Compound Interest

Percentage Accurate: 28.5% → 95.8%

Time: 9.2s

Alternatives: 12

Speedup: 24.3×

• could not determine a ground truth

  1. i = 1.145082272837234e-250
  2. n = -1.1115712930133625e+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: 28.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -1e-28)
     t_0
     (if (<= t_0 0.0)
       (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
       (if (<= t_0 INFINITY)
         (* 100.0 (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) n))
         (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -1e-28) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -1e-28) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * Math.expm1((Math.log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= -1e-28:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (100.0 * math.expm1((math.log1p((i / n)) * n))) / (i / n)
	elif t_0 <= math.inf:
		tmp = 100.0 * (((math.pow(((i / n) + 1.0), n) - 1.0) / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -1e-28)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-28], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -9.99999999999999971e-29

   1. Initial program 98.2%

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

   if -9.99999999999999971e-29 < (*.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 23.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log1p.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.6

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

   3. Applied rewrites 99.6%

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

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

   1. Initial program 97.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 59.0

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

   3. Applied rewrites 59.0%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)} \]
   4. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1}}{i} \cdot n\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1}{i} \cdot n\right) \]

      3. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1}{i} \cdot n\right) \]

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 79.8%

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

Alternative 2: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(100 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\right) \cdot n\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -2e+15)
     t_0
     (if (<= t_0 0.0)
       (* (* 100.0 (/ (expm1 (* (log1p (/ i n)) n)) i)) n)
       (if (<= t_0 INFINITY)
         (* 100.0 (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) n))
         (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * (expm1((log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (100.0 * (Math.expm1((Math.log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= -2e+15:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (100.0 * (math.expm1((math.log1p((i / n)) * n)) / i)) * n
	elif t_0 <= math.inf:
		tmp = 100.0 * (((math.pow(((i / n) + 1.0), n) - 1.0) / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -2e+15)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(100.0 * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+15], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(100.0 * N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -2e15

   1. Initial program 99.5%

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

   if -2e15 < (*.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 25.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 98.6

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

   3. Applied rewrites 98.6%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1}}{i} \cdot n\right) \]

      5. lift-*.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1}{i} \cdot n\right) \]

      6. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1}{i} \cdot n\right) \]

      7. lift-log1p.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 98.7%

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

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

   1. Initial program 97.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 59.0

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

   3. Applied rewrites 59.0%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)} \]
   4. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1}}{i} \cdot n\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1}{i} \cdot n\right) \]

      3. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1}{i} \cdot n\right) \]

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 79.8%

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

Alternative 3: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 -2e+15)
     t_0
     (if (<= t_0 0.0)
       (* 100.0 (* (/ (expm1 (* (log1p (/ i n)) n)) i) n))
       (if (<= t_0 INFINITY)
         (* 100.0 (* (/ (- (pow (+ (/ i n) 1.0) n) 1.0) i) n))
         (* 100.0 n))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * ((expm1((log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * (((pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * ((Math.expm1((Math.log1p((i / n)) * n)) / i) * n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (((Math.pow(((i / n) + 1.0), n) - 1.0) / i) * n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= -2e+15:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = 100.0 * ((math.expm1((math.log1p((i / n)) * n)) / i) * n)
	elif t_0 <= math.inf:
		tmp = 100.0 * (((math.pow(((i / n) + 1.0), n) - 1.0) / i) * n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= -2e+15)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i) * n));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0) / i) * n));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+15], t$95$0, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < -2e15

   1. Initial program 99.5%

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

   if -2e15 < (*.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 25.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 98.6

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

   3. Applied rewrites 98.6%

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

   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 97.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 59.0

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

   3. Applied rewrites 59.0%

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i} \cdot n\right)} \]
   4. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1}}{i} \cdot n\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1}{i} \cdot n\right) \]

      3. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n} - 1}{i} \cdot n\right) \]

