Compound Interest

Percentage Accurate: 27.8% → 93.4%

Time: 11.9s

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: 27.8% 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: 93.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.4%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 97.3%

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

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

   1. Initial program 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. +-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-/.f64 99.9

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

   8. Applied rewrites 99.9%

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

   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

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

      1. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

Alternative 2: 93.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.4%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 96.7%

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

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

   1. Initial program 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. +-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-/.f64 99.9

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

   8. Applied rewrites 99.9%

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

   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

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

      1. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

Alternative 3: 92.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.4%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.2

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

   6. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 95.5

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

   8. Applied rewrites 95.5%

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

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

   1. Initial program 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. +-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-/.f64 99.9

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

   8. Applied rewrites 99.9%

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

   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

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

      1. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

Alternative 4: 84.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.4%

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

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 26.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. rem-exp-log N/A

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

      13. rem-exp-log N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 87.2

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

   4. Applied rewrites 87.2%

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

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

   1. Initial program 99.8%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.0

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

   6. Applied rewrites 75.0%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. +-commutative N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-/.f64 99.9

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

   8. Applied rewrites 99.9%

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

   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

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

      1. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

Alternative 5: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(100 \cdot \left(n \cdot n\right)\right) \cdot \frac{\log \left(-i\right) - \log \left(-n\right)}{i}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-137}:\\ \;\;\;\;\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.25e-77)
     t_0
     (if (<= n -5e-310)
       (* (* 100.0 (* n n)) (/ (- (log (- i)) (log (- n))) i))
       (if (<= n 8e-137)
         (* (* n (/ (- (log i) (log n)) i)) (* n 100.0))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.25e-77) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = (100.0 * (n * n)) * ((log(-i) - log(-n)) / i);
	} else if (n <= 8e-137) {
		tmp = (n * ((log(i) - log(n)) / i)) * (n * 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.25e-77) {
		tmp = t_0;
	} else if (n <= -5e-310) {
		tmp = (100.0 * (n * n)) * ((Math.log(-i) - Math.log(-n)) / i);
	} else if (n <= 8e-137) {
		tmp = (n * ((Math.log(i) - Math.log(n)) / i)) * (n * 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -1.25e-77:
		tmp = t_0
	elif n <= -5e-310:
		tmp = (100.0 * (n * n)) * ((math.log(-i) - math.log(-n)) / i)
	elif n <= 8e-137:
		tmp = (n * ((math.log(i) - math.log(n)) / i)) * (n * 100.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.25e-77)
		tmp = t_0;
	elseif (n <= -5e-310)
		tmp = Float64(Float64(100.0 * Float64(n * n)) * Float64(Float64(log(Float64(-i)) - log(Float64(-n))) / i));
	elseif (n <= 8e-137)
		tmp = Float64(Float64(n * Float64(Float64(log(i) - log(n)) / i)) * Float64(n * 100.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.25e-77], t$95$0, If[LessEqual[n, -5e-310], N[(N[(100.0 * N[(n * n), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Log[(-i)], $MachinePrecision] - N[Log[(-n)], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-137], N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.24999999999999991e-77 or 7.99999999999999982e-137 < n

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -1.24999999999999991e-77 < n < -4.999999999999985e-310

   1. Initial program 61.8%

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

   3. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 79.3%

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

   if -4.999999999999985e-310 < n < 7.99999999999999982e-137

   1. Initial program 25.0%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 19.5

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

   7. Applied rewrites 19.5%

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

   8. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. associate-*r/ N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. div-add N/A

