Compound Interest

Percentage Accurate: 29.0% → 82.0%

Time: 7.8s

Alternatives: 13

Speedup: 8.9×

• could not determine a ground truth

  1. i = 1.2464350070628156e-173
  2. n = -2.8028621276838946e+28

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 29.0% 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: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 100, \frac{e^{i} \cdot i}{n} \cdot -50\right) \cdot n\\ \mathbf{elif}\;n \leq 8.4 \cdot 10^{-284}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-58}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -4.7e-70)
     (* (fma t_0 100.0 (* (/ (* (exp i) i) n) -50.0)) n)
     (if (<= n 8.4e-284)
       (/ (* 100.0 (expm1 (* (log (+ (/ i n) 1.0)) n))) (/ i n))
       (if (<= n 6.6e-58)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.05e-30)
           (* (* (expm1 (* (log (/ i n)) n)) n) (/ 100.0 i))
           (* (* t_0 n) 100.0)))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -4.7e-70) {
		tmp = fma(t_0, 100.0, (((exp(i) * i) / n) * -50.0)) * n;
	} else if (n <= 8.4e-284) {
		tmp = (100.0 * expm1((log(((i / n) + 1.0)) * n))) / (i / n);
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = (expm1((log((i / n)) * n)) * n) * (100.0 / i);
	} else {
		tmp = (t_0 * n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -4.7e-70)
		tmp = Float64(fma(t_0, 100.0, Float64(Float64(Float64(exp(i) * i) / n) * -50.0)) * n);
	elseif (n <= 8.4e-284)
		tmp = Float64(Float64(100.0 * expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n))) / Float64(i / n));
	elseif (n <= 6.6e-58)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.05e-30)
		tmp = Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) * n) * Float64(100.0 / i));
	else
		tmp = Float64(Float64(t_0 * n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -4.7e-70], N[(N[(t$95$0 * 100.0 + N[(N[(N[(N[Exp[i], $MachinePrecision] * i), $MachinePrecision] / n), $MachinePrecision] * -50.0), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 8.4e-284], N[(N[(100.0 * N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.6e-58], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-30], N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -4.6999999999999998e-70

   1. Initial program 26.9%

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

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{i} - 1}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{e^{i} - 1}{i}, 100, -50 \cdot \frac{i \cdot e^{i}}{n}\right) \cdot n \]

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 85.8

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

   4. Applied rewrites 85.8%

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

   if -4.6999999999999998e-70 < n < 8.39999999999999965e-284

   1. Initial program 54.2%

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

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

   3. Applied rewrites 70.8%

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

   if 8.39999999999999965e-284 < n < 6.60000000000000052e-58

   1. Initial program 23.5%

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

   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 59.3%

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

   if 6.60000000000000052e-58 < n < 1.0500000000000001e-30

   1. Initial program 10.2%

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

   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 37.4%

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

   6. Applied rewrites 37.3%

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

   if 1.0500000000000001e-30 < n

   1. Initial program 22.5%

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

   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 70.5

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

   4. Applied rewrites 70.5%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 70.5

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

      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 92.9

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

   6. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.4 \cdot 10^{-284}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-58}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -4.7e-70)
     t_0
     (if (<= n 8.4e-284)
       (/ (* 100.0 (expm1 (* (log (+ (/ i n) 1.0)) n))) (/ i n))
       (if (<= n 6.6e-58)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.05e-30)
           (* (* (expm1 (* (log (/ i n)) n)) n) (/ 100.0 i))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -4.7e-70) {
		tmp = t_0;
	} else if (n <= 8.4e-284) {
		tmp = (100.0 * expm1((log(((i / n) + 1.0)) * n))) / (i / n);
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = (expm1((log((i / n)) * n)) * n) * (100.0 / i);
	} 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 <= -4.7e-70) {
		tmp = t_0;
	} else if (n <= 8.4e-284) {
		tmp = (100.0 * Math.expm1((Math.log(((i / n) + 1.0)) * n))) / (i / n);
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = (Math.expm1((Math.log((i / n)) * n)) * n) * (100.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -4.7e-70:
		tmp = t_0
	elif n <= 8.4e-284:
		tmp = (100.0 * math.expm1((math.log(((i / n) + 1.0)) * n))) / (i / n)
	elif n <= 6.6e-58:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.05e-30:
		tmp = (math.expm1((math.log((i / n)) * n)) * n) * (100.0 / i)
	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 <= -4.7e-70)
		tmp = t_0;
	elseif (n <= 8.4e-284)
		tmp = Float64(Float64(100.0 * expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n))) / Float64(i / n));
	elseif (n <= 6.6e-58)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.05e-30)
		tmp = Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) * n) * Float64(100.0 / i));
	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, -4.7e-70], t$95$0, If[LessEqual[n, 8.4e-284], N[(N[(100.0 * N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.6e-58], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-30], N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.6999999999999998e-70 or 1.0500000000000001e-30 < n