      4. lift-log1p.f64 N/A

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

      5. pow-to-exp N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 79.8%

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

Alternative 4: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -3.35 \cdot 10^{-62}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-310}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log \left(-i\right) - \log \left(-n\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-75}:\\ \;\;\;\;100 \cdot \frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -3.35e-62)
     (* 100.0 (* t_0 n))
     (if (<= n -4e-310)
       (* 100.0 (* (* n n) (/ (- (log (- i)) (log (- n))) i)))
       (if (<= n 2.25e-75)
         (* 100.0 (/ (* (- (log i) (log n)) n) (/ i n)))
         (* (* 100.0 t_0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -3.35e-62) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -4e-310) {
		tmp = 100.0 * ((n * n) * ((log(-i) - log(-n)) / i));
	} else if (n <= 2.25e-75) {
		tmp = 100.0 * (((log(i) - log(n)) * n) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -3.35e-62) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= -4e-310) {
		tmp = 100.0 * ((n * n) * ((Math.log(-i) - Math.log(-n)) / i));
	} else if (n <= 2.25e-75) {
		tmp = 100.0 * (((Math.log(i) - Math.log(n)) * n) / (i / n));
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -3.35e-62:
		tmp = 100.0 * (t_0 * n)
	elif n <= -4e-310:
		tmp = 100.0 * ((n * n) * ((math.log(-i) - math.log(-n)) / i))
	elif n <= 2.25e-75:
		tmp = 100.0 * (((math.log(i) - math.log(n)) * n) / (i / n))
	else:
		tmp = (100.0 * t_0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -3.35e-62)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= -4e-310)
		tmp = Float64(100.0 * Float64(Float64(n * n) * Float64(Float64(log(Float64(-i)) - log(Float64(-n))) / i)));
	elseif (n <= 2.25e-75)
		tmp = Float64(100.0 * Float64(Float64(Float64(log(i) - log(n)) * n) / Float64(i / n)));
	else
		tmp = Float64(Float64(100.0 * t_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, -3.35e-62], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4e-310], N[(100.0 * N[(N[(n * n), $MachinePrecision] * N[(N[(N[Log[(-i)], $MachinePrecision] - N[Log[(-n)], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.25e-75], N[(100.0 * N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.34999999999999996e-62

   1. Initial program 26.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 69.1

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

   3. Applied rewrites 69.1%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 85.7%

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

   if -3.34999999999999996e-62 < n < -3.999999999999988e-310

   1. Initial program 52.1%

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

   2. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. log-pow-rev N/A

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

      9. unpow1 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. frac-2neg N/A

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

      6. log-div N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 60.9

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

   6. Applied rewrites 60.9%

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

   if -3.999999999999988e-310 < n < 2.2500000000000002e-75

   1. Initial program 26.6%

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

   2. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. log-pow-rev N/A

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

      6. unpow1 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-log.f64 66.9

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

   4. Applied rewrites 66.9%

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

   if 2.2500000000000002e-75 < n

   1. Initial program 20.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 74.9

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 90.0

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

   8. Applied rewrites 90.0%

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

Alternative 5: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right) \cdot n}{i}\\ t_1 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-205}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 8.1 \cdot 10^{-203}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{elif}\;n \leq 1.96:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* (expm1 i) n) i))) (t_1 (* 100.0 (/ i (/ i n)))))
   (if (<= n -1.35e-30)
     t_0
     (if (<= n -3.5e-205)
       t_1
       (if (<= n 8.1e-203)
         (* 100.0 (* (/ (- 1.0 1.0) i) n))
         (if (<= n 1.96) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.35e-30) {
		tmp = t_0;
	} else if (n <= -3.5e-205) {
		tmp = t_1;
	} else if (n <= 8.1e-203) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.96) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.expm1(i) * n) / i);
	double t_1 = 100.0 * (i / (i / n));
	double tmp;
	if (n <= -1.35e-30) {
		tmp = t_0;
	} else if (n <= -3.5e-205) {
		tmp = t_1;
	} else if (n <= 8.1e-203) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else if (n <= 1.96) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) * n) / i)
	t_1 = 100.0 * (i / (i / n))
	tmp = 0
	if n <= -1.35e-30:
		tmp = t_0
	elif n <= -3.5e-205:
		tmp = t_1
	elif n <= 8.1e-203:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	elif n <= 1.96:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) * n) / i))
	t_1 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (n <= -1.35e-30)
		tmp = t_0;
	elseif (n <= -3.5e-205)
		tmp = t_1;
	elseif (n <= 8.1e-203)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	elseif (n <= 1.96)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.35e-30], t$95$0, If[LessEqual[n, -3.5e-205], t$95$1, If[LessEqual[n, 8.1e-203], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.96], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.34999999999999994e-30 or 1.96 < n

   1. Initial program 25.1%

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

   2. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 90.9

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

   4. Applied rewrites 90.9%

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

   if -1.34999999999999994e-30 < n < -3.5e-205 or 8.1000000000000005e-203 < n < 1.96

   1. Initial program 24.1%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 62.3%

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

   if -3.5e-205 < n < 8.1000000000000005e-203

   1. Initial program 57.1%

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

   2. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   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 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 30.5

