Compound Interest

Percentage Accurate: 27.7% → 94.7%

Time: 7.6s

Alternatives: 12

Speedup: 8.9×

• could not determine a ground truth

  1. i = 1.780455611258085e-103
  2. n = 1.0787582823467024e+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: 27.7% 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: 94.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 5e-219)
     (* (/ (expm1 (* (log1p (/ i n)) n)) (/ i n)) 100.0)
     (if (<= t_0 INFINITY)
       (/ (* (- (pow (+ (/ i n) 1.0) n) 1.0) 100.0) (/ i n))
       (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 5e-219) {
		tmp = (expm1((log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((pow(((i / n) + 1.0), n) - 1.0) * 100.0) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 5e-219) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 5e-219:
		tmp = (math.expm1((math.log1p((i / n)) * n)) / (i / n)) * 100.0
	elif t_0 <= math.inf:
		tmp = ((math.pow(((i / n) + 1.0), n) - 1.0) * 100.0) / (i / n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 5e-219)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / Float64(i / n)) * 100.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) * 100.0) / 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, 5e-219], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 5.0000000000000002e-219

   1. Initial program 26.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-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 36.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-log1p.f64 N/A

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

      6. lift-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 5.0000000000000002e-219 < (*.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 98.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 60.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. pow-to-exp N/A

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

      9. +-commutative N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 98.6

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

   7. Applied rewrites 98.6%

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

   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 80.5%

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

Alternative 2: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 5e-219)
     (* (/ (expm1 (* (log1p (/ i n)) n)) (/ i n)) 100.0)
     (if (<= t_0 5e-17)
       (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
       (* (* (/ (expm1 i) i) 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 <= 5e-219) {
		tmp = (expm1((log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= 5e-17) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else {
		tmp = ((expm1(i) / i) * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 5e-219) {
		tmp = (Math.expm1((Math.log1p((i / n)) * n)) / (i / n)) * 100.0;
	} else if (t_0 <= 5e-17) {
		tmp = 100.0 * ((Math.pow((i / n), n) - 1.0) / (i / n));
	} else {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 5e-219:
		tmp = (math.expm1((math.log1p((i / n)) * n)) / (i / n)) * 100.0
	elif t_0 <= 5e-17:
		tmp = 100.0 * ((math.pow((i / n), n) - 1.0) / (i / n))
	else:
		tmp = ((math.expm1(i) / i) * 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 <= 5e-219)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / Float64(i / n)) * 100.0);
	elseif (t_0 <= 5e-17)
		tmp = Float64(100.0 * Float64(Float64((Float64(i / n) ^ n) - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(Float64(expm1(i) / i) * 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, 5e-219], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, 5e-17], N[(100.0 * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 5.0000000000000002e-219

   1. Initial program 26.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-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 36.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-log1p.f64 N/A

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

      6. lift-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 5.0000000000000002e-219 < (*.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))) < 4.9999999999999999e-17

   1. Initial program 96.5%

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

   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 96.5

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

   4. Applied rewrites 96.5%

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

   if 4.9999999999999999e-17 < (*.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 21.9%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 63.3

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 82.0

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

   7. Applied rewrites 82.0%

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

Alternative 3: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right) \cdot \frac{100}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(\left(\log i - \log n\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -9.5e-186)
     t_0
     (if (<= n -2e-311)
       (* (expm1 (* (log (+ (/ i n) 1.0)) n)) (/ 100.0 (/ i n)))
       (if (<= n 2.5e-134)
         (/ (* (* (- (log i) (log n)) n) 100.0) (/ i n))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -9.5e-186) {
		tmp = t_0;
	} else if (n <= -2e-311) {
		tmp = expm1((log(((i / n) + 1.0)) * n)) * (100.0 / (i / n));
	} else if (n <= 2.5e-134) {
		tmp = (((log(i) - log(n)) * n) * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -9.5e-186) {
		tmp = t_0;
	} else if (n <= -2e-311) {
		tmp = Math.expm1((Math.log(((i / n) + 1.0)) * n)) * (100.0 / (i / n));
	} else if (n <= 2.5e-134) {
		tmp = (((Math.log(i) - Math.log(n)) * n) * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -9.5e-186:
		tmp = t_0
	elif n <= -2e-311:
		tmp = math.expm1((math.log(((i / n) + 1.0)) * n)) * (100.0 / (i / n))
	elif n <= 2.5e-134:
		tmp = (((math.log(i) - math.log(n)) * n) * 100.0) / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -9.5e-186)
		tmp = t_0;
	elseif (n <= -2e-311)
		tmp = Float64(expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n)) * Float64(100.0 / Float64(i / n)));
	elseif (n <= 2.5e-134)
		tmp = Float64(Float64(Float64(Float64(log(i) - log(n)) * n) * 100.0) / 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] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -9.5e-186], t$95$0, If[LessEqual[n, -2e-311], N[(N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-134], N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.4999999999999998e-186 or 2.5000000000000002e-134 < n