   1. Initial program 24.9%

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

   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 68.8

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

   4. Applied rewrites 68.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 68.8

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

      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 89.1

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

   6. Applied rewrites 89.1%

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

   if -4.6999999999999998e-70 < n < 8.39999999999999965e-284

   1. Initial program 54.2%

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

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

   3. Applied rewrites 70.8%

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

   if 8.39999999999999965e-284 < n < 6.60000000000000052e-58

   1. Initial program 23.5%

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

   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 59.3%

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

   if 6.60000000000000052e-58 < n < 1.0500000000000001e-30

   1. Initial program 10.2%

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

   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 37.4%

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

   6. Applied rewrites 37.3%

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

Alternative 3: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 8.4 \cdot 10^{-284}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-58}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -4.7e-70)
     t_0
     (if (<= n 8.4e-284)
       (* (* (/ (expm1 (* (log (+ (/ i n) 1.0)) n)) i) n) 100.0)
       (if (<= n 6.6e-58)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.05e-30)
           (* (* (expm1 (* (log (/ i n)) n)) n) (/ 100.0 i))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -4.7e-70) {
		tmp = t_0;
	} else if (n <= 8.4e-284) {
		tmp = ((expm1((log(((i / n) + 1.0)) * n)) / i) * n) * 100.0;
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = (expm1((log((i / n)) * n)) * n) * (100.0 / i);
	} 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 <= -4.7e-70) {
		tmp = t_0;
	} else if (n <= 8.4e-284) {
		tmp = ((Math.expm1((Math.log(((i / n) + 1.0)) * n)) / i) * n) * 100.0;
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = (Math.expm1((Math.log((i / n)) * n)) * n) * (100.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -4.7e-70:
		tmp = t_0
	elif n <= 8.4e-284:
		tmp = ((math.expm1((math.log(((i / n) + 1.0)) * n)) / i) * n) * 100.0
	elif n <= 6.6e-58:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.05e-30:
		tmp = (math.expm1((math.log((i / n)) * n)) * n) * (100.0 / i)
	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 <= -4.7e-70)
		tmp = t_0;
	elseif (n <= 8.4e-284)
		tmp = Float64(Float64(Float64(expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n)) / i) * n) * 100.0);
	elseif (n <= 6.6e-58)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.05e-30)
		tmp = Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) * n) * Float64(100.0 / i));
	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, -4.7e-70], t$95$0, If[LessEqual[n, 8.4e-284], N[(N[(N[(N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 6.6e-58], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-30], N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.6999999999999998e-70 or 1.0500000000000001e-30 < n

   1. Initial program 24.9%

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

   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 68.8

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

   4. Applied rewrites 68.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 68.8

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

      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 89.1

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

   6. Applied rewrites 89.1%

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

   if -4.6999999999999998e-70 < n < 8.39999999999999965e-284

   1. Initial program 54.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 69.4%

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

   if 8.39999999999999965e-284 < n < 6.60000000000000052e-58

   1. Initial program 23.5%

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

   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 59.3%

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

   if 6.60000000000000052e-58 < n < 1.0500000000000001e-30

   1. Initial program 10.2%

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

   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 37.4%

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

   6. Applied rewrites 37.3%

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

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -1.95 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\left(\left(1 - 1\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-58}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -1.95e-193)
     t_0
     (if (<= n 4.2e-240)
       (/ (* (* (- 1.0 1.0) n) 100.0) i)
       (if (<= n 6.6e-58)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.05e-30)
           (* (* (* (log (/ i n)) n) n) (/ 100.0 i))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -1.95e-193) {
		tmp = t_0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = ((log((i / n)) * n) * n) * (100.0 / i);
	} 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 <= -1.95e-193) {
		tmp = t_0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = ((Math.log((i / n)) * n) * n) * (100.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -1.95e-193:
		tmp = t_0
	elif n <= 4.2e-240:
		tmp = (((1.0 - 1.0) * n) * 100.0) / i
	elif n <= 6.6e-58:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.05e-30:
		tmp = ((math.log((i / n)) * n) * n) * (100.0 / i)
	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 <= -1.95e-193)
		tmp = t_0;
	elseif (n <= 4.2e-240)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) * n) * 100.0) / i);
	elseif (n <= 6.6e-58)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.05e-30)
		tmp = Float64(Float64(Float64(log(Float64(i / n)) * n) * n) * Float64(100.0 / i));
	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, -1.95e-193], t$95$0, If[LessEqual[n, 4.2e-240], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 6.6e-58], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-30], N[(N[(N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.9499999999999999e-193 or 1.0500000000000001e-30 < n

