Compound Interest

Percentage Accurate: 28.7% → 95.1%

Time: 10.4s

Alternatives: 14

Speedup: 8.9×

• could not determine a ground truth

  1. i = 3.0026896939105474e-104
  2. n = 4.9210101004159837e+30

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (/
        (*
         (fma
          (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
          i
          1.0)
         i)
        i)
       n)
      100.0)
     (if (<= t_0 0.0)
       (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
       (if (<= t_0 INFINITY)
         (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
         (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * n) * 100.0;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * n) * 100.0);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64((Float64(i / n) ^ n) - 1.0) / Float64(i / n)));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   2. 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 21.4

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

   4. Applied rewrites 21.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 21.4

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

      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 21.4

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

      9. +-commutative 21.4

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

      10. exp-to-pow 21.4

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

   6. Applied rewrites 21.4%

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

   7. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 83.5

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

   9. Applied rewrites 83.5%

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

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

   1. Initial program 25.7%

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

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{\frac{i}{n}}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-log1p.f64 N/A

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

      6. lift-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 98.0%

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

   2. Taylor expanded in i around inf

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

      1. lift-/.f64 92.7

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

   4. Applied rewrites 92.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 79.8%

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

Alternative 2: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log i, n, \left(-n\right) \cdot \log n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -2.6e-8)
     t_0
     (if (<= n 1.1e-276)
       (/ (* 100.0 (expm1 (* (log (+ (/ i n) 1.0)) n))) (/ i n))
       (if (<= n 3.9e-159)
         (* (* (/ (fma (log i) n (* (- n) (log n))) i) n) 100.0)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -2.6e-8) {
		tmp = t_0;
	} else if (n <= 1.1e-276) {
		tmp = (100.0 * expm1((log(((i / n) + 1.0)) * n))) / (i / n);
	} else if (n <= 3.9e-159) {
		tmp = ((fma(log(i), n, (-n * log(n))) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -2.6e-8)
		tmp = t_0;
	elseif (n <= 1.1e-276)
		tmp = Float64(Float64(100.0 * expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n))) / Float64(i / n));
	elseif (n <= 3.9e-159)
		tmp = Float64(Float64(Float64(fma(log(i), n, Float64(Float64(-n) * log(n))) / i) * n) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -2.6e-8], t$95$0, If[LessEqual[n, 1.1e-276], 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, 3.9e-159], N[(N[(N[(N[(N[Log[i], $MachinePrecision] * n + N[((-n) * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.6000000000000001e-8 or 3.89999999999999977e-159 < n

   1. Initial program 23.8%

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

   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 65.7

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

   4. Applied rewrites 65.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 65.7

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

      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 85.0

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

      9. +-commutative 85.0

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

      10. exp-to-pow 85.0

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

   6. Applied rewrites 85.0%

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

   if -2.6000000000000001e-8 < n < 1.0999999999999999e-276

   1. Initial program 44.5%

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

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 65.7%

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

   if 1.0999999999999999e-276 < n < 3.89999999999999977e-159

   1. Initial program 31.5%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 42.4

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.4

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 42.9

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

      9. +-commutative 42.9

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

      10. exp-to-pow 42.9

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

   6. Applied rewrites 42.9%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sum-log N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. neg-log N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lift-log.f64 71.8

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

   8. Applied rewrites 71.8%

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

Alternative 3: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-276}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log i, n, \left(-n\right) \cdot \log n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -2.6e-8)
     t_0
     (if (<= n 1.1e-276)
       (* 100.0 (/ (expm1 (* (log (+ (/ i n) 1.0)) n)) (/ i n)))
       (if (<= n 3.9e-159)
         (* (* (/ (fma (log i) n (* (- n) (log n))) i) n) 100.0)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -2.6e-8) {
		tmp = t_0;
	} else if (n <= 1.1e-276) {
		tmp = 100.0 * (expm1((log(((i / n) + 1.0)) * n)) / (i / n));
	} else if (n <= 3.9e-159) {
		tmp = ((fma(log(i), n, (-n * log(n))) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -2.6e-8)
		tmp = t_0;
	elseif (n <= 1.1e-276)
		tmp = Float64(100.0 * Float64(expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n)) / Float64(i / n)));
	elseif (n <= 3.9e-159)
		tmp = Float64(Float64(Float64(fma(log(i), n, Float64(Float64(-n) * log(n))) / i) * n) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -2.6e-8], t$95$0, If[LessEqual[n, 1.1e-276], N[(100.0 * N[(N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.9e-159], N[(N[(N[(N[(N[Log[i], $MachinePrecision] * n + N[((-n) * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.6000000000000001e-8 or 3.89999999999999977e-159 < n