   1. Initial program 23.6%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -9.4999999999999998e-186 < n < -1.9999999999999e-311

   1. Initial program 74.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-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 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 88.7%

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

   if -1.9999999999999e-311 < n < 2.5000000000000002e-134

   1. Initial program 29.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. negate-sub N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 70.8

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

   8. Applied rewrites 70.8%

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

Alternative 4: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(\left(\log i - \log n\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -9.5e-186)
     t_0
     (if (<= n -2e-311)
       (* (/ (expm1 (* (log (+ (/ i n) 1.0)) n)) (/ i n)) 100.0)
       (if (<= n 2.5e-134)
         (/ (* (* (- (log i) (log n)) n) 100.0) (/ i n))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -9.5e-186) {
		tmp = t_0;
	} else if (n <= -2e-311) {
		tmp = (expm1((log(((i / n) + 1.0)) * n)) / (i / n)) * 100.0;
	} else if (n <= 2.5e-134) {
		tmp = (((log(i) - log(n)) * n) * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -9.5e-186) {
		tmp = t_0;
	} else if (n <= -2e-311) {
		tmp = (Math.expm1((Math.log(((i / n) + 1.0)) * n)) / (i / n)) * 100.0;
	} else if (n <= 2.5e-134) {
		tmp = (((Math.log(i) - Math.log(n)) * n) * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -9.5e-186:
		tmp = t_0
	elif n <= -2e-311:
		tmp = (math.expm1((math.log(((i / n) + 1.0)) * n)) / (i / n)) * 100.0
	elif n <= 2.5e-134:
		tmp = (((math.log(i) - math.log(n)) * n) * 100.0) / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -9.5e-186)
		tmp = t_0;
	elseif (n <= -2e-311)
		tmp = Float64(Float64(expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n)) / Float64(i / n)) * 100.0);
	elseif (n <= 2.5e-134)
		tmp = Float64(Float64(Float64(Float64(log(i) - log(n)) * n) * 100.0) / 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] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -9.5e-186], t$95$0, If[LessEqual[n, -2e-311], N[(N[(N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 2.5e-134], N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.4999999999999998e-186 or 2.5000000000000002e-134 < n

   1. Initial program 23.6%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -9.4999999999999998e-186 < n < -1.9999999999999e-311

   1. Initial program 74.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. *-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 88.6%

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

   if -1.9999999999999e-311 < n < 2.5000000000000002e-134

   1. Initial program 29.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. negate-sub N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 70.8

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

   8. Applied rewrites 70.8%

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

Alternative 5: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.65 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-307}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(\left(\log i - \log n\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -2.65e-189)
     t_0
     (if (<= n 5e-307)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (if (<= n 2.5e-134)
         (/ (* (* (- (log i) (log n)) n) 100.0) (/ i n))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -2.65e-189) {
		tmp = t_0;
	} else if (n <= 5e-307) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 2.5e-134) {
		tmp = (((log(i) - log(n)) * n) * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -2.65e-189) {
		tmp = t_0;
	} else if (n <= 5e-307) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 2.5e-134) {
		tmp = (((Math.log(i) - Math.log(n)) * n) * 100.0) / (i / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -2.65e-189:
		tmp = t_0
	elif n <= 5e-307:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	elif n <= 2.5e-134:
		tmp = (((math.log(i) - math.log(n)) * n) * 100.0) / (i / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -2.65e-189)
		tmp = t_0;
	elseif (n <= 5e-307)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 2.5e-134)
		tmp = Float64(Float64(Float64(Float64(log(i) - log(n)) * n) * 100.0) / 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] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.65e-189], t$95$0, If[LessEqual[n, 5e-307], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-134], N[(N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.6499999999999999e-189 or 2.5000000000000002e-134 < n

   1. Initial program 23.7%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -2.6499999999999999e-189 < n < 5.00000000000000014e-307

   1. Initial program 74.9%

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

   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 76.0%

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

   if 5.00000000000000014e-307 < n < 2.5000000000000002e-134

   1. Initial program 29.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. negate-sub N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 70.5

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

   8. Applied rewrites 70.5%

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

Alternative 6: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.65 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-307}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-134}:\\ \;\;\;\;n \cdot \left(n \cdot \left(\frac{\log i - \log n}{i} \cdot 100\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -2.65e-189)
     t_0
     (if (<= n 5e-307)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (if (<= n 2.5e-134)
         (* n (* n (* (/ (- (log i) (log n)) i) 100.0)))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -2.65e-189) {
		tmp = t_0;
	} else if (n <= 5e-307) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 2.5e-134) {
		tmp = n * (n * (((log(i) - log(n)) / i) * 100.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -2.65e-189) {
		tmp = t_0;
	} else if (n <= 5e-307) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 2.5e-134) {
		tmp = n * (n * (((Math.log(i) - Math.log(n)) / i) * 100.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -2.65e-189:
		tmp = t_0
	elif n <= 5e-307:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	elif n <= 2.5e-134:
		tmp = n * (n * (((math.log(i) - math.log(n)) / i) * 100.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -2.65e-189)
		tmp = t_0;
	elseif (n <= 5e-307)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 2.5e-134)
		tmp = Float64(n * Float64(n * Float64(Float64(Float64(log(i) - log(n)) / i) * 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] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.65e-189], t$95$0, If[LessEqual[n, 5e-307], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.5e-134], N[(n * N[(n * N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.6499999999999999e-189 or 2.5000000000000002e-134 < n

