <= 1.55e-142:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (100.0 * t_0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -9.5e-35)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= -7.6e-267)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(n * Float64(-log(Float64(n / i))))) / i) * n));
	elseif (n <= 1.55e-142)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(100.0 * t_0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -9.5e-35], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -7.6e-267], N[(100.0 * N[(N[(N[(Exp[N[(n * (-N[Log[N[(n / i), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55e-142], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.5000000000000003e-35

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 75.4%

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

   if -9.5000000000000003e-35 < n < -7.60000000000000006e-267

   1. Initial program 29.4%

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

   2. Taylor expanded in i around -inf

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

   4. Applied rewrites 14.7%

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

   6. Applied rewrites 28.6%

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

   if -7.60000000000000006e-267 < n < 1.55e-142

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

   if 1.55e-142 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-35}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq -7.6 \cdot 10^{-267}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \left(-\log \left(\frac{n}{i}\right)\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -9.5e-35)
     (* 100.0 (* t_0 n))
     (if (<= n -7.6e-267)
       (* 100.0 (* (expm1 (* n (- (log (/ n i))))) (/ n i)))
       (if (<= n 1.55e-142)
         (* 100.0 (/ (- 1.0 1.0) (/ i n)))
         (* (* 100.0 t_0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -9.5e-35) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -7.6e-267) {
		tmp = 100.0 * (expm1((n * -log((n / i)))) * (n / i));
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -9.5e-35) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -7.6e-267) {
		tmp = 100.0 * (Math.expm1((n * -Math.log((n / i)))) * (n / i));
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -9.5e-35:
		tmp = 100.0 * (t_0 * n)
	elif n <= -7.6e-267:
		tmp = 100.0 * (math.expm1((n * -math.log((n / i)))) * (n / i))
	elif n <= 1.55e-142:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (100.0 * t_0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -9.5e-35)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= -7.6e-267)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * Float64(-log(Float64(n / i))))) * Float64(n / i)));
	elseif (n <= 1.55e-142)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(100.0 * t_0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -9.5e-35], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -7.6e-267], N[(100.0 * N[(N[(Exp[N[(n * (-N[Log[N[(n / i), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55e-142], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -9.5000000000000003e-35

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 75.4%

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

   if -9.5000000000000003e-35 < n < -7.60000000000000006e-267

   1. Initial program 29.4%

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

   2. Taylor expanded in i around -inf

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

   4. Applied rewrites 14.7%

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

   6. Applied rewrites 28.6%

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

   if -7.60000000000000006e-267 < n < 1.55e-142

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

   if 1.55e-142 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

Alternative 8: 80.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-193}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -6.5e-193)
     (* 100.0 (* t_0 n))
     (if (<= n 1.55e-142)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (* (* 100.0 t_0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -6.5e-193) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -6.5e-193) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -6.5e-193:
		tmp = 100.0 * (t_0 * n)
	elif n <= 1.55e-142:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (100.0 * t_0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -6.5e-193)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= 1.55e-142)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(100.0 * t_0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -6.5e-193], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55e-142], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.5000000000000004e-193

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 75.4%

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

   if -6.5000000000000004e-193 < n < 1.55e-142

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

   if 1.55e-142 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

Alternative 9: 80.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* (/ (expm1 i) i) n))))
   (if (<= n -6.5e-193)
     t_0
     (if (<= n 1.55e-142) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) / i) * n);
	double tmp;
	if (n <= -6.5e-193) {
		tmp = t_0;
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.expm1(i) / i) * n);
	double tmp;
	if (n <= -6.5e-193) {
		tmp = t_0;
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) / i) * n)
	tmp = 0
	if n <= -6.5e-193:
		tmp = t_0
	elif n <= 1.55e-142:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) / i) * n))
	tmp = 0.0
	if (n <= -6.5e-193)
		tmp = t_0;
	elseif (n <= 1.55e-142)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e-193], t$95$0, If[LessEqual[n, 1.55e-142], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.5000000000000004e-193 or 1.55e-142 < n