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

      9. lower-/.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-log.f64 76.6

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

   10. Applied rewrites 76.6%

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

Alternative 6: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -1.3 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 6.5 \cdot 10^{-229}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-137}:\\ \;\;\;\;\left(n \cdot \frac{\log i - \log n}{i}\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -1.3e-127)
     t_0
     (if (<= n 6.5e-229)
       (* 0.0 100.0)
       (if (<= n 8e-137)
         (* (* n (/ (- (log i) (log n)) i)) (* n 100.0))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.3e-127) {
		tmp = t_0;
	} else if (n <= 6.5e-229) {
		tmp = 0.0 * 100.0;
	} else if (n <= 8e-137) {
		tmp = (n * ((log(i) - log(n)) / i)) * (n * 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -1.3e-127) {
		tmp = t_0;
	} else if (n <= 6.5e-229) {
		tmp = 0.0 * 100.0;
	} else if (n <= 8e-137) {
		tmp = (n * ((Math.log(i) - Math.log(n)) / i)) * (n * 100.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -1.3e-127:
		tmp = t_0
	elif n <= 6.5e-229:
		tmp = 0.0 * 100.0
	elif n <= 8e-137:
		tmp = (n * ((math.log(i) - math.log(n)) / i)) * (n * 100.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -1.3e-127)
		tmp = t_0;
	elseif (n <= 6.5e-229)
		tmp = Float64(0.0 * 100.0);
	elseif (n <= 8e-137)
		tmp = Float64(Float64(n * Float64(Float64(log(i) - log(n)) / i)) * Float64(n * 100.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.3e-127], t$95$0, If[LessEqual[n, 6.5e-229], N[(0.0 * 100.0), $MachinePrecision], If[LessEqual[n, 8e-137], N[(N[(n * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.29999999999999995e-127 or 7.99999999999999982e-137 < n

   1. Initial program 22.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -1.29999999999999995e-127 < n < 6.5e-229

   1. Initial program 65.0%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 28.6

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 75.9

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 75.9%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 75.9

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 75.9%

      \[\leadsto \color{blue}{0 \cdot 100} \]

   if 6.5e-229 < n < 7.99999999999999982e-137

   1. Initial program 25.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 9.7

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

   7. Applied rewrites 9.7%

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

   8. Taylor expanded in n around 0

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

      1. associate-/l* N/A

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

      2. div-add N/A

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

      3. associate-*r/ N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. div-add N/A

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

      9. lower-/.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-log.f64 82.3

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

   10. Applied rewrites 82.3%

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

Alternative 7: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-127} \lor \neg \left(n \leq 6.9 \cdot 10^{-146}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.3e-127) (not (<= n 6.9e-146)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 0.0 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-127) || !(n <= 6.9e-146)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 0.0 * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-127) || !(n <= 6.9e-146)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 0.0 * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.3e-127) or not (n <= 6.9e-146):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 0.0 * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.3e-127) || !(n <= 6.9e-146))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(0.0 * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.3e-127], N[Not[LessEqual[n, 6.9e-146]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(0.0 * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.29999999999999995e-127 or 6.9000000000000002e-146 < n

   1. Initial program 21.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if -1.29999999999999995e-127 < n < 6.9000000000000002e-146

   1. Initial program 55.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 25.2

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 69.1

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 69.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 69.1

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 69.1%

      \[\leadsto \color{blue}{0 \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-127} \lor \neg \left(n \leq 6.9 \cdot 10^{-146}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5\right) - \frac{0.5}{n}, i, 1\right) \cdot i}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.75e-127)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 5.3e-139)
     (* 0.0 100.0)
     (*
      (/
       (*
        (fma
         (-
          (fma
           (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
           i
           0.5)
          (/ 0.5 n))
         i
         1.0)
        i)
       i)
      (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.75e-127) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 5.3e-139) {
		tmp = 0.0 * 100.0;
	} else {
		tmp = ((fma((fma((((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), i, 0.5) - (0.5 / n)), i, 1.0) * i) / i) * (n * 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.75e-127)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 5.3e-139)
		tmp = Float64(0.0 * 100.0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), i, 0.5) - Float64(0.5 / n)), i, 1.0) * i) / i) * Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.75e-127], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.3e-139], N[(0.0 * 100.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + 0.5), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.74999999999999995e-127

   1. Initial program 22.9%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.5%

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

   if -1.74999999999999995e-127 < n < 5.2999999999999997e-139

   1. Initial program 54.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 24.8

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

   4. Applied rewrites 24.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 67.8

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 67.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 67.8%

      \[\leadsto \color{blue}{0 \cdot 100} \]

   if 5.2999999999999997e-139 < n

   1. Initial program 21.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 73.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5\right) - \frac{0.5}{n}, i, 1\right) \cdot i}}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5\right) - \frac{0.5}{n}, i, 1\right) \cdot i}{i} \cdot \left(n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.75e-127)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 5.3e-139)
     (* 0.0 100.0)
     (*
      100.0
      (fma
       (fma
        (* n i)
        (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
        (* (- 0.5 (/ 0.5 n)) n))
       i
       n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.75e-127) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 5.3e-139) {
		tmp = 0.0 * 100.0;
	} else {
		tmp = 100.0 * fma(fma((n * i), (((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), ((0.5 - (0.5 / n)) * n)), i, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.75e-127)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 5.3e-139)
		tmp = Float64(0.0 * 100.0);
	else
		tmp = Float64(100.0 * fma(fma(Float64(n * i), Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), Float64(Float64(0.5 - Float64(0.5 / n)) * n)), i, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.75e-127], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.3e-139], N[(0.0 * 100.0), $MachinePrecision], N[(100.0 * N[(N[(N[(n * i), $MachinePrecision] * N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.74999999999999995e-127