   1. Initial program 23.8%

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

   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 65.7

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

   4. Applied rewrites 65.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 65.7

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

      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 85.0

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

      9. +-commutative 85.0

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

      10. exp-to-pow 85.0

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

   6. Applied rewrites 85.0%

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

   if -2.6000000000000001e-8 < n < 1.0999999999999999e-276

   1. Initial program 44.5%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-/.f64 65.5

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

   3. Applied rewrites 65.5%

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

   if 1.0999999999999999e-276 < n < 3.89999999999999977e-159

   1. Initial program 31.5%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 42.4

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.4

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 42.9

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

      9. +-commutative 42.9

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

      10. exp-to-pow 42.9

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

   6. Applied rewrites 42.9%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sum-log N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. neg-log N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lift-log.f64 71.8

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

   8. Applied rewrites 71.8%

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

Alternative 4: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-276}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log i, n, \left(-n\right) \cdot \log n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -2.5e-68)
     t_0
     (if (<= n 1.1e-276)
       (* (* (/ (expm1 (* (log (+ (/ i n) 1.0)) n)) i) n) 100.0)
       (if (<= n 3.9e-159)
         (* (* (/ (fma (log i) n (* (- n) (log n))) i) n) 100.0)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -2.5e-68) {
		tmp = t_0;
	} else if (n <= 1.1e-276) {
		tmp = ((expm1((log(((i / n) + 1.0)) * n)) / i) * n) * 100.0;
	} else if (n <= 3.9e-159) {
		tmp = ((fma(log(i), n, (-n * log(n))) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -2.5e-68)
		tmp = t_0;
	elseif (n <= 1.1e-276)
		tmp = Float64(Float64(Float64(expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n)) / i) * n) * 100.0);
	elseif (n <= 3.9e-159)
		tmp = Float64(Float64(Float64(fma(log(i), n, Float64(Float64(-n) * log(n))) / i) * n) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -2.5e-68], t$95$0, If[LessEqual[n, 1.1e-276], 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, 3.9e-159], N[(N[(N[(N[(N[Log[i], $MachinePrecision] * n + N[((-n) * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.49999999999999986e-68 or 3.89999999999999977e-159 < 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

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

      1. lower-expm1.f64 65.6

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

   4. Applied rewrites 65.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 65.6

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

      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 83.8

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

      9. +-commutative 83.8

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

      10. exp-to-pow 83.8

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

   6. Applied rewrites 83.8%

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

   if -2.49999999999999986e-68 < n < 1.0999999999999999e-276

   1. Initial program 52.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 69.8%

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

   if 1.0999999999999999e-276 < n < 3.89999999999999977e-159

   1. Initial program 31.5%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 42.4

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.4

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 42.9

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

      9. +-commutative 42.9

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

      10. exp-to-pow 42.9

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

   6. Applied rewrites 42.9%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sum-log N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. neg-log N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lift-log.f64 71.8

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

   8. Applied rewrites 71.8%

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

Alternative 5: 79.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -1 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-277}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log i, n, \left(-n\right) \cdot \log n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -1e-68)
     t_0
     (if (<= n 4.1e-277)
       (/ (* 100.0 (expm1 (* (log (/ i n)) n))) (/ i n))
       (if (<= n 3.9e-159)
         (* (* (/ (fma (log i) n (* (- n) (log n))) i) n) 100.0)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -1e-68) {
		tmp = t_0;
	} else if (n <= 4.1e-277) {
		tmp = (100.0 * expm1((log((i / n)) * n))) / (i / n);
	} else if (n <= 3.9e-159) {
		tmp = ((fma(log(i), n, (-n * log(n))) / i) * n) * 100.0;
	} 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 <= -1e-68)
		tmp = t_0;
	elseif (n <= 4.1e-277)
		tmp = Float64(Float64(100.0 * expm1(Float64(log(Float64(i / n)) * n))) / Float64(i / n));
	elseif (n <= 3.9e-159)
		tmp = Float64(Float64(Float64(fma(log(i), n, Float64(Float64(-n) * log(n))) / i) * n) * 100.0);
	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, -1e-68], t$95$0, If[LessEqual[n, 4.1e-277], N[(N[(100.0 * N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.9e-159], N[(N[(N[(N[(N[Log[i], $MachinePrecision] * n + N[((-n) * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.00000000000000007e-68 or 3.89999999999999977e-159 < 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