   1. Initial program 23.7%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -2.6499999999999999e-189 < n < 5.00000000000000014e-307

   1. Initial program 74.9%

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

   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 76.0%

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

   if 5.00000000000000014e-307 < n < 2.5000000000000002e-134

   1. Initial program 29.0%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. log-pow-rev N/A

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

      11. unpow1 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 61.1

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

   4. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 61.1

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

   6. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. negate-sub N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. lower-*.f64 N/A

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

      16. associate-*r/ N/A

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

   8. Applied rewrites 70.5%

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

Alternative 7: 80.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.65 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{-135}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -2.65e-189)
     t_0
     (if (<= n 3.6e-135) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -2.65e-189) {
		tmp = t_0;
	} else if (n <= 3.6e-135) {
		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) * 100.0) * n;
	double tmp;
	if (n <= -2.65e-189) {
		tmp = t_0;
	} else if (n <= 3.6e-135) {
		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) * 100.0) * n
	tmp = 0
	if n <= -2.65e-189:
		tmp = t_0
	elif n <= 3.6e-135:
		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) * 100.0) * n)
	tmp = 0.0
	if (n <= -2.65e-189)
		tmp = t_0;
	elseif (n <= 3.6e-135)
		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] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.65e-189], t$95$0, If[LessEqual[n, 3.6e-135], 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 < -2.6499999999999999e-189 or 3.59999999999999978e-135 < n

   1. Initial program 23.7%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -2.6499999999999999e-189 < n < 3.59999999999999978e-135

   1. Initial program 47.2%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 66.0%

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

Alternative 8: 65.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.75:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -1.9e+17) t_0 (if (<= n 1.75) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -1.9e+17) {
		tmp = t_0;
	} else if (n <= 1.75) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -1.9e+17)
		tmp = t_0;
	elseif (n <= 1.75)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.9e+17], t$95$0, If[LessEqual[n, 1.75], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9e17 or 1.75 < n

   1. Initial program 24.4%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 78.5

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(50 + \frac{50}{3} \cdot i, i, 100\right) \cdot n \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{50}{3} \cdot i + 50, i, 100\right) \cdot n \]

      5. lower-fma.f64 67.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   10. Applied rewrites 67.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   if -1.9e17 < n < 1.75

   1. Initial program 32.2%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 61.0%

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

Alternative 9: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+16}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{fma}\left(0.5, i, 1\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e+16)
   (* 100.0 (/ (* i n) i))
   (if (<= n 1.5) (* 100.0 (/ i (/ i n))) (* 100.0 (* (fma 0.5 i 1.0) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e+16) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * (fma(0.5, i, 1.0) * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e+16)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(fma(0.5, i, 1.0) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e+16], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(0.5 * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5e16

   1. Initial program 27.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 57.0%

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

   if -5e16 < n < 1.5

   1. Initial program 32.2%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 61.0%

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

   if 1.5 < n

   1. Initial program 21.6%

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

   2. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 71.1

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 71.1

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

   7. Applied rewrites 71.1%

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

Alternative 10: 62.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+16}:\\ \;\;\;\;100 \cdot \frac{i \cdot n}{i}\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5e+16)
   (* 100.0 (/ (* i n) i))
   (if (<= n 1.5) (* 100.0 (/ i (/ i n))) (* (fma 50.0 i 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5e+16) {
		tmp = 100.0 * ((i * n) / i);
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = fma(50.0, i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5e+16)
		tmp = Float64(100.0 * Float64(Float64(i * n) / i));
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(fma(50.0, i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5e+16], N[(100.0 * N[(N[(i * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5e16

   1. Initial program 27.3%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in i around 0

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

   7. Applied rewrites 57.0%

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

   if -5e16 < n < 1.5

   1. Initial program 32.2%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 61.0%

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

   if 1.5 < n

   1. Initial program 21.6%

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

   2. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 68.6

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 71.1%

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

Alternative 11: 55.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.7%

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

2. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-exp.f64 67.8

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

4. Applied rewrites 67.8%

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

5. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 55.0

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

7. Applied rewrites 55.0%

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

8. Taylor expanded in n around inf

   \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]
9. Step-by-step derivation

10. Applied rewrites 55.3%

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

Alternative 12: 49.5% 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 27.7%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 49.5%

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

Developer Target 1: 34.4% 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 2025116 
(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))))