Compound Interest

Percentage Accurate: 28.1% → 94.7%

Time: 7.8s

Alternatives: 15

Speedup: 8.9×

• could not determine a ground truth

  1. i = 2.2076068996602806e-249
  2. n = -1.6868660087107509e+31

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.1% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ t_1 := \frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
        (t_1 (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) (* n 100.0))))
   (if (<= t_0 -1e-19)
     t_1
     (if (<= t_0 0.0)
       (* (* (/ (expm1 (* (log1p (/ i n)) n)) i) n) 100.0)
       (if (<= t_0 INFINITY) t_1 (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double t_1 = ((pow(((i / n) + 1.0), n) - 1.0) / i) * (n * 100.0);
	double tmp;
	if (t_0 <= -1e-19) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((expm1((log1p((i / n)) * n)) / i) * n) * 100.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double t_1 = ((Math.pow(((i / n) + 1.0), n) - 1.0) / i) * (n * 100.0);
	double tmp;
	if (t_0 <= -1e-19) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n) * 100.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	t_1 = ((math.pow(((i / n) + 1.0), n) - 1.0) / i) * (n * 100.0)
	tmp = 0
	if t_0 <= -1e-19:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = ((math.expm1((math.log1p((i / n)) * n)) / i) * n) * 100.0
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	t_1 = Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / i) * Float64(n * 100.0))
	tmp = 0.0
	if (t_0 <= -1e-19)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n) * 100.0);
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-19], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(100.0 * n), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -9.9999999999999998e-20 or 0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 97.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 56.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow-to-exp N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 97.9

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

   7. Applied rewrites 97.9%

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

   if -9.9999999999999998e-20 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 0.0

   1. Initial program 24.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 35.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-log1p.f64 N/A

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

      6. lift-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

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

   1. Initial program 97.8%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 3.7%

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

Alternative 2: 94.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n)
     (if (<= t_0 0.0)
       (* (* (/ (expm1 (* (log1p (/ i n)) n)) i) n) 100.0)
       (if (<= t_0 INFINITY)
         (* (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) n) 100.0)
         (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else if (t_0 <= 0.0) {
		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) / i) * n) * 100.0;
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n) * 100.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / i) * n) * 100.0);
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]
   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 13.4

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

   4. Applied rewrites 13.4%

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

   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 13.4

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + 100\right) \cdot n \]

      2. *-commutative N/A

         \[\leadsto \left(\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) \cdot i + 100\right) \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right), i, 100\right) \cdot n \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50, i, 100\right) \cdot n \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{50}{3} + \frac{25}{6} \cdot i\right) \cdot i + 50, i, 100\right) \cdot n \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{50}{3} + \frac{25}{6} \cdot i, i, 50\right), i, 100\right) \cdot n \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{6} \cdot i + \frac{50}{3}, i, 50\right), i, 100\right) \cdot n \]

      8. lower-fma.f64 86.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   10. Applied rewrites 86.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

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

   1. Initial program 25.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 36.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-log1p.f64 N/A

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

      6. lift-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

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

   1. Initial program 97.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 59.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow-to-exp N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 97.8

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

   5. Applied rewrites 97.8%

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

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

   1. Initial program 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 78.7%

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

Alternative 3: 94.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n)
     (if (<= t_0 0.0)
       (* (* (/ (expm1 (* (log1p (/ i n)) n)) i) n) 100.0)
       (if (<= t_0 2e-6)
         (* 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 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else if (t_0 <= 0.0) {
		tmp = ((expm1((log1p((i / n)) * n)) / i) * n) * 100.0;
	} else if (t_0 <= 2e-6) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else {
		tmp = ((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 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n) * 100.0);
	elseif (t_0 <= 2e-6)
		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, (-Infinity)], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], 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 4 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf

      \[\leadsto \color{blue}{n \cdot \left(-50 \cdot \frac{i \cdot e^{i}}{n} + 100 \cdot \frac{e^{i} - 1}{i}\right)} \]
   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 13.4

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

   4. Applied rewrites 13.4%

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

   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 13.4

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) + 100\right) \cdot n \]

      2. *-commutative N/A

         \[\leadsto \left(\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right) \cdot i + 100\right) \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right), i, 100\right) \cdot n \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right) + 50, i, 100\right) \cdot n \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{50}{3} + \frac{25}{6} \cdot i\right) \cdot i + 50, i, 100\right) \cdot n \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{50}{3} + \frac{25}{6} \cdot i, i, 50\right), i, 100\right) \cdot n \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{6} \cdot i + \frac{50}{3}, i, 50\right), i, 100\right) \cdot n \]

