Compound Interest

Percentage Accurate: 28.7% → 91.2%

Time: 8.8s

Alternatives: 16

Speedup: 24.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 32.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 97.6

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

   4. Applied rewrites 97.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 82.0%

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

Alternative 2: 90.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[i_, n_] := If[LessEqual[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], Infinity], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 32.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 97.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 82.0%

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

Alternative 3: 90.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 32.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 96.9

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

   4. Applied rewrites 96.9%

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

   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. Add Preprocessing

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 82.0%

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

Alternative 4: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{-199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-309}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.3e-199)
     t_0
     (if (<= n -4e-309)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (if (<= n 9.2e-39)
         (* 100.0 (/ (* (- (log i) (log n)) n) (/ i n)))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.3e-199) {
		tmp = t_0;
	} else if (n <= -4e-309) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 9.2e-39) {
		tmp = 100.0 * (((log(i) - log(n)) * n) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.3e-199) {
		tmp = t_0;
	} else if (n <= -4e-309) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 9.2e-39) {
		tmp = 100.0 * (((Math.log(i) - Math.log(n)) * n) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -1.3e-199:
		tmp = t_0
	elif n <= -4e-309:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	elif n <= 9.2e-39:
		tmp = 100.0 * (((math.log(i) - math.log(n)) * n) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.3e-199)
		tmp = t_0;
	elseif (n <= -4e-309)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 9.2e-39)
		tmp = Float64(100.0 * Float64(Float64(Float64(log(i) - log(n)) * n) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.3e-199], t$95$0, If[LessEqual[n, -4e-309], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.2e-39], N[(100.0 * N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.3e-199 or 9.20000000000000033e-39 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 93.6

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

   8. Applied rewrites 93.6%

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

   if -1.3e-199 < n < -3.9999999999999977e-309

   1. Initial program 90.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 90.2%

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

   if -3.9999999999999977e-309 < n < 9.20000000000000033e-39

   1. Initial program 27.3%

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

   3. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(\log i + -1 \cdot \log n\right)}}{\frac{i}{n}} \]
   4. 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 79.1

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

   5. Applied rewrites 79.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-199}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-309}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{-199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-309}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \left(\frac{\mathsf{fma}\left(\log n, -1, \log i\right) \cdot n}{i} \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.3e-199)
     t_0
     (if (<= n -4e-309)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (if (<= n 9.2e-39)
         (* 100.0 (* (/ (* (fma (log n) -1.0 (log i)) n) i) n))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.3e-199) {
		tmp = t_0;
	} else if (n <= -4e-309) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 9.2e-39) {
		tmp = 100.0 * (((fma(log(n), -1.0, log(i)) * n) / i) * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.3e-199)
		tmp = t_0;
	elseif (n <= -4e-309)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 9.2e-39)
		tmp = Float64(100.0 * Float64(Float64(Float64(fma(log(n), -1.0, log(i)) * n) / i) * n));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.3e-199], t$95$0, If[LessEqual[n, -4e-309], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.2e-39], N[(100.0 * N[(N[(N[(N[(N[Log[n], $MachinePrecision] * -1.0 + N[Log[i], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.3e-199 or 9.20000000000000033e-39 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 93.6

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

   8. Applied rewrites 93.6%

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

   if -1.3e-199 < n < -3.9999999999999977e-309

   1. Initial program 90.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 90.2%

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

   if -3.9999999999999977e-309 < n < 9.20000000000000033e-39

   1. Initial program 27.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 79.1

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

   7. Applied rewrites 79.1%

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

Alternative 6: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{-199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{-183}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \left(\left(n \cdot n\right) \cdot \frac{\log i - \log n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.3e-199)
     t_0
     (if (<= n 1.32e-183)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (if (<= n 9.2e-39)
         (* 100.0 (* (* n n) (/ (- (log i) (log n)) i)))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.3e-199) {
		tmp = t_0;
	} else if (n <= 1.32e-183) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 9.2e-39) {
		tmp = 100.0 * ((n * n) * ((log(i) - log(n)) / i));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.3e-199) {
		tmp = t_0;
	} else if (n <= 1.32e-183) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else if (n <= 9.2e-39) {
		tmp = 100.0 * ((n * n) * ((Math.log(i) - Math.log(n)) / i));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -1.3e-199:
		tmp = t_0
	elif n <= 1.32e-183:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	elif n <= 9.2e-39:
		tmp = 100.0 * ((n * n) * ((math.log(i) - math.log(n)) / i))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.3e-199)
		tmp = t_0;
	elseif (n <= 1.32e-183)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	elseif (n <= 9.2e-39)
		tmp = Float64(100.0 * Float64(Float64(n * n) * Float64(Float64(log(i) - log(n)) / i)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.3e-199], t$95$0, If[LessEqual[n, 1.32e-183], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 9.2e-39], N[(100.0 * N[(N[(n * n), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.3e-199 or 9.20000000000000033e-39 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 93.6