   1. Initial program 22.9%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.5%

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

   if -1.74999999999999995e-127 < n < 5.2999999999999997e-139

   1. Initial program 54.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 24.8

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

   4. Applied rewrites 24.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 67.8

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 67.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 67.8%

      \[\leadsto \color{blue}{0 \cdot 100} \]

   if 5.2999999999999997e-139 < n

   1. Initial program 21.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 71.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(n \cdot i, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), i, n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.75e-127)
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (if (<= n 5.3e-139)
     (* 0.0 100.0)
     (*
      (fma
       (*
        100.0
        (fma
         (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
         i
         (- 0.5 (/ 0.5 n))))
       i
       100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.75e-127) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else if (n <= 5.3e-139) {
		tmp = 0.0 * 100.0;
	} else {
		tmp = fma((100.0 * fma((((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), i, (0.5 - (0.5 / n)))), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.75e-127)
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	elseif (n <= 5.3e-139)
		tmp = Float64(0.0 * 100.0);
	else
		tmp = Float64(fma(Float64(100.0 * fma(Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), i, Float64(0.5 - Float64(0.5 / n)))), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.75e-127], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.3e-139], N[(0.0 * 100.0), $MachinePrecision], N[(N[(N[(100.0 * N[(N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.74999999999999995e-127

   1. Initial program 22.9%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.5%

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

   if -1.74999999999999995e-127 < n < 5.2999999999999997e-139

   1. Initial program 54.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 24.8

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

   4. Applied rewrites 24.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 67.8

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 67.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 67.8%

      \[\leadsto \color{blue}{0 \cdot 100} \]

   if 5.2999999999999997e-139 < n

   1. Initial program 21.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), i, 100\right)} \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127} \lor \neg \left(n \leq 6.9 \cdot 10^{-146}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.75e-127) (not (<= n 6.9e-146)))
   (* 100.0 (fma (* (fma 0.16666666666666666 i 0.5) n) i n))
   (* 0.0 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e-127) || !(n <= 6.9e-146)) {
		tmp = 100.0 * fma((fma(0.16666666666666666, i, 0.5) * n), i, n);
	} else {
		tmp = 0.0 * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.75e-127) || !(n <= 6.9e-146))
		tmp = Float64(100.0 * fma(Float64(fma(0.16666666666666666, i, 0.5) * n), i, n));
	else
		tmp = Float64(0.0 * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.75e-127], N[Not[LessEqual[n, 6.9e-146]], $MachinePrecision]], N[(100.0 * N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision], N[(0.0 * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.74999999999999995e-127 or 6.9000000000000002e-146 < n

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 67.3%

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

   if -1.74999999999999995e-127 < n < 6.9000000000000002e-146

   1. Initial program 55.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 25.2

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 69.1

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 69.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 69.1

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 69.1%

      \[\leadsto \color{blue}{0 \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127} \lor \neg \left(n \leq 6.9 \cdot 10^{-146}\right):\\ \;\;\;\;100 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right) \cdot n, i, n\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.075:\\ \;\;\;\;0 \cdot 100\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.075)
   (* 0.0 100.0)
   (if (<= i 1.4e-13)
     (* (fma 0.5 i 1.0) (* n 100.0))
     (* 100.0 (/ (* n n) n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.075) {
		tmp = 0.0 * 100.0;
	} else if (i <= 1.4e-13) {
		tmp = fma(0.5, i, 1.0) * (n * 100.0);
	} else {
		tmp = 100.0 * ((n * n) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.075)
		tmp = Float64(0.0 * 100.0);
	elseif (i <= 1.4e-13)
		tmp = Float64(fma(0.5, i, 1.0) * Float64(n * 100.0));
	else
		tmp = Float64(100.0 * Float64(Float64(n * n) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.075], N[(0.0 * 100.0), $MachinePrecision], If[LessEqual[i, 1.4e-13], N[(N[(0.5 * i + 1.0), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -0.0749999999999999972