   1. Initial program 25.9%

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

   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 68.2

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

   4. Applied rewrites 68.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 68.2

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

      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 86.2

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

   6. Applied rewrites 86.2%

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

   if -1.9499999999999999e-193 < n < 4.19999999999999987e-240

   1. Initial program 66.3%

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

   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 65.3%

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

   6. Applied rewrites 65.3%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 79.1%

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

   if 4.19999999999999987e-240 < n < 6.60000000000000052e-58

   1. Initial program 20.2%

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

   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 60.4%

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

   if 6.60000000000000052e-58 < n < 1.0500000000000001e-30

   1. Initial program 10.2%

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

   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 37.4%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. log-pow-rev N/A

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

      6. unpow1 N/A

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

      7. log-div N/A

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

      8. lower-log.f64 N/A

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

      9. lift-/.f64 37.3

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

   7. Applied rewrites 37.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 37.3

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

   9. Applied rewrites 37.3%

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

Alternative 5: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -1.95 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\left(\left(1 - 1\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-58}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot n\right) \cdot \frac{100}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -1.95e-193)
     t_0
     (if (<= n 4.2e-240)
       (/ (* (* (- 1.0 1.0) n) 100.0) i)
       (if (<= n 6.6e-58)
         (* 100.0 (/ i (/ i n)))
         (if (<= n 1.05e-30)
           (* (* (expm1 (* (log (/ i n)) n)) n) (/ 100.0 i))
           t_0))))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -1.95e-193) {
		tmp = t_0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = (expm1((log((i / n)) * n)) * n) * (100.0 / i);
	} 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 <= -1.95e-193) {
		tmp = t_0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 6.6e-58) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-30) {
		tmp = (Math.expm1((Math.log((i / n)) * n)) * n) * (100.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -1.95e-193:
		tmp = t_0
	elif n <= 4.2e-240:
		tmp = (((1.0 - 1.0) * n) * 100.0) / i
	elif n <= 6.6e-58:
		tmp = 100.0 * (i / (i / n))
	elif n <= 1.05e-30:
		tmp = (math.expm1((math.log((i / n)) * n)) * n) * (100.0 / i)
	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 <= -1.95e-193)
		tmp = t_0;
	elseif (n <= 4.2e-240)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) * n) * 100.0) / i);
	elseif (n <= 6.6e-58)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.05e-30)
		tmp = Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) * n) * Float64(100.0 / i));
	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, -1.95e-193], t$95$0, If[LessEqual[n, 4.2e-240], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 6.6e-58], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-30], N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * n), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.9499999999999999e-193 or 1.0500000000000001e-30 < n

