Compound Interest

Percentage Accurate: 28.4% → 84.0%

Time: 6.3s

Alternatives: 14

Speedup: 8.9×

• could not determine a ground truth

  1. i = 1.0420142163403124e-195
  2. n = -1.7740444161622383e+27

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.4% 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: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-280}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\mathsf{fma}\left(-0.5, \frac{i}{n}, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -4e-280)
     (/ (* 100.0 (- (pow (+ (/ i n) 1.0) n) 1.0)) (/ i n))
     (if (<= t_0 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_0 INFINITY)
         t_0
         (* 100.0 (/ 1.0 (fma -0.5 (/ i n) (/ 1.0 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 <= -4e-280) {
		tmp = (100.0 * (pow(((i / n) + 1.0), n) - 1.0)) / (i / n);
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (1.0 / fma(-0.5, (i / n), (1.0 / 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 <= -4e-280)
		tmp = Float64(Float64(100.0 * Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0)) / Float64(i / n));
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(100.0 * Float64(1.0 / fma(-0.5, Float64(i / n), Float64(1.0 / 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, -4e-280], N[(N[(100.0 * N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(100.0 * N[(1.0 / N[(-0.5 * N[(i / n), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -3.9999999999999998e-280

   1. Initial program 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 28.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 28.4

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

   3. Applied rewrites 28.4%

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

   if -3.9999999999999998e-280 < (*.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 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

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

   1. Initial program 28.4%

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

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. division-flip N/A

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

      3. lower-special-/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-special-/ N/A

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

      6. lower-/.f64 49.1

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{1}{\frac{-1}{2} \cdot \frac{i}{n} + \color{blue}{\frac{1}{n}}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-/.f64 58.0

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

   12. Applied rewrites 58.0%

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

Alternative 2: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n} - 1\\ t_1 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-280}:\\ \;\;\;\;\frac{100 \cdot t\_0}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{t\_0}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\mathsf{fma}\left(-0.5, \frac{i}{n}, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ (/ i n) 1.0) n) 1.0))
        (t_1 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_1 -4e-280)
     (/ (* 100.0 t_0) (/ i n))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_1 INFINITY)
         (* (* (/ t_0 i) n) 100.0)
         (* 100.0 (/ 1.0 (fma -0.5 (/ i n) (/ 1.0 n)))))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n) - 1.0;
	double t_1 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_1 <= -4e-280) {
		tmp = (100.0 * t_0) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 / i) * n) * 100.0;
	} else {
		tmp = 100.0 * (1.0 / fma(-0.5, (i / n), (1.0 / n)));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0)
	t_1 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_1 <= -4e-280)
		tmp = Float64(Float64(100.0 * t_0) / Float64(i / n));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 / i) * n) * 100.0);
	else
		tmp = Float64(100.0 * Float64(1.0 / fma(-0.5, Float64(i / n), Float64(1.0 / n))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-280], N[(N[(100.0 * t$95$0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(100.0 * N[(1.0 / N[(-0.5 * N[(i / n), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -3.9999999999999998e-280

   1. Initial program 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift--.f64 28.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 28.4

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

   3. Applied rewrites 28.4%

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

   if -3.9999999999999998e-280 < (*.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 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

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

   1. Initial program 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 28.5%

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

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. division-flip N/A

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

      3. lower-special-/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-special-/ N/A

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

      6. lower-/.f64 49.1

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{1}{\frac{-1}{2} \cdot \frac{i}{n} + \color{blue}{\frac{1}{n}}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-/.f64 58.0

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

   12. Applied rewrites 58.0%

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

Alternative 3: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ t_1 := \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\mathsf{fma}\left(-0.5, \frac{i}{n}, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

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) i) n) 100.0)))
   (if (<= t_0 -4e-280)
     t_1
     (if (<= t_0 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_0 INFINITY)
         t_1
         (* 100.0 (/ 1.0 (fma -0.5 (/ i n) (/ 1.0 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) / i) * n) * 100.0;
	double tmp;
	if (t_0 <= -4e-280) {
		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 * (1.0 / fma(-0.5, (i / n), (1.0 / n)));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	t_1 = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / i) * n) * 100.0)
	tmp = 0.0
	if (t_0 <= -4e-280)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(100.0 * Float64(1.0 / fma(-0.5, Float64(i / n), Float64(1.0 / n))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-280], t$95$1, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(100.0 * N[(1.0 / N[(-0.5 * N[(i / n), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -3.9999999999999998e-280 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 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 28.5%