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 71.0%

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

   6. Applied rewrites 75.4%

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

   if -6.5000000000000004e-193 < n < 1.55e-142

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

Alternative 10: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right) \cdot n\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (+ 100.0 (* i (+ 50.0 (* 16.666666666666668 i)))) n)))
   (if (<= n -1.15e-47)
     t_0
     (if (<= n 1.55e-142) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (100.0 + (i * (50.0 + (16.666666666666668 * i)))) * n;
	double tmp;
	if (n <= -1.15e-47) {
		tmp = t_0;
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (100.0d0 + (i * (50.0d0 + (16.666666666666668d0 * i)))) * n
    if (n <= (-1.15d-47)) then
        tmp = t_0
    else if (n <= 1.55d-142) then
        tmp = 100.0d0 * ((1.0d0 - 1.0d0) / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (100.0 + (i * (50.0 + (16.666666666666668 * i)))) * n;
	double tmp;
	if (n <= -1.15e-47) {
		tmp = t_0;
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 + (i * (50.0 + (16.666666666666668 * i)))) * n
	tmp = 0
	if n <= -1.15e-47:
		tmp = t_0
	elif n <= 1.55e-142:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 + Float64(i * Float64(50.0 + Float64(16.666666666666668 * i)))) * n)
	tmp = 0.0
	if (n <= -1.15e-47)
		tmp = t_0;
	elseif (n <= 1.55e-142)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (100.0 + (i * (50.0 + (16.666666666666668 * i)))) * n;
	tmp = 0.0;
	if (n <= -1.15e-47)
		tmp = t_0;
	elseif (n <= 1.55e-142)
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 + N[(i * N[(50.0 + N[(16.666666666666668 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.15e-47], t$95$0, If[LessEqual[n, 1.55e-142], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.14999999999999991e-47 or 1.55e-142 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 57.0%

       \[\leadsto \left(100 + \color{blue}{i \cdot \left(50 + 16.666666666666668 \cdot i\right)}\right) \cdot n \]

   if -1.14999999999999991e-47 < n < 1.55e-142

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

Alternative 11: 61.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n \cdot 0.5 - 0.5, i, n\right) \cdot 100\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (- (* n 0.5) 0.5) i n) 100.0)))
   (if (<= n -1.15e-47)
     t_0
     (if (<= n 1.55e-142) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(((n * 0.5) - 0.5), i, n) * 100.0;
	double tmp;
	if (n <= -1.15e-47) {
		tmp = t_0;
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(Float64(Float64(n * 0.5) - 0.5), i, n) * 100.0)
	tmp = 0.0
	if (n <= -1.15e-47)
		tmp = t_0;
	elseif (n <= 1.55e-142)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(n * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * i + n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -1.15e-47], t$95$0, If[LessEqual[n, 1.55e-142], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.14999999999999991e-47 or 1.55e-142 < n

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   6. Applied rewrites 53.9%

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

   if -1.14999999999999991e-47 < n < 1.55e-142

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

Alternative 12: 61.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;100 \cdot \frac{n \cdot i}{i}\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left(1 + 0.5 \cdot i\right)\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.15e-47)
   (* 100.0 (/ (* n i) i))
   (if (<= n 1.55e-142)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (* (* 100.0 (+ 1.0 (* 0.5 i))) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.15e-47) {
		tmp = 100.0 * ((n * i) / i);
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * (1.0 + (0.5 * i))) * n;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.15d-47)) then
        tmp = 100.0d0 * ((n * i) / i)
    else if (n <= 1.55d-142) then
        tmp = 100.0d0 * ((1.0d0 - 1.0d0) / (i / n))
    else
        tmp = (100.0d0 * (1.0d0 + (0.5d0 * i))) * n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.15e-47) {
		tmp = 100.0 * ((n * i) / i);
	} else if (n <= 1.55e-142) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (100.0 * (1.0 + (0.5 * i))) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.15e-47:
		tmp = 100.0 * ((n * i) / i)
	elif n <= 1.55e-142:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (100.0 * (1.0 + (0.5 * i))) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.15e-47)
		tmp = Float64(100.0 * Float64(Float64(n * i) / i));
	elseif (n <= 1.55e-142)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(100.0 * Float64(1.0 + Float64(0.5 * i))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.15e-47)
		tmp = 100.0 * ((n * i) / i);
	elseif (n <= 1.55e-142)
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	else
		tmp = (100.0 * (1.0 + (0.5 * i))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.15e-47], N[(100.0 * N[(N[(n * i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.55e-142], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(1.0 + N[(0.5 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.14999999999999991e-47