      8. lower-fma.f64 86.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

   10. Applied rewrites 86.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]

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

   1. Initial program 25.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 36.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-log1p.f64 N/A

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

      6. lift-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.0 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 1.99999999999999991e-6

   1. Initial program 95.8%

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

   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 95.7

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

   4. Applied rewrites 95.7%

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

   if 1.99999999999999991e-6 < (*.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 21.2%

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

   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 62.3

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

   4. Applied rewrites 62.3%

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

   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 81.2

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

   7. Applied rewrites 81.2%

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

Alternative 4: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0 \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(n \cdot \left(\frac{\log i}{i} + \frac{-\log n}{i}\right)\right) \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.5e-123)
     (* t_0 (* n 100.0))
     (if (<= n -5e-310)
       (* (* (/ (expm1 (* (log (+ (/ i n) 1.0)) n)) i) n) 100.0)
       (if (<= n 3.1e-140)
         (* (* (* n (+ (/ (log i) i) (/ (- (log n)) i))) n) 100.0)
         (* (* t_0 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -5e-310) {
		tmp = ((expm1((log(((i / n) + 1.0)) * n)) / i) * n) * 100.0;
	} else if (n <= 3.1e-140) {
		tmp = ((n * ((log(i) / i) + (-log(n) / i))) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -5e-310) {
		tmp = ((Math.expm1((Math.log(((i / n) + 1.0)) * n)) / i) * n) * 100.0;
	} else if (n <= 3.1e-140) {
		tmp = ((n * ((Math.log(i) / i) + (-Math.log(n) / i))) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.5e-123:
		tmp = t_0 * (n * 100.0)
	elif n <= -5e-310:
		tmp = ((math.expm1((math.log(((i / n) + 1.0)) * n)) / i) * n) * 100.0
	elif n <= 3.1e-140:
		tmp = ((n * ((math.log(i) / i) + (-math.log(n) / i))) * n) * 100.0
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.5e-123)
		tmp = Float64(t_0 * Float64(n * 100.0));
	elseif (n <= -5e-310)
		tmp = Float64(Float64(Float64(expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n)) / i) * n) * 100.0);
	elseif (n <= 3.1e-140)
		tmp = Float64(Float64(Float64(n * Float64(Float64(log(i) / i) + Float64(Float64(-log(n)) / i))) * n) * 100.0);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.5e-123], N[(t$95$0 * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5e-310], N[(N[(N[(N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.1e-140], N[(N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] / i), $MachinePrecision] + N[((-N[Log[n], $MachinePrecision]) / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.50000000000000015e-123

   1. Initial program 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 22.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 83.2%

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

   if -2.50000000000000015e-123 < n < -4.999999999999985e-310

   1. Initial program 63.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 76.4%

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

   if -4.999999999999985e-310 < n < 3.0999999999999999e-140

   1. Initial program 31.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 46.4%

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

   4. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. sum-log N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 42.5