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

   if -3.9999999999999998e-280 < (*.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 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. division-flip N/A

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

      3. lower-special-/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-special-/ N/A

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

      6. lower-/.f64 49.1

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{1}{\frac{-1}{2} \cdot \frac{i}{n} + \color{blue}{\frac{1}{n}}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-/.f64 58.0

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

   12. Applied rewrites 58.0%

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

Alternative 4: 82.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;100 \cdot \frac{\left(\mathsf{fma}\left(0.5, i, 1\right) \cdot i\right) \cdot n}{i}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\left(n \cdot 100\right) \cdot \mathsf{expm1}\left(\log \left(\frac{i}{\left|n\right|}\right) \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\mathsf{fma}\left(-0.5, \frac{i}{n}, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (* 100.0 (/ (* (* (fma 0.5 i 1.0) i) n) i))
     (if (<= t_0 0.0)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (if (<= t_0 INFINITY)
         (/ (* (* n 100.0) (expm1 (* (log (/ i (fabs n))) n))) i)
         (* 100.0 (/ 1.0 (fma -0.5 (/ i n) (/ 1.0 n)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 100.0 * (((fma(0.5, i, 1.0) * i) * n) / i);
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((n * 100.0) * expm1((log((i / fabs(n))) * n))) / i;
	} else {
		tmp = 100.0 * (1.0 / fma(-0.5, (i / n), (1.0 / n)));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(Float64(Float64(fma(0.5, i, 1.0) * i) * n) / i));
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(n * 100.0) * expm1(Float64(log(Float64(i / abs(n))) * n))) / i);
	else
		tmp = Float64(100.0 * Float64(1.0 / fma(-0.5, Float64(i / n), Float64(1.0 / n))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(100.0 * N[(N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * i), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(n * 100.0), $MachinePrecision] * N[(Exp[N[(N[Log[N[(i / N[Abs[n], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 * N[(1.0 / N[(-0.5 * N[(i / n), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 52.0

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

   7. Applied rewrites 52.0%

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

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

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

   1. Initial program 28.4%

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

   2. Taylor expanded in i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 15.1%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. add-flip-rev N/A

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

      7. +-commutative N/A

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

      8. sub-flip N/A

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

      9. log-fabs N/A

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

      10. diff-log N/A

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

      11. lower-log.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-fabs.f64 19.9

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

   6. Applied rewrites 19.9%

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

   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 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. division-flip N/A

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

      3. lower-special-/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-special-/ N/A

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

      6. lower-/.f64 49.1

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{1}{\frac{-1}{2} \cdot \frac{i}{n} + \color{blue}{\frac{1}{n}}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-/.f64 58.0

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

   12. Applied rewrites 58.0%

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

Alternative 5: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;100 \cdot \frac{\left(\mathsf{fma}\left(0.5, i, 1\right) \cdot i\right) \cdot n}{i}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\mathsf{fma}\left(-0.5, \frac{i}{n}, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (* 100.0 (/ (* (* (fma 0.5 i 1.0) i) n) i))
     (if (<= t_0 INFINITY)
       (* 100.0 (/ (expm1 i) (/ i n)))
       (* 100.0 (/ 1.0 (fma -0.5 (/ i n) (/ 1.0 n))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 100.0 * (((fma(0.5, i, 1.0) * i) * n) / i);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = 100.0 * (1.0 / fma(-0.5, (i / n), (1.0 / n)));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(100.0 * Float64(Float64(Float64(fma(0.5, i, 1.0) * i) * n) / i));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(1.0 / fma(-0.5, Float64(i / n), Float64(1.0 / n))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(100.0 * N[(N[(N[(N[(0.5 * i + 1.0), $MachinePrecision] * i), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(1.0 / N[(-0.5 * N[(i / n), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 52.0

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

   7. Applied rewrites 52.0%

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

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

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

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. division-flip N/A

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

      3. lower-special-/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-special-/ N/A

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

      6. lower-/.f64 49.1

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{1}{\frac{-1}{2} \cdot \frac{i}{n} + \color{blue}{\frac{1}{n}}} \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-/.f64 58.0