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

   8. Applied rewrites 93.6%

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

   if -1.3e-199 < n < 1.3199999999999999e-183

   1. Initial program 58.1%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 81.6%

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

   if 1.3199999999999999e-183 < n < 9.20000000000000033e-39

   1. Initial program 23.2%

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

   3. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \color{blue}{\frac{{n}^{2} \cdot \left(\log i + -1 \cdot \log n\right)}{i}} \]
   4. 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 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 7: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-199} \lor \neg \left(n \leq 1.05 \cdot 10^{-73}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.3e-199) (not (<= n 1.05e-73)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-199) || !(n <= 1.05e-73)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-199) || !(n <= 1.05e-73)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.3e-199) or not (n <= 1.05e-73):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.3e-199) || !(n <= 1.05e-73))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.3e-199], N[Not[LessEqual[n, 1.05e-73]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.3e-199 or 1.0499999999999999e-73 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 91.5

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

   8. Applied rewrites 91.5%

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

   if -1.3e-199 < n < 1.0499999999999999e-73

   1. Initial program 46.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-199} \lor \neg \left(n \leq 1.05 \cdot 10^{-73}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(4.166666666666667 - \frac{25}{n}\right) \cdot n, i, \left(16.666666666666668 - \frac{50}{n}\right) \cdot n\right), i, \left(50 - \frac{50}{n}\right) \cdot n\right), i, n \cdot 100\right)\\ \mathbf{elif}\;n \leq -1.55 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.9e+44)
   (fma
    (fma
     (fma
      (* (- 4.166666666666667 (/ 25.0 n)) n)
      i
      (* (- 16.666666666666668 (/ 50.0 n)) n))
     i
     (* (- 50.0 (/ 50.0 n)) n))
    i
    (* n 100.0))
   (if (<= n -1.55e-199)
     (* 100.0 (/ i (/ i n)))
     (if (<= n 1.05e-73)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (*
        (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
        n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.9e+44) {
		tmp = fma(fma(fma(((4.166666666666667 - (25.0 / n)) * n), i, ((16.666666666666668 - (50.0 / n)) * n)), i, ((50.0 - (50.0 / n)) * n)), i, (n * 100.0));
	} else if (n <= -1.55e-199) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-73) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.9e+44)
		tmp = fma(fma(fma(Float64(Float64(4.166666666666667 - Float64(25.0 / n)) * n), i, Float64(Float64(16.666666666666668 - Float64(50.0 / n)) * n)), i, Float64(Float64(50.0 - Float64(50.0 / n)) * n)), i, Float64(n * 100.0));
	elseif (n <= -1.55e-199)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.05e-73)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.9e+44], N[(N[(N[(N[(N[(4.166666666666667 - N[(25.0 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * i + N[(N[(16.666666666666668 - N[(50.0 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + N[(N[(50.0 - N[(50.0 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.55e-199], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-73], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.9000000000000001e44

   1. Initial program 15.9%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot n + \color{blue}{i \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\frac{25}{6} - 25 \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{50}{3} - 50 \cdot \frac{1}{n}\right)\right) + n \cdot \left(50 - 50 \cdot \frac{1}{n}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto i \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(\frac{25}{6} - 25 \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{50}{3} - 50 \cdot \frac{1}{n}\right)\right) + n \cdot \left(50 - 50 \cdot \frac{1}{n}\right)\right) + 100 \cdot \color{blue}{n} \]

      2. *-commutative N/A

         \[\leadsto \left(i \cdot \left(i \cdot \left(n \cdot \left(\frac{25}{6} - 25 \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{50}{3} - 50 \cdot \frac{1}{n}\right)\right) + n \cdot \left(50 - 50 \cdot \frac{1}{n}\right)\right) \cdot i + 100 \cdot n \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i \cdot \left(i \cdot \left(n \cdot \left(\frac{25}{6} - 25 \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{50}{3} - 50 \cdot \frac{1}{n}\right)\right) + n \cdot \left(50 - 50 \cdot \frac{1}{n}\right), i, 100 \cdot n\right) \]