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

      1. lower-expm1.f64 65.6

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

   4. Applied rewrites 65.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 65.6

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

      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 83.8

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

      9. +-commutative 83.8

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

      10. exp-to-pow 83.8

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

   6. Applied rewrites 83.8%

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

   if -1.00000000000000007e-68 < n < 4.09999999999999989e-277

   1. Initial program 52.6%

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

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 71.5%

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

   4. Taylor expanded in i around inf

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

      1. lift-/.f64 67.5

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

   6. Applied rewrites 67.5%

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

   if 4.09999999999999989e-277 < n < 3.89999999999999977e-159

   1. Initial program 31.5%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 42.4

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.4

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 42.9

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

      9. +-commutative 42.9

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

      10. exp-to-pow 42.9

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

   6. Applied rewrites 42.9%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sum-log N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. neg-log N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lift-log.f64 71.9

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

   8. Applied rewrites 71.9%

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

Alternative 6: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -1 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{-277}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\left(\left(-\log n\right) + \log i\right) \cdot n}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -1e-68)
     t_0
     (if (<= n 4.1e-277)
       (/ (* 100.0 (expm1 (* (log (/ i n)) n))) (/ i n))
       (if (<= n 3.9e-159)
         (* (* (/ (* (+ (- (log n)) (log i)) n) i) n) 100.0)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -1e-68) {
		tmp = t_0;
	} else if (n <= 4.1e-277) {
		tmp = (100.0 * expm1((log((i / n)) * n))) / (i / n);
	} else if (n <= 3.9e-159) {
		tmp = ((((-log(n) + log(i)) * n) / i) * n) * 100.0;
	} 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 <= -1e-68) {
		tmp = t_0;
	} else if (n <= 4.1e-277) {
		tmp = (100.0 * Math.expm1((Math.log((i / n)) * n))) / (i / n);
	} else if (n <= 3.9e-159) {
		tmp = ((((-Math.log(n) + Math.log(i)) * n) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -1e-68:
		tmp = t_0
	elif n <= 4.1e-277:
		tmp = (100.0 * math.expm1((math.log((i / n)) * n))) / (i / n)
	elif n <= 3.9e-159:
		tmp = ((((-math.log(n) + math.log(i)) * n) / i) * n) * 100.0
	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 <= -1e-68)
		tmp = t_0;
	elseif (n <= 4.1e-277)
		tmp = Float64(Float64(100.0 * expm1(Float64(log(Float64(i / n)) * n))) / Float64(i / n));
	elseif (n <= 3.9e-159)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-log(n)) + log(i)) * n) / i) * n) * 100.0);
	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, -1e-68], t$95$0, If[LessEqual[n, 4.1e-277], N[(N[(100.0 * N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.9e-159], N[(N[(N[(N[(N[((-N[Log[n], $MachinePrecision]) + N[Log[i], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.00000000000000007e-68 or 3.89999999999999977e-159 < 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