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

   6. Applied rewrites 42.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sum-log N/A

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

      7. div-add N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-log.f64 76.4

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

   8. Applied rewrites 76.4%

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

   if 3.0999999999999999e-140 < n

   1. Initial program 19.9%

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

   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 65.0

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

   4. Applied rewrites 65.0%

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

   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 84.7

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

   7. Applied rewrites 84.7%

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

Alternative 5: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0 \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(n \cdot \left(\frac{\log i}{i} + \frac{-\log n}{i}\right)\right) \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.5e-123)
     (* t_0 (* n 100.0))
     (if (<= n -5e-310)
       (* (* (/ (expm1 (* (log (/ i n)) n)) i) n) 100.0)
       (if (<= n 3.1e-140)
         (* (* (* n (+ (/ (log i) i) (/ (- (log n)) i))) n) 100.0)
         (* (* t_0 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -5e-310) {
		tmp = ((expm1((log((i / n)) * n)) / i) * n) * 100.0;
	} else if (n <= 3.1e-140) {
		tmp = ((n * ((log(i) / i) + (-log(n) / i))) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -5e-310) {
		tmp = ((Math.expm1((Math.log((i / n)) * n)) / i) * n) * 100.0;
	} else if (n <= 3.1e-140) {
		tmp = ((n * ((Math.log(i) / i) + (-Math.log(n) / i))) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.5e-123:
		tmp = t_0 * (n * 100.0)
	elif n <= -5e-310:
		tmp = ((math.expm1((math.log((i / n)) * n)) / i) * n) * 100.0
	elif n <= 3.1e-140:
		tmp = ((n * ((math.log(i) / i) + (-math.log(n) / i))) * n) * 100.0
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.5e-123)
		tmp = Float64(t_0 * Float64(n * 100.0));
	elseif (n <= -5e-310)
		tmp = Float64(Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) / i) * n) * 100.0);
	elseif (n <= 3.1e-140)
		tmp = Float64(Float64(Float64(n * Float64(Float64(log(i) / i) + Float64(Float64(-log(n)) / i))) * n) * 100.0);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.5e-123], N[(t$95$0 * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5e-310], N[(N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.1e-140], N[(N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] / i), $MachinePrecision] + N[((-N[Log[n], $MachinePrecision]) / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.50000000000000015e-123

   1. Initial program 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 22.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 83.2%

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

   if -2.50000000000000015e-123 < n < -4.999999999999985e-310

   1. Initial program 63.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 76.4%

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

   4. Taylor expanded in i around inf

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

      1. lift-/.f64 72.5

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

   6. Applied rewrites 72.5%

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

   if -4.999999999999985e-310 < n < 3.0999999999999999e-140

   1. Initial program 31.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 46.4%

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

   4. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. sum-log N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 42.5

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

   6. Applied rewrites 42.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sum-log N/A

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

      7. div-add N/A

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

      8. log-rec N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-log.f64 76.4

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

   8. Applied rewrites 76.4%

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

   if 3.0999999999999999e-140 < n

   1. Initial program 19.9%

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

   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 65.0

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

   4. Applied rewrites 65.0%

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

   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 84.7

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

   7. Applied rewrites 84.7%

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

Alternative 6: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0 \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq -2.25 \cdot 10^{-269}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.5e-123)
     (* t_0 (* n 100.0))
     (if (<= n -2.25e-269)
       (* (* (/ (expm1 (* (log (/ i n)) n)) i) n) 100.0)
       (if (<= n 5.2e-178)
         (* (* (/ (- 1.0 1.0) i) n) 100.0)
         (* (* t_0 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -2.25e-269) {
		tmp = ((expm1((log((i / n)) * n)) / i) * n) * 100.0;
	} else if (n <= 5.2e-178) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -2.25e-269) {
		tmp = ((Math.expm1((Math.log((i / n)) * n)) / i) * n) * 100.0;
	} else if (n <= 5.2e-178) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.5e-123:
		tmp = t_0 * (n * 100.0)
	elif n <= -2.25e-269:
		tmp = ((math.expm1((math.log((i / n)) * n)) / i) * n) * 100.0
	elif n <= 5.2e-178:
		tmp = (((1.0 - 1.0) / i) * n) * 100.0
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.5e-123)
		tmp = Float64(t_0 * Float64(n * 100.0));
	elseif (n <= -2.25e-269)
		tmp = Float64(Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) / i) * n) * 100.0);
	elseif (n <= 5.2e-178)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.5e-123], N[(t$95$0 * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -2.25e-269], N[(N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 5.2e-178], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.50000000000000015e-123

   1. Initial program 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 22.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 83.2%