   1. Initial program 25.9%

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

   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 68.2

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

   4. Applied rewrites 68.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 68.2

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

      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 86.2

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

   6. Applied rewrites 86.2%

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

   if -1.9499999999999999e-193 < n < 4.19999999999999987e-240

   1. Initial program 66.3%

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

   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 65.3%

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

   6. Applied rewrites 65.3%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 79.1%

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

   if 4.19999999999999987e-240 < n < 6.60000000000000052e-58

   1. Initial program 20.2%

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

   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 60.4%

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

   if 6.60000000000000052e-58 < n < 1.0500000000000001e-30

   1. Initial program 10.2%

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

   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 37.4%

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

   6. Applied rewrites 37.3%

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

Alternative 6: 80.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{-193}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\left(\left(1 - 1\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-13}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.95e-193)
   (* (* (/ (expm1 i) i) n) 100.0)
   (if (<= n 4.2e-240)
     (/ (* (* (- 1.0 1.0) n) 100.0) i)
     (if (<= n 5.3e-13)
       (* 100.0 (/ i (/ i n)))
       (* 100.0 (/ (* (expm1 i) n) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.95e-193) {
		tmp = ((expm1(i) / i) * n) * 100.0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 5.3e-13) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((expm1(i) * n) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.95e-193) {
		tmp = ((Math.expm1(i) / i) * n) * 100.0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 5.3e-13) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((Math.expm1(i) * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.95e-193:
		tmp = ((math.expm1(i) / i) * n) * 100.0
	elif n <= 4.2e-240:
		tmp = (((1.0 - 1.0) * n) * 100.0) / i
	elif n <= 5.3e-13:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((math.expm1(i) * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.95e-193)
		tmp = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0);
	elseif (n <= 4.2e-240)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) * n) * 100.0) / i);
	elseif (n <= 5.3e-13)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(expm1(i) * n) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.95e-193], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 4.2e-240], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 5.3e-13], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.9499999999999999e-193

   1. Initial program 28.3%

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

   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 66.6

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

   4. Applied rewrites 66.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 66.6

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

      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 81.5

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

   6. Applied rewrites 81.5%

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

   if -1.9499999999999999e-193 < n < 4.19999999999999987e-240

   1. Initial program 66.3%

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

   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 65.3%

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

   6. Applied rewrites 65.3%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 79.1%

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

   if 4.19999999999999987e-240 < n < 5.2999999999999996e-13

   1. Initial program 18.0%

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

   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 61.6%

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

   if 5.2999999999999996e-13 < n

   1. Initial program 23.1%

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

   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 93.9

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

   4. Applied rewrites 93.9%

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

Alternative 7: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\left(\left(1 - 1\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-13}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))))
   (if (<= n -3.8e-84)
     t_0
     (if (<= n 4.2e-240)
       (/ (* (* (- 1.0 1.0) n) 100.0) i)
       (if (<= n 5.3e-13) (* 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-84) {
		tmp = t_0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 5.3e-13) {
		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-84) {
		tmp = t_0;
	} else if (n <= 4.2e-240) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else if (n <= 5.3e-13) {
		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-84:
		tmp = t_0
	elif n <= 4.2e-240:
		tmp = (((1.0 - 1.0) * n) * 100.0) / i
	elif n <= 5.3e-13:
		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-84)
		tmp = t_0;
	elseif (n <= 4.2e-240)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) * n) * 100.0) / i);
	elseif (n <= 5.3e-13)
		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-84], t$95$0, If[LessEqual[n, 4.2e-240], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 5.3e-13], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.79999999999999986e-84 or 5.2999999999999996e-13 < n

   1. Initial program 25.0%

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

   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 88.4

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

   4. Applied rewrites 88.4%

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

   if -3.79999999999999986e-84 < n < 4.19999999999999987e-240

   1. Initial program 54.8%

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

   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 58.4%

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

   6. Applied rewrites 53.1%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 62.4%

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

   if 4.19999999999999987e-240 < n < 5.2999999999999996e-13

   1. Initial program 18.0%

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

   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 61.6%

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

Alternative 8: 62.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.12 \cdot 10^{-67}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{-212}:\\ \;\;\;\;\frac{\left(\left(1 - 1\right) \cdot n\right) \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.12e-67)
   (* 100.0 (/ (* i n) i))
   (if (<= n 2.6e-212)
     (/ (* (* (- 1.0 1.0) n) 100.0) i)
     (* 100.0 (fma (* n i) 0.5 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.12e-67) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 2.6e-212) {
		tmp = (((1.0 - 1.0) * n) * 100.0) / i;
	} else {
		tmp = 100.0 * fma((n * i), 0.5, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.12e-67)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 2.6e-212)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) * n) * 100.0) / i);
	else
		tmp = Float64(100.0 * fma(Float64(n * i), 0.5, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.12e-67], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.6e-212], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision], N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5 + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.12e-67

   1. Initial program 26.9%

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

   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 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 57.2%

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

   if -1.12e-67 < n < 2.6e-212

   1. Initial program 50.5%

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

   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 54.7%

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

   6. Applied rewrites 49.0%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 59.5%

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

   if 2.6e-212 < n

   1. Initial program 21.0%

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

   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 73.2

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

   4. Applied rewrites 73.2%

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

   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 63.8

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

   7. Applied rewrites 63.8%

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

Alternative 9: 61.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.55 \cdot 10^{+56}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.55e+56)
   (* 100.0 (/ (* i n) i))
   (if (<= n 3.5e-29) (* 100.0 (/ i (/ i n))) (* 100.0 (fma (* n i) 0.5 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.55e+56) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 3.5e-29) {
		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 <= -3.55e+56)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 3.5e-29)
		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, -3.55e+56], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-29], 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 < -3.55e56