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

      1. lower-expm1.f64 65.6

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

   4. Applied rewrites 65.6%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 65.6

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

      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 83.8

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

      9. +-commutative 83.8

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

      10. exp-to-pow 83.8

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

   6. Applied rewrites 83.8%

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

   if -1.00000000000000007e-68 < n < 4.09999999999999989e-277

   1. Initial program 52.6%

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

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 71.5%

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

   4. Taylor expanded in i around inf

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

      1. lift-/.f64 67.5

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

   6. Applied rewrites 67.5%

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

   if 4.09999999999999989e-277 < n < 3.89999999999999977e-159

   1. Initial program 31.5%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 42.4

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

   4. Applied rewrites 42.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.4

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 42.9

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

      9. +-commutative 42.9

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

      10. exp-to-pow 42.9

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

   6. Applied rewrites 42.9%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. log-prod N/A

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

      5. neg-log N/A

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

      6. mul-1-neg N/A

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

      7. lower-+.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 72.0

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

   8. Applied rewrites 72.0%

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

Alternative 7: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ \mathbf{if}\;n \leq -2.2 \cdot 10^{-110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-242}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))))
   (if (<= n -2.2e-110)
     t_0
     (if (<= n 1.06e-242)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (if (<= n 4.4e-21) (* 100.0 (* i (/ n i))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double tmp;
	if (n <= -2.2e-110) {
		tmp = t_0;
	} else if (n <= 1.06e-242) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 4.4e-21) {
		tmp = 100.0 * (i * (n / i));
	} 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 <= -2.2e-110) {
		tmp = t_0;
	} else if (n <= 1.06e-242) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 4.4e-21) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) * n) / i)
	tmp = 0
	if n <= -2.2e-110:
		tmp = t_0
	elif n <= 1.06e-242:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	elif n <= 4.4e-21:
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	tmp = 0.0
	if (n <= -2.2e-110)
		tmp = t_0;
	elseif (n <= 1.06e-242)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 4.4e-21)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.2e-110], t$95$0, If[LessEqual[n, 1.06e-242], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.4e-21], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.1999999999999999e-110 or 4.4000000000000001e-21 < n

   1. Initial program 24.7%

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

   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 86.5

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

   4. Applied rewrites 86.5%

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

   if -2.1999999999999999e-110 < n < 1.06000000000000008e-242

   1. Initial program 55.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 64.6%

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

   if 1.06000000000000008e-242 < n < 4.4000000000000001e-21

   1. Initial program 19.1%

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

   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 25.1

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

   4. Applied rewrites 25.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 26.4%

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]
   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 58.7

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

   9. Applied rewrites 58.7%

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

Alternative 8: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -9.2 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-161}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) n) 100.0)))
   (if (<= n -9.2e-176)
     t_0
     (if (<= n 2.7e-161) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -9.2e-176) {
		tmp = t_0;
	} else if (n <= 2.7e-161) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * n) * 100.0;
	double tmp;
	if (n <= -9.2e-176) {
		tmp = t_0;
	} else if (n <= 2.7e-161) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * n) * 100.0
	tmp = 0
	if n <= -9.2e-176:
		tmp = t_0
	elif n <= 2.7e-161:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * n) * 100.0)
	tmp = 0.0
	if (n <= -9.2e-176)
		tmp = t_0;
	elseif (n <= 2.7e-161)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -9.2e-176], t$95$0, If[LessEqual[n, 2.7e-161], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.2000000000000005e-176 or 2.6999999999999999e-161 < n

   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

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

      1. lower-expm1.f64 65.4

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

   4. Applied rewrites 65.4%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
   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 65.4

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

      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 81.8

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

      9. +-commutative 81.8

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

      10. exp-to-pow 81.8

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

   6. Applied rewrites 81.8%

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

   if -9.2000000000000005e-176 < n < 2.6999999999999999e-161

   1. Initial program 52.1%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 69.5%

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

Alternative 9: 61.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right)\\ \mathbf{if}\;n \leq -2.3 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-161}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (fma (* n i) 0.5 n))))
   (if (<= n -2.3e-70)
     t_0
     (if (<= n 2.7e-161) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * fma((n * i), 0.5, n);
	double tmp;
	if (n <= -2.3e-70) {
		tmp = t_0;
	} else if (n <= 2.7e-161) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * fma(Float64(n * i), 0.5, n))
	tmp = 0.0
	if (n <= -2.3e-70)
		tmp = t_0;
	elseif (n <= 2.7e-161)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * i), $MachinePrecision] * 0.5 + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.3e-70], t$95$0, If[LessEqual[n, 2.7e-161], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.30000000000000001e-70 or 2.6999999999999999e-161 < 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 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

   if -2.30000000000000001e-70 < n < 2.6999999999999999e-161

   1. Initial program 46.6%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 58.8%

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

Alternative 10: 60.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \mathsf{fma}\left(n \cdot i, 0.5, n\right)\\ \mathbf{if}\;n \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (fma (* n i) 0.5 n))))
   (if (<= n -4.8e+76) t_0 (if (<= n 1.45e-21) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * fma((n * i), 0.5, n);
	double tmp;
	if (n <= -4.8e+76) {
		tmp = t_0;
	} else if (n <= 1.45e-21) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * fma(Float64(n * i), 0.5, n))
	tmp = 0.0
	if (n <= -4.8e+76)
		tmp = t_0;
	elseif (n <= 1.45e-21)
		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 * i), $MachinePrecision] * 0.5 + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.8e+76], t$95$0, If[LessEqual[n, 1.45e-21], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.8e76 or 1.45e-21 < 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 91.8