   1. Initial program 29.4%

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

   2. Taylor expanded in n around inf

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.5%

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

   if -1.14999999999999991e-47 < n < 1.55e-142

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

   if 1.55e-142 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 54.1%

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

Alternative 13: 61.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-190}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{elif}\;n \leq 9.6 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot \left(1 + 0.5 \cdot i\right)\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.5e-190)
   (* n (fma i 50.0 100.0))
   (if (<= n 9.6e-175)
     (* 100.0 (/ (* n n) n))
     (* (* 100.0 (+ 1.0 (* 0.5 i))) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.5e-190) {
		tmp = n * fma(i, 50.0, 100.0);
	} else if (n <= 9.6e-175) {
		tmp = 100.0 * ((n * n) / n);
	} else {
		tmp = (100.0 * (1.0 + (0.5 * i))) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.5e-190)
		tmp = Float64(n * fma(i, 50.0, 100.0));
	elseif (n <= 9.6e-175)
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	else
		tmp = Float64(Float64(100.0 * Float64(1.0 + Float64(0.5 * i))) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.5e-190], N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.6e-175], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(1.0 + N[(0.5 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.50000000000000048e-190

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   7. Applied rewrites 54.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   9. Applied rewrites 54.1%

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

   if -5.50000000000000048e-190 < n < 9.6e-175

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.0%

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

   6. Applied rewrites 48.6%

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

   if 9.6e-175 < n

   1. Initial program 29.4%

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

   3. Applied rewrites 23.8%

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 54.1%

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

Alternative 14: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{if}\;n \leq -5.5 \cdot 10^{-190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9.6 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i 50.0 100.0))))
   (if (<= n -5.5e-190) t_0 (if (<= n 9.6e-175) (* 100.0 (/ (* n n) n)) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, 50.0, 100.0);
	double tmp;
	if (n <= -5.5e-190) {
		tmp = t_0;
	} else if (n <= 9.6e-175) {
		tmp = 100.0 * ((n * n) / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, 50.0, 100.0))
	tmp = 0.0
	if (n <= -5.5e-190)
		tmp = t_0;
	elseif (n <= 9.6e-175)
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.5e-190], t$95$0, If[LessEqual[n, 9.6e-175], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -5.50000000000000048e-190 or 9.6e-175 < n

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   7. Applied rewrites 54.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   9. Applied rewrites 54.1%

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

   if -5.50000000000000048e-190 < n < 9.6e-175

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.0%

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

   6. Applied rewrites 48.6%

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

Alternative 15: 60.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.5 \cdot 10^{-190}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{elif}\;n \leq 9.6 \cdot 10^{-175}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100, n, 50 \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.5e-190)
   (* n (fma i 50.0 100.0))
   (if (<= n 9.6e-175)
     (* 100.0 (/ (* n n) n))
     (fma 100.0 n (* 50.0 (* i n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.5e-190) {
		tmp = n * fma(i, 50.0, 100.0);
	} else if (n <= 9.6e-175) {
		tmp = 100.0 * ((n * n) / n);
	} else {
		tmp = fma(100.0, n, (50.0 * (i * n)));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.5e-190)
		tmp = Float64(n * fma(i, 50.0, 100.0));
	elseif (n <= 9.6e-175)
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	else
		tmp = fma(100.0, n, Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.5e-190], N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.6e-175], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.50000000000000048e-190

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   7. Applied rewrites 54.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   9. Applied rewrites 54.1%