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

   12. Applied rewrites 58.0%

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

Alternative 6: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.25 \cdot 10^{-188}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(i\right)}{i} \cdot n\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.25e-188)
   (* (/ (* 100.0 (expm1 i)) i) n)
   (if (<= n 5.2e-107)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (* (* (/ (expm1 i) i) n) 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.25e-188) {
		tmp = ((100.0 * expm1(i)) / i) * n;
	} else if (n <= 5.2e-107) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = ((expm1(i) / i) * n) * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.25e-188) {
		tmp = ((100.0 * Math.expm1(i)) / i) * n;
	} else if (n <= 5.2e-107) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = ((Math.expm1(i) / i) * n) * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.25e-188:
		tmp = ((100.0 * math.expm1(i)) / i) * n
	elif n <= 5.2e-107:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = ((math.expm1(i) / i) * n) * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.25e-188)
		tmp = Float64(Float64(Float64(100.0 * expm1(i)) / i) * n);
	elseif (n <= 5.2e-107)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.25e-188], N[(N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 5.2e-107], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.24999999999999997e-188

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 61.2

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

   6. Applied rewrites 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 74.8

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

   8. Applied rewrites 74.8%

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

   if -2.24999999999999997e-188 < n < 5.2000000000000001e-107

   1. Initial program 28.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 17.8%

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

   if 5.2000000000000001e-107 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.3

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.9

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

   6. Applied rewrites 74.9%

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

Alternative 7: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -2.25 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-107}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -2.25e-188)
     t_0
     (if (<= n 5.2e-107) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -2.25e-188) {
		tmp = t_0;
	} else if (n <= 5.2e-107) {
		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 = ((Math.expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -2.25e-188) {
		tmp = t_0;
	} else if (n <= 5.2e-107) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -2.25e-188:
		tmp = t_0
	elif n <= 5.2e-107:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -2.25e-188)
		tmp = t_0;
	elseif (n <= 5.2e-107)
		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[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -2.25e-188], t$95$0, If[LessEqual[n, 5.2e-107], 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 < -2.24999999999999997e-188 or 5.2000000000000001e-107 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.3

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.9

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

   6. Applied rewrites 74.9%

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

   if -2.24999999999999997e-188 < n < 5.2000000000000001e-107

   1. Initial program 28.4%

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

   2. Taylor expanded in i around 0

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

   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 8: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))))
   (if (<= n -3.8e-21) t_0 (if (<= n 4.5e-24) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double tmp;
	if (n <= -3.8e-21) {
		tmp = t_0;
	} else if (n <= 4.5e-24) {
		tmp = 100.0 * (i / (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) * n) / i);
	double tmp;
	if (n <= -3.8e-21) {
		tmp = t_0;
	} else if (n <= 4.5e-24) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) * n) / i)
	tmp = 0
	if n <= -3.8e-21:
		tmp = t_0
	elif n <= 4.5e-24:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	tmp = 0.0
	if (n <= -3.8e-21)
		tmp = t_0;
	elseif (n <= 4.5e-24)
		tmp = Float64(100.0 * Float64(i / 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] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.8e-21], t$95$0, If[LessEqual[n, 4.5e-24], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.7999999999999998e-21 or 4.4999999999999997e-24 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   if -3.7999999999999998e-21 < n < 4.4999999999999997e-24

   1. Initial program 28.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 42.4%

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

Alternative 9: 78.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}\\ \mathbf{if}\;n \leq -5 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 68000000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* 100.0 (* (expm1 i) n)) i)))
   (if (<= n -5e-21)
     t_0
     (if (<= n 68000000000.0) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (100.0 * (expm1(i) * n)) / i;
	double tmp;
	if (n <= -5e-21) {
		tmp = t_0;
	} else if (n <= 68000000000.0) {
		tmp = 100.0 * (i / (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) * n)) / i;
	double tmp;
	if (n <= -5e-21) {
		tmp = t_0;
	} else if (n <= 68000000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (100.0 * (math.expm1(i) * n)) / i
	tmp = 0
	if n <= -5e-21:
		tmp = t_0
	elif n <= 68000000000.0:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * Float64(expm1(i) * n)) / i)
	tmp = 0.0
	if (n <= -5e-21)
		tmp = t_0;
	elseif (n <= 68000000000.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5e-21], t$95$0, If[LessEqual[n, 68000000000.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.99999999999999973e-21 or 6.8e10 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 61.3

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.3

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.9

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

   6. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 74.7

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in n around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lift-expm1.f64 69.8