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 30.5

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

   6. Applied rewrites 30.5%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 76.4%

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

Alternative 6: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -3.5 \cdot 10^{-205}:\\ \;\;\;\;100 \cdot \left(t\_0 \cdot n\right)\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot t\_0\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -3.5e-205)
     (* 100.0 (* t_0 n))
     (if (<= n 3.3e-78)
       (* 100.0 (* (/ (- 1.0 1.0) i) n))
       (* (* 100.0 t_0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -3.5e-205) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 3.3e-78) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -3.5e-205) {
		tmp = 100.0 * (t_0 * n);
	} else if (n <= 3.3e-78) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = (100.0 * t_0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -3.5e-205:
		tmp = 100.0 * (t_0 * n)
	elif n <= 3.3e-78:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	else:
		tmp = (100.0 * t_0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -3.5e-205)
		tmp = Float64(100.0 * Float64(t_0 * n));
	elseif (n <= 3.3e-78)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(Float64(100.0 * t_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, -3.5e-205], N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3e-78], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * t$95$0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.5e-205

   1. Initial program 28.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 75.2

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

   3. Applied rewrites 75.2%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 80.6%

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

   if -3.5e-205 < n < 3.29999999999999982e-78

   1. Initial program 41.4%

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

   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 49.3

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

   4. Applied rewrites 49.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 20.2

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 20.2

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

   6. Applied rewrites 20.2%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 60.2%

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

   if 3.29999999999999982e-78 < n

   1. Initial program 20.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 75.0

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

   3. Applied rewrites 75.0%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 89.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.8

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

   8. Applied rewrites 89.8%

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

Alternative 7: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot n\right)\\ \mathbf{if}\;n \leq -3.5 \cdot 10^{-205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* (/ (expm1 i) i) n))))
   (if (<= n -3.5e-205)
     t_0
     (if (<= n 3.3e-78) (* 100.0 (* (/ (- 1.0 1.0) i) n)) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((expm1(i) / i) * n);
	double tmp;
	if (n <= -3.5e-205) {
		tmp = t_0;
	} else if (n <= 3.3e-78) {
		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 = 100.0 * ((Math.expm1(i) / i) * n);
	double tmp;
	if (n <= -3.5e-205) {
		tmp = t_0;
	} else if (n <= 3.3e-78) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.expm1(i) / i) * n)
	tmp = 0
	if n <= -3.5e-205:
		tmp = t_0
	elif n <= 3.3e-78:
		tmp = 100.0 * (((1.0 - 1.0) / i) * n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(expm1(i) / i) * n))
	tmp = 0.0
	if (n <= -3.5e-205)
		tmp = t_0;
	elseif (n <= 3.3e-78)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.5e-205], t$95$0, If[LessEqual[n, 3.3e-78], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.5e-205 or 3.29999999999999982e-78 < n

   1. Initial program 25.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log1p.f64 N/A

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

      14. lift-/.f64 75.1

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

   3. Applied rewrites 75.1%

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

   4. Taylor expanded in i around 0

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

   6. Applied rewrites 84.6%

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

   if -3.5e-205 < n < 3.29999999999999982e-78

   1. Initial program 41.4%

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

   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 49.3

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

   4. Applied rewrites 49.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 20.2

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 20.2

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

   6. Applied rewrites 20.2%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 60.2%

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

Alternative 8: 63.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(100 \cdot \left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n\right), i, n \cdot 100\right)\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.5e-100)
   (fma (* 100.0 (* (fma 0.16666666666666666 i 0.5) n)) i (* n 100.0))
   (if (<= n 3.3e-78)
     (* 100.0 (* (/ (- 1.0 1.0) i) n))
     (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.5e-100) {
		tmp = fma((100.0 * (fma(0.16666666666666666, i, 0.5) * n)), i, (n * 100.0));
	} else if (n <= 3.3e-78) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.5e-100)
		tmp = fma(Float64(100.0 * Float64(fma(0.16666666666666666, i, 0.5) * n)), i, Float64(n * 100.0));
	elseif (n <= 3.3e-78)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.5e-100], N[(N[(100.0 * N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3e-78], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.5e-100

   1. Initial program 26.1%

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

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + \color{blue}{100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i + \color{blue}{100} \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{i}, 100 \cdot n\right) \]

   4. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n \cdot 100\right)} \]

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 60.0

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

   7. Applied rewrites 60.0%

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

   if -2.5e-100 < n < 3.29999999999999982e-78

   1. Initial program 42.1%

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

   2. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   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 43.3

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

   4. Applied rewrites 43.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 16.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 16.7

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 56.1%

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

   if 3.29999999999999982e-78 < n

   1. Initial program 20.8%

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

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + \color{blue}{100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i + \color{blue}{100} \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{i}, 100 \cdot n\right) \]

   4. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n \cdot 100\right)} \]