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

   if -5.50000000000000048e-190 < n < 9.6e-175

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.0%

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

   6. Applied rewrites 48.6%

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

   if 9.6e-175 < n

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around inf

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

   7. Applied rewrites 54.1%

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

Alternative 16: 60.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, 50, 100\right)\\ \mathbf{if}\;n \leq -7 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.11:\\ \;\;\;\;i \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i 50.0 100.0))))
   (if (<= n -7e+15) t_0 (if (<= n 0.11) (* i (* 100.0 (/ n i))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, 50.0, 100.0);
	double tmp;
	if (n <= -7e+15) {
		tmp = t_0;
	} else if (n <= 0.11) {
		tmp = i * (100.0 * (n / i));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, 50.0, 100.0))
	tmp = 0.0
	if (n <= -7e+15)
		tmp = t_0;
	elseif (n <= 0.11)
		tmp = Float64(i * Float64(100.0 * Float64(n / i)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * 50.0 + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7e+15], t$95$0, If[LessEqual[n, 0.11], N[(i * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7e15 or 0.110000000000000001 < n

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   7. Applied rewrites 54.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   9. Applied rewrites 54.1%

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

   if -7e15 < n < 0.110000000000000001

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in i around inf

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

   7. Applied rewrites 39.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 40.6%

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

Alternative 17: 54.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.4%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 54.0%

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

5. Taylor expanded in n around inf

   \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

7. Applied rewrites 54.1%

   \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

9. Applied rewrites 54.1%

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

Alternative 18: 53.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 0.9:\\ \;\;\;\;\mathsf{fma}\left(-0.5, i, n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 0.9) (* (fma -0.5 i n) 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 0.9) {
		tmp = fma(-0.5, i, n) * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 0.9)
		tmp = Float64(fma(-0.5, i, n) * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 0.9], N[(N[(-0.5 * i + n), $MachinePrecision] * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 0.900000000000000022

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   6. Applied rewrites 53.9%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 47.2%

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

   if 0.900000000000000022 < i

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   7. Applied rewrites 54.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   8. Taylor expanded in i around inf

      \[\leadsto 50 \cdot \left(i \cdot \color{blue}{n}\right) \]

   10. Applied rewrites 12.5%

       \[\leadsto 50 \cdot \left(i \cdot \color{blue}{n}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 53.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 9.2 \cdot 10^{+41}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 9.2e+41) (* 100.0 n) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 9.2e+41) {
		tmp = 100.0 * n;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 9.2d+41) then
        tmp = 100.0d0 * n
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 9.2e+41) {
		tmp = 100.0 * n;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 9.2e+41:
		tmp = 100.0 * n
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 9.2e+41)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 9.2e+41)
		tmp = 100.0 * n;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 9.2e+41], N[(100.0 * n), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 9.1999999999999994e41

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.0%

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

   if 9.1999999999999994e41 < i

   1. Initial program 29.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   7. Applied rewrites 54.1%

      \[\leadsto n \cdot \color{blue}{\left(100 + 50 \cdot i\right)} \]

   8. Taylor expanded in i around inf

      \[\leadsto 50 \cdot \left(i \cdot \color{blue}{n}\right) \]

   10. Applied rewrites 12.5%

       \[\leadsto 50 \cdot \left(i \cdot \color{blue}{n}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 48.0% accurate, 8.9× 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. Taylor expanded in i around 0

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

4. Applied rewrites 48.0%

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

Alternative 21: 2.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ -50 \cdot i \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* -50.0 i))

C

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

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 = (-50.0d0) * i
end function

Java

public static double code(double i, double n) {
	return -50.0 * i;
}

Python

def code(i, n):
	return -50.0 * i

Julia

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

MATLAB

function tmp = code(i, n)
	tmp = -50.0 * i;
end

Wolfram

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

Derivation

1. Initial program 29.4%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 54.0%

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

5. Taylor expanded in n around 0

   \[\leadsto -50 \cdot \color{blue}{i} \]

7. Applied rewrites 2.8%

   \[\leadsto -50 \cdot \color{blue}{i} \]
8. Add Preprocessing

Developer Target 1: 35.2% 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 2025153 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform c (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))))