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

   11. Applied rewrites 69.8%

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

   if -4.99999999999999973e-21 < n < 6.8e10

   1. Initial program 28.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 42.4%

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

Alternative 10: 62.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-20}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-30}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9e-20)
   (* 100.0 (/ (* i n) i))
   (if (<= n 5e-30) (* 100.0 (/ i (/ i n))) (* 100.0 (fma (* n i) 0.5 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9e-20) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 5e-30) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * fma((n * i), 0.5, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9e-20)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 5e-30)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * fma(Float64(n * i), 0.5, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9e-20], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e-30], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5 + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.0000000000000003e-20

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

   if -9.0000000000000003e-20 < n < 4.99999999999999972e-30

   1. Initial program 28.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 42.4%

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

   if 4.99999999999999972e-30 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 54.2

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

   7. Applied rewrites 54.2%

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

Alternative 11: 61.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -9 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -9e-20) t_0 (if (<= n 4e-24) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -9e-20) {
		tmp = t_0;
	} else if (n <= 4e-24) {
		tmp = 100.0 * (i / (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 * n) / i)
    if (n <= (-9d-20)) then
        tmp = t_0
    else if (n <= 4d-24) then
        tmp = 100.0d0 * (i / (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 * n) / i);
	double tmp;
	if (n <= -9e-20) {
		tmp = t_0;
	} else if (n <= 4e-24) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((i * n) / i)
	tmp = 0
	if n <= -9e-20:
		tmp = t_0
	elif n <= 4e-24:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(i * n) / i))
	tmp = 0.0
	if (n <= -9e-20)
		tmp = t_0;
	elseif (n <= 4e-24)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * ((i * n) / i);
	tmp = 0.0;
	if (n <= -9e-20)
		tmp = t_0;
	elseif (n <= 4e-24)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9e-20], t$95$0, If[LessEqual[n, 4e-24], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.0000000000000003e-20 or 3.99999999999999969e-24 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

   if -9.0000000000000003e-20 < n < 3.99999999999999969e-24

   1. Initial program 28.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 42.4%

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

Alternative 12: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -7 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 28000000000000:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -7e+25)
     t_0
     (if (<= n 28000000000000.0) (* 100.0 (* i (/ n i))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((i * n) / i);
	double tmp;
	if (n <= -7e+25) {
		tmp = t_0;
	} else if (n <= 28000000000000.0) {
		tmp = 100.0 * (i * (n / i));
	} 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 * n) / i)
    if (n <= (-7d+25)) then
        tmp = t_0
    else if (n <= 28000000000000.0d0) then
        tmp = 100.0d0 * (i * (n / i))
    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 * n) / i);
	double tmp;
	if (n <= -7e+25) {
		tmp = t_0;
	} else if (n <= 28000000000000.0) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((i * n) / i)
	tmp = 0
	if n <= -7e+25:
		tmp = t_0
	elif n <= 28000000000000.0:
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(i * n) / i))
	tmp = 0.0
	if (n <= -7e+25)
		tmp = t_0;
	elseif (n <= 28000000000000.0)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * ((i * n) / i);
	tmp = 0.0;
	if (n <= -7e+25)
		tmp = t_0;
	elseif (n <= 28000000000000.0)
		tmp = 100.0 * (i * (n / i));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7e+25], t$95$0, If[LessEqual[n, 28000000000000.0], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.99999999999999999e25 or 2.8e13 < n

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

   if -6.99999999999999999e25 < n < 2.8e13

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 40.8

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

   9. Applied rewrites 40.8%

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

Alternative 13: 51.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.5e-21) (* 100.0 n) (* 100.0 (* i (/ n i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.5e-21) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.5d-21) then
        tmp = 100.0d0 * n
    else
        tmp = 100.0d0 * (i * (n / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.5e-21) {
		tmp = 100.0 * n;
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.5e-21:
		tmp = 100.0 * n
	else:
		tmp = 100.0 * (i * (n / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.5e-21)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.5e-21)
		tmp = 100.0 * n;
	else
		tmp = 100.0 * (i * (n / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.5e-21], N[(100.0 * n), $MachinePrecision], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.49999999999999996e-21

   1. Initial program 28.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 48.6%

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

   if 1.49999999999999996e-21 < i

   1. Initial program 28.4%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 40.8

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

   9. Applied rewrites 40.8%

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

Alternative 14: 48.6% 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 28.4%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 48.6%

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

Developer Target 1: 34.1% 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 2025132 
(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))))