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 72.6

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

   7. Applied rewrites 72.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot \color{blue}{n} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-78}:\\ \;\;\;\;100 \cdot \left(\frac{1 - 1}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n)))
   (if (<= n -2.5e-100)
     t_0
     (if (<= n 3.3e-78) (* 100.0 (* (/ (- 1.0 1.0) i) n)) t_0))))

C

double code(double i, double n) {
	double t_0 = fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
	double tmp;
	if (n <= -2.5e-100) {
		tmp = t_0;
	} else if (n <= 3.3e-78) {
		tmp = 100.0 * (((1.0 - 1.0) / i) * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n)
	tmp = 0.0
	if (n <= -2.5e-100)
		tmp = t_0;
	elseif (n <= 3.3e-78)
		tmp = Float64(100.0 * Float64(Float64(Float64(1.0 - 1.0) / i) * n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.5e-100], t$95$0, If[LessEqual[n, 3.3e-78], N[(100.0 * N[(N[(N[(1.0 - 1.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.5e-100 or 3.29999999999999982e-78 < n

   1. Initial program 23.6%

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

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + \color{blue}{100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i + \color{blue}{100} \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{i}, 100 \cdot n\right) \]

   4. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n \cdot 100\right)} \]

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 66.0

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

   7. Applied rewrites 66.0%

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

   if -2.5e-100 < n < 3.29999999999999982e-78

   1. Initial program 42.1%

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

   2. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + i\right)} - 1}{\frac{i}{n}} \]
   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 43.3

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

   4. Applied rewrites 43.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 16.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 16.7

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in i around 0

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

   9. Applied rewrites 56.1%

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

Alternative 10: 53.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot i}{n} \cdot 33.333333333333336\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.35e+154) (* 100.0 n) (* (/ (* i i) n) 33.333333333333336)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.35e+154) {
		tmp = 100.0 * n;
	} else {
		tmp = ((i * i) / n) * 33.333333333333336;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.35d+154) then
        tmp = 100.0d0 * n
    else
        tmp = ((i * i) / n) * 33.333333333333336d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.35e+154) {
		tmp = 100.0 * n;
	} else {
		tmp = ((i * i) / n) * 33.333333333333336;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.35e+154:
		tmp = 100.0 * n
	else:
		tmp = ((i * i) / n) * 33.333333333333336
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.35e+154)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(Float64(Float64(i * i) / n) * 33.333333333333336);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.35e+154)
		tmp = 100.0 * n;
	else
		tmp = ((i * i) / n) * 33.333333333333336;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.35e+154], N[(100.0 * n), $MachinePrecision], N[(N[(N[(i * i), $MachinePrecision] / n), $MachinePrecision] * 33.333333333333336), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.35000000000000003e154

   1. Initial program 24.3%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 55.3%

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

   if 1.35000000000000003e154 < i

   1. Initial program 58.2%

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

   2. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + \color{blue}{100 \cdot n} \]

      2. *-commutative N/A

         \[\leadsto \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i + \color{blue}{100} \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{i}, 100 \cdot n\right) \]

   4. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n \cdot 100\right)} \]

   5. Taylor expanded in n around 0

      \[\leadsto \frac{100}{3} \cdot \color{blue}{\frac{{i}^{2}}{n}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{i}^{2}}{n} \cdot \frac{100}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{i}^{2}}{n} \cdot \frac{100}{3} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{i}^{2}}{n} \cdot \frac{100}{3} \]

      4. unpow2 N/A

         \[\leadsto \frac{i \cdot i}{n} \cdot \frac{100}{3} \]

      5. lower-*.f64 42.7

         \[\leadsto \frac{i \cdot i}{n} \cdot 33.333333333333336 \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{i \cdot i}{n} \cdot \color{blue}{33.333333333333336} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* (fma (* (fma 0.16666666666666666 i 0.5) i) 100.0 100.0) n))

C

double code(double i, double n) {
	return fma((fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(Float64(fma(0.16666666666666666, i, 0.5) * i), 100.0, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 28.5%

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

2. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto i \cdot \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + \color{blue}{100 \cdot n} \]

   2. *-commutative N/A

      \[\leadsto \left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) \cdot i + \color{blue}{100} \cdot n \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(100 \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right) + 100 \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right), \color{blue}{i}, 100 \cdot n\right) \]

4. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left({n}^{-2}, 0.3333333333333333, 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n \cdot 100\right)} \]

5. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 56.9

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

7. Applied rewrites 56.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot i, 100, 100\right) \cdot \color{blue}{n} \]
8. Add Preprocessing

Alternative 12: 49.2% accurate, 24.3× 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 28.5%

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

2. Taylor expanded in i around 0

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

4. Applied rewrites 49.2%

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

Developer Target 1: 34.2% 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 2025095 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))