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

   4. Applied rewrites 91.8%

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

   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 64.2

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

   7. Applied rewrites 64.2%

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

   if -4.8e76 < n < 1.45e-21

   1. Initial program 36.1%

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

   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 57.4%

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

Alternative 11: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+34}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -2e-8) t_0 (if (<= n 3.8e+34) (* 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 <= -2e-8) {
		tmp = t_0;
	} else if (n <= 3.8e+34) {
		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 <= (-2d-8)) then
        tmp = t_0
    else if (n <= 3.8d+34) 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 <= -2e-8) {
		tmp = t_0;
	} else if (n <= 3.8e+34) {
		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 <= -2e-8:
		tmp = t_0
	elif n <= 3.8e+34:
		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 <= -2e-8)
		tmp = t_0;
	elseif (n <= 3.8e+34)
		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 <= -2e-8)
		tmp = t_0;
	elseif (n <= 3.8e+34)
		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, -2e-8], t$95$0, If[LessEqual[n, 3.8e+34], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2e-8 or 3.8000000000000001e34 < n

   1. Initial program 24.2%

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

   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 91.5

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 61.8%

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

   if -2e-8 < n < 3.8000000000000001e34

   1. Initial program 34.7%

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

   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 59.4%

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

Alternative 12: 59.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i \cdot n}{i}\\ \mathbf{if}\;n \leq -2 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{+34}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* i n) i))))
   (if (<= n -2e+77) t_0 (if (<= n 3.8e+34) (* 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 <= -2e+77) {
		tmp = t_0;
	} else if (n <= 3.8e+34) {
		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 <= (-2d+77)) then
        tmp = t_0
    else if (n <= 3.8d+34) 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 <= -2e+77) {
		tmp = t_0;
	} else if (n <= 3.8e+34) {
		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 <= -2e+77:
		tmp = t_0
	elif n <= 3.8e+34:
		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 <= -2e+77)
		tmp = t_0;
	elseif (n <= 3.8e+34)
		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 <= -2e+77)
		tmp = t_0;
	elseif (n <= 3.8e+34)
		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, -2e+77], t$95$0, If[LessEqual[n, 3.8e+34], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.99999999999999997e77 or 3.8000000000000001e34 < n

   1. Initial program 20.8%

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

   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 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 63.4%

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

   if -1.99999999999999997e77 < n < 3.8000000000000001e34

   1. Initial program 36.5%

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

   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 47.1

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

   4. Applied rewrites 47.1%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.5

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

   9. Applied rewrites 55.5%

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

Alternative 13: 54.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{if}\;i \leq -3 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 5 \cdot 10^{-9}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* i (/ n i)))))
   (if (<= i -3e-28) t_0 (if (<= i 5e-9) (* 100.0 n) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i * (n / i));
	double tmp;
	if (i <= -3e-28) {
		tmp = t_0;
	} else if (i <= 5e-9) {
		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 <= (-3d-28)) then
        tmp = t_0
    else if (i <= 5d-9) 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 <= -3e-28) {
		tmp = t_0;
	} else if (i <= 5e-9) {
		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 <= -3e-28:
		tmp = t_0
	elif i <= 5e-9:
		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 <= -3e-28)
		tmp = t_0;
	elseif (i <= 5e-9)
		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 <= -3e-28)
		tmp = t_0;
	elseif (i <= 5e-9)
		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, -3e-28], t$95$0, If[LessEqual[i, 5e-9], N[(100.0 * n), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -3.00000000000000003e-28 or 5.0000000000000001e-9 < i

   1. Initial program 49.9%

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

   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 62.6

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 18.7%

      \[\leadsto 100 \cdot \frac{i \cdot n}{i} \]
   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 22.2

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

   9. Applied rewrites 22.2%

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

   if -3.00000000000000003e-28 < i < 5.0000000000000001e-9

   1. Initial program 9.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 85.4%

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

Alternative 14: 48.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 48.0%

   \[\leadsto 100 \cdot \color{blue}{n} \]
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 2025120 
(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))))