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

   if -2.50000000000000015e-123 < n < -2.2500000000000001e-269

   1. Initial program 56.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 72.1%

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

   4. Taylor expanded in i around inf

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

      1. lift-/.f64 67.9

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

   6. Applied rewrites 67.9%

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

   if -2.2500000000000001e-269 < n < 5.19999999999999997e-178

   1. Initial program 48.9%

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

   2. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 67.7

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

   4. Applied rewrites 67.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.7

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 31.2

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

   6. Applied rewrites 31.2%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 77.6%

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

   if 5.19999999999999997e-178 < n

   1. Initial program 20.0%

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

   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 63.2

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

   4. Applied rewrites 63.2%

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

   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 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 7: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -3 \cdot 10^{-132}:\\ \;\;\;\;t\_0 \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-203}:\\ \;\;\;\;\left(\left(n \cdot \frac{\log \left(\frac{i}{n}\right)}{i}\right) \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -3e-132)
     (* t_0 (* n 100.0))
     (if (<= n -7.2e-203)
       (* (* (* n (/ (log (/ i n)) i)) n) 100.0)
       (if (<= n 5.2e-178)
         (* (* (/ (- 1.0 1.0) i) n) 100.0)
         (* (* t_0 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -3e-132) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -7.2e-203) {
		tmp = ((n * (log((i / n)) / i)) * n) * 100.0;
	} else if (n <= 5.2e-178) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -3e-132) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= -7.2e-203) {
		tmp = ((n * (Math.log((i / n)) / i)) * n) * 100.0;
	} else if (n <= 5.2e-178) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -3e-132:
		tmp = t_0 * (n * 100.0)
	elif n <= -7.2e-203:
		tmp = ((n * (math.log((i / n)) / i)) * n) * 100.0
	elif n <= 5.2e-178:
		tmp = (((1.0 - 1.0) / i) * n) * 100.0
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -3e-132)
		tmp = Float64(t_0 * Float64(n * 100.0));
	elseif (n <= -7.2e-203)
		tmp = Float64(Float64(Float64(n * Float64(log(Float64(i / n)) / i)) * n) * 100.0);
	elseif (n <= 5.2e-178)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -3e-132], N[(t$95$0 * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -7.2e-203], N[(N[(N[(n * N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 5.2e-178], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3e-132

   1. Initial program 25.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 23.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 23.1%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 83.1%

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

   if -3e-132 < n < -7.19999999999999958e-203

   1. Initial program 48.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. sum-log N/A

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 41.6

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

   6. Applied rewrites 41.6%

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

   7. Taylor expanded in i around 0

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

      1. lift-/.f64 41.6

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

   9. Applied rewrites 41.6%

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

   if -7.19999999999999958e-203 < n < 5.19999999999999997e-178

   1. Initial program 54.8%

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

   2. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 65.5

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

   4. Applied rewrites 65.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.5

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 30.3

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

   6. Applied rewrites 30.3%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 75.5%

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

   if 5.19999999999999997e-178 < n

   1. Initial program 20.0%

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

   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 63.2

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

   4. Applied rewrites 63.2%

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

   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 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 8: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0 \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -2.5e-123)
     (* t_0 (* n 100.0))
     (if (<= n 5.2e-178)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= 5.2e-178) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0 * (n * 100.0);
	} else if (n <= 5.2e-178) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -2.5e-123:
		tmp = t_0 * (n * 100.0)
	elif n <= 5.2e-178:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -2.5e-123)
		tmp = Float64(t_0 * Float64(n * 100.0));
	elseif (n <= 5.2e-178)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2.5e-123], N[(t$95$0 * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.2e-178], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.50000000000000015e-123

   1. Initial program 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 22.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 83.2%

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

   if -2.50000000000000015e-123 < n < 5.19999999999999997e-178

   1. Initial program 52.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 67.2%

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

   if 5.19999999999999997e-178 < n

   1. Initial program 20.0%

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

   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 63.2

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

   4. Applied rewrites 63.2%

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

   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 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 9: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -2.5e-123)
     t_0
     (if (<= n 5.2e-178) (* 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 <= -2.5e-123) {
		tmp = t_0;
	} else if (n <= 5.2e-178) {
		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 <= -2.5e-123) {
		tmp = t_0;
	} else if (n <= 5.2e-178) {
		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 <= -2.5e-123:
		tmp = t_0
	elif n <= 5.2e-178:
		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 <= -2.5e-123)
		tmp = t_0;
	elseif (n <= 5.2e-178)
		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, -2.5e-123], t$95$0, If[LessEqual[n, 5.2e-178], 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 < -2.50000000000000015e-123 or 5.19999999999999997e-178 < n

   1. Initial program 22.7%

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

   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 73.1

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

   4. Applied rewrites 73.1%

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

   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 82.6

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

   7. Applied rewrites 82.6%

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

   if -2.50000000000000015e-123 < n < 5.19999999999999997e-178

   1. Initial program 52.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 67.2%

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

Alternative 10: 64.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -2.5e-123)
     t_0
     (if (<= n 5.2e-178) (* 100.0 (/ (- 1.0 1.0) (/ 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 <= -2.5e-123) {
		tmp = t_0;
	} else if (n <= 5.2e-178) {
		tmp = 100.0 * ((1.0 - 1.0) / (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 <= -2.5e-123)
		tmp = t_0;
	elseif (n <= 5.2e-178)
		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[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.5e-123], t$95$0, If[LessEqual[n, 5.2e-178], 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 < -2.50000000000000015e-123 or 5.19999999999999997e-178 < n