   1. Initial program 24.0%

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

   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 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 58.5%

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

   if -3.55e56 < n < 3.4999999999999997e-29

   1. Initial program 36.7%

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

   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 58.4%

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

   if 3.4999999999999997e-29 < n

   1. Initial program 22.5%

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

   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 92.9

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

   4. Applied rewrites 92.9%

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

   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 69.8

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

   7. Applied rewrites 69.8%

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

Alternative 10: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -3.55 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{+28}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -3.55e+56) t_0 (if (<= n 2.9e+28) (* 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 <= -3.55e+56) {
		tmp = t_0;
	} else if (n <= 2.9e+28) {
		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 <= (-3.55d+56)) then
        tmp = t_0
    else if (n <= 2.9d+28) 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 <= -3.55e+56) {
		tmp = t_0;
	} else if (n <= 2.9e+28) {
		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 <= -3.55e+56:
		tmp = t_0
	elif n <= 2.9e+28:
		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 <= -3.55e+56)
		tmp = t_0;
	elseif (n <= 2.9e+28)
		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 <= -3.55e+56)
		tmp = t_0;
	elseif (n <= 2.9e+28)
		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, -3.55e+56], t$95$0, If[LessEqual[n, 2.9e+28], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.55e56 or 2.9000000000000001e28 < n

   1. Initial program 22.5%

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

   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 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 63.5%

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

   if -3.55e56 < n < 2.9000000000000001e28

   1. Initial program 36.3%

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

   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 58.8%

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

Alternative 11: 60.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 760000:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -1e+59) t_0 (if (<= n 760000.0) (* 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 <= -1e+59) {
		tmp = t_0;
	} else if (n <= 760000.0) {
		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 <= (-1d+59)) then
        tmp = t_0
    else if (n <= 760000.0d0) 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 <= -1e+59) {
		tmp = t_0;
	} else if (n <= 760000.0) {
		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 <= -1e+59:
		tmp = t_0
	elif n <= 760000.0:
		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 <= -1e+59)
		tmp = t_0;
	elseif (n <= 760000.0)
		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 <= -1e+59)
		tmp = t_0;
	elseif (n <= 760000.0)
		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, -1e+59], t$95$0, If[LessEqual[n, 760000.0], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.99999999999999972e58 or 7.6e5 < n

   1. Initial program 23.2%

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

   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 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 63.4%

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

   if -9.99999999999999972e58 < n < 7.6e5

   1. Initial program 36.0%

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

   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 44.9

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

   4. Applied rewrites 44.9%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 33.6%

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]
   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 57.1

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

   9. Applied rewrites 57.1%

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

Alternative 12: 55.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{if}\;i \leq -5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+44}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* i (/ n i)))))
   (if (<= i -5e+17) t_0 (if (<= i 4e+44) (* 100.0 n) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i * (n / i));
	double tmp;
	if (i <= -5e+17) {
		tmp = t_0;
	} else if (i <= 4e+44) {
		tmp = 100.0 * 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 (i <= (-5d+17)) then
        tmp = t_0
    else if (i <= 4d+44) then
        tmp = 100.0d0 * 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 (i <= -5e+17) {
		tmp = t_0;
	} else if (i <= 4e+44) {
		tmp = 100.0 * n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i * (n / i))
	tmp = 0
	if i <= -5e+17:
		tmp = t_0
	elif i <= 4e+44:
		tmp = 100.0 * n
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i * Float64(n / i)))
	tmp = 0.0
	if (i <= -5e+17)
		tmp = t_0;
	elseif (i <= 4e+44)
		tmp = Float64(100.0 * 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 (i <= -5e+17)
		tmp = t_0;
	elseif (i <= 4e+44)
		tmp = 100.0 * n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5e+17], t$95$0, If[LessEqual[i, 4e+44], N[(100.0 * n), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -5e17 or 4.0000000000000004e44 < i

   1. Initial program 57.5%

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

   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 63.5

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

   4. Applied rewrites 63.5%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 18.1%

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]
   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 20.6

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

   9. Applied rewrites 20.6%

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

   if -5e17 < i < 4.0000000000000004e44

   1. Initial program 10.5%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 78.3%

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

Alternative 13: 49.6% 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 29.0%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 49.6%

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

Developer Target 1: 34.5% 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 2025112 
(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))))