   8. Applied rewrites 70.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(4.166666666666667 - \frac{25}{n}\right) \cdot n, i, \left(16.666666666666668 - \frac{50}{n}\right) \cdot n\right), i, \left(50 - \frac{50}{n}\right) \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]

   if -1.9000000000000001e44 < n < -1.55000000000000006e-199

   1. Initial program 37.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.7%

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

   if -1.55000000000000006e-199 < n < 1.0499999999999999e-73

   1. Initial program 46.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.8%

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

   if 1.0499999999999999e-73 < n

   1. Initial program 12.6%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 89.9

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 79.6

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

   11. Applied rewrites 79.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(4.166666666666667 - \frac{25}{n}\right) \cdot n, i, \left(16.666666666666668 - \frac{50}{n}\right) \cdot n\right), i, \left(50 - \frac{50}{n}\right) \cdot n\right), i, n \cdot 100\right)\\ \mathbf{elif}\;n \leq -1.55 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.55 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)
           i
           100.0)
          n)))
   (if (<= n -1.9e+44)
     t_0
     (if (<= n -1.55e-199)
       (* 100.0 (/ i (/ i n)))
       (if (<= n 1.05e-73) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -1.9e+44) {
		tmp = t_0;
	} else if (n <= -1.55e-199) {
		tmp = 100.0 * (i / (i / n));
	} else if (n <= 1.05e-73) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -1.9e+44)
		tmp = t_0;
	elseif (n <= -1.55e-199)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (n <= 1.05e-73)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.9e+44], t$95$0, If[LessEqual[n, -1.55e-199], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-73], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.9000000000000001e44 or 1.0499999999999999e-73 < n

   1. Initial program 13.9%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 92.8

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 75.9

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

   11. Applied rewrites 75.9%

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

   if -1.9000000000000001e44 < n < -1.55000000000000006e-199

   1. Initial program 37.2%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 76.7%

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

   if -1.55000000000000006e-199 < n < 1.0499999999999999e-73

   1. Initial program 46.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -1.55 \cdot 10^{-199}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+44} \lor \neg \left(n \leq 2.1\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.9e+44) (not (<= n 2.1)))
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e+44) || !(n <= 2.1)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.9e+44) || !(n <= 2.1))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.9e+44], N[Not[LessEqual[n, 2.1]], $MachinePrecision]], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.9000000000000001e44 or 2.10000000000000009 < n

   1. Initial program 13.3%

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

   3. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-/.f64 97.3

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 78.8

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

   11. Applied rewrites 78.8%

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

   if -1.9000000000000001e44 < n < 2.10000000000000009

   1. Initial program 39.7%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 60.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+44} \lor \neg \left(n \leq 2.1\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0) n))

C

double code(double i, double n) {
	return fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-exp.f64 72.0

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

5. Applied rewrites 72.0%

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

6. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-expm1.f64 N/A

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

   4. lift-/.f64 78.7

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

8. Applied rewrites 78.7%

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

9. Taylor expanded in i around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. +-commutative N/A

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

    8. lower-fma.f64 60.9

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

11. Applied rewrites 60.9%

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

Alternative 12: 57.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) 100.0) n))

C

double code(double i, double n) {
	return (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * 100.0) * n;
}

Julia

function code(i, n)
	return Float64(Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-exp.f64 72.0

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

5. Applied rewrites 72.0%

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

6. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-expm1.f64 N/A

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

   4. lift-/.f64 78.7

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

8. Applied rewrites 78.7%

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

9. Taylor expanded in i around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-fma.f64 59.5

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

11. Applied rewrites 59.5%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n \]
12. Add Preprocessing

Alternative 13: 57.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n))

C

double code(double i, double n) {
	return fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-exp.f64 72.0

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

5. Applied rewrites 72.0%

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

6. Taylor expanded in i around 0

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

   1. +-commutative N/A

      \[\leadsto \left(i \cdot \left(\left(50 + i \cdot \left(\frac{50}{3} - 50 \cdot \frac{1}{n}\right)\right) - 50 \cdot \frac{1}{n}\right) + 100\right) \cdot n \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\left(50 + i \cdot \left(\frac{50}{3} - 50 \cdot \frac{1}{n}\right)\right) - 50 \cdot \frac{1}{n}\right) \cdot i + 100\right) \cdot n \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(50 + i \cdot \left(\frac{50}{3} - 50 \cdot \frac{1}{n}\right)\right) - 50 \cdot \frac{1}{n}, i, 100\right) \cdot n \]

8. Applied rewrites 59.0%

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

9. Taylor expanded in n around inf

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

    1. +-commutative N/A

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

    2. lower-fma.f64 59.5

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

11. Applied rewrites 59.5%

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

Alternative 14: 54.7% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in n around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-expm1.f64 73.5

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

5. Applied rewrites 73.5%

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

6. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 57.3

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

8. Applied rewrites 57.3%

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

Alternative 15: 54.7% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in n around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-exp.f64 72.0

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

5. Applied rewrites 72.0%

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

6. Taylor expanded in i around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 56.8

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

8. Applied rewrites 56.8%

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

9. Taylor expanded in n around inf

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

11. Applied rewrites 57.3%

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

Alternative 16: 48.8% 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 24.3%

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

3. Taylor expanded in i around 0

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

5. Applied rewrites 53.5%

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

Developer Target 1: 35.0% 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 2025051 
(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))))