   1. Initial program 22.7%

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

   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 73.1

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

   4. Applied rewrites 73.1%

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

   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 82.6

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

   7. Applied rewrites 82.6%

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

   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 64.0

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

   10. Applied rewrites 64.0%

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

   if -2.50000000000000015e-123 < n < 5.19999999999999997e-178

   1. Initial program 52.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 67.2%

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

Alternative 11: 62.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;\left(\frac{1 - 1}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 50.0 i 100.0) n)))
   (if (<= n -2.5e-123)
     t_0
     (if (<= n 5.2e-178) (* (* (/ (- 1.0 1.0) i) n) 100.0) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(50.0, i, 100.0) * n;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0;
	} else if (n <= 5.2e-178) {
		tmp = (((1.0 - 1.0) / i) * n) * 100.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(50.0, i, 100.0) * n)
	tmp = 0.0
	if (n <= -2.5e-123)
		tmp = t_0;
	elseif (n <= 5.2e-178)
		tmp = Float64(Float64(Float64(Float64(1.0 - 1.0) / i) * n) * 100.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.5e-123], t$95$0, If[LessEqual[n, 5.2e-178], N[(N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.50000000000000015e-123 or 5.19999999999999997e-178 < n

   1. Initial program 22.7%

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

   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 73.1

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

   4. Applied rewrites 73.1%

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

   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 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto \left(50 \cdot i + 100\right) \cdot n \]

      2. lower-fma.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -2.50000000000000015e-123 < n < 5.19999999999999997e-178

   1. Initial program 52.4%

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

   2. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 55.2

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

   4. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.2

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 23.7

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

   6. Applied rewrites 23.7%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 67.2%

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

Alternative 12: 62.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 50.0 i 100.0) n)))
   (if (<= n -2.5e-123)
     t_0
     (if (<= n 5.2e-178) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(50.0, i, 100.0) * n;
	double tmp;
	if (n <= -2.5e-123) {
		tmp = t_0;
	} else if (n <= 5.2e-178) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(50.0, i, 100.0) * n)
	tmp = 0.0
	if (n <= -2.5e-123)
		tmp = t_0;
	elseif (n <= 5.2e-178)
		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[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.5e-123], t$95$0, If[LessEqual[n, 5.2e-178], 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 < -2.50000000000000015e-123 or 5.19999999999999997e-178 < n

   1. Initial program 22.7%

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

   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 73.1

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

   4. Applied rewrites 73.1%

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

   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 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto \left(50 \cdot i + 100\right) \cdot n \]

      2. lower-fma.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -2.50000000000000015e-123 < n < 5.19999999999999997e-178

   1. Initial program 52.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 67.2%

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

Alternative 13: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.52:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 50.0 i 100.0) n)))
   (if (<= n -2.15e+79) t_0 (if (<= n 0.52) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(50.0, i, 100.0) * n;
	double tmp;
	if (n <= -2.15e+79) {
		tmp = t_0;
	} else if (n <= 0.52) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(50.0, i, 100.0) * n)
	tmp = 0.0
	if (n <= -2.15e+79)
		tmp = t_0;
	elseif (n <= 0.52)
		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[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.15e+79], t$95$0, If[LessEqual[n, 0.52], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.1500000000000002e79 or 0.52000000000000002 < n

   1. Initial program 22.4%

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

   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 78.2

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

   4. Applied rewrites 78.2%

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

   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 92.8

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

         \[\leadsto \left(50 \cdot i + 100\right) \cdot n \]

      2. lower-fma.f64 64.4

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

   10. Applied rewrites 64.4%

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

   if -2.1500000000000002e79 < n < 0.52000000000000002

   1. Initial program 34.4%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 59.6%

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

Alternative 14: 54.5% 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 28.1%

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

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 67.5

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

4. Applied rewrites 67.5%

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

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 75.4

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

7. Applied rewrites 75.4%

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

8. Taylor expanded in i around 0

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

   1. +-commutative N/A

      \[\leadsto \left(50 \cdot i + 100\right) \cdot n \]

   2. lower-fma.f64 54.5

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

10. Applied rewrites 54.5%

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

Alternative 15: 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 28.1%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 49.5%

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

Developer Target 1: 33.9% 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 2025119 
(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))))