   1. Initial program 60.3%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 31.3

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 31.3%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 31.3

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 31.3%

      \[\leadsto \color{blue}{0 \cdot 100} \]

   if -0.0749999999999999972 < i < 1.4000000000000001e-13

   1. Initial program 7.2%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 83.3%

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

   if 1.4000000000000001e-13 < i

   1. Initial program 48.8%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 42.4%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 23.1%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.075:\\ \;\;\;\;0 \cdot 100\\ \mathbf{elif}\;i \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(0.5 \cdot n - 0.5, i, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.75e-127)
   (* (fma 0.5 i 1.0) (* n 100.0))
   (if (<= n 5.3e-139) (* 0.0 100.0) (* 100.0 (fma (- (* 0.5 n) 0.5) i n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.75e-127) {
		tmp = fma(0.5, i, 1.0) * (n * 100.0);
	} else if (n <= 5.3e-139) {
		tmp = 0.0 * 100.0;
	} else {
		tmp = 100.0 * fma(((0.5 * n) - 0.5), i, n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.75e-127)
		tmp = Float64(fma(0.5, i, 1.0) * Float64(n * 100.0));
	elseif (n <= 5.3e-139)
		tmp = Float64(0.0 * 100.0);
	else
		tmp = Float64(100.0 * fma(Float64(Float64(0.5 * n) - 0.5), i, n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.75e-127], N[(N[(0.5 * i + 1.0), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.3e-139], N[(0.0 * 100.0), $MachinePrecision], N[(100.0 * N[(N[(N[(0.5 * n), $MachinePrecision] - 0.5), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.74999999999999995e-127

   1. Initial program 22.9%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 61.3

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 61.6%

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

   if -1.74999999999999995e-127 < n < 5.2999999999999997e-139

   1. Initial program 54.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 24.8

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

   4. Applied rewrites 24.8%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 67.8

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 67.8%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 67.8%

      \[\leadsto \color{blue}{0 \cdot 100} \]

   if 5.2999999999999997e-139 < n

   1. Initial program 21.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 50.2%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right), n, \left(-0.5 \cdot i - 0.5\right) \cdot i\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 66.9%

       \[\leadsto 100 \cdot \mathsf{fma}\left(0.5 \cdot n - 0.5, i, n\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-139}:\\ \;\;\;\;0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(0.5 \cdot n - 0.5, i, n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127} \lor \neg \left(n \leq 6.9 \cdot 10^{-146}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.75e-127) (not (<= n 6.9e-146)))
   (* (fma 0.5 i 1.0) (* n 100.0))
   (* 0.0 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e-127) || !(n <= 6.9e-146)) {
		tmp = fma(0.5, i, 1.0) * (n * 100.0);
	} else {
		tmp = 0.0 * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.75e-127) || !(n <= 6.9e-146))
		tmp = Float64(fma(0.5, i, 1.0) * Float64(n * 100.0));
	else
		tmp = Float64(0.0 * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.75e-127], N[Not[LessEqual[n, 6.9e-146]], $MachinePrecision]], N[(N[(0.5 * i + 1.0), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], N[(0.0 * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.74999999999999995e-127 or 6.9000000000000002e-146 < n

   1. Initial program 21.9%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 63.9

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

   7. Applied rewrites 63.9%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 64.0%

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

   if -1.74999999999999995e-127 < n < 6.9000000000000002e-146

   1. Initial program 55.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 25.2

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 69.1

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 69.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 69.1

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 69.1%

      \[\leadsto \color{blue}{0 \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127} \lor \neg \left(n \leq 6.9 \cdot 10^{-146}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(n \cdot 100\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127} \lor \neg \left(n \leq 5.6 \cdot 10^{-149}\right):\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.75e-127) (not (<= n 5.6e-149))) (* 100.0 n) (* 0.0 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e-127) || !(n <= 5.6e-149)) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0 * 100.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.75d-127)) .or. (.not. (n <= 5.6d-149))) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0 * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.75e-127) || !(n <= 5.6e-149)) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0 * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.75e-127) or not (n <= 5.6e-149):
		tmp = 100.0 * n
	else:
		tmp = 0.0 * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.75e-127) || !(n <= 5.6e-149))
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(0.0 * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.75e-127) || ~((n <= 5.6e-149)))
		tmp = 100.0 * n;
	else
		tmp = 0.0 * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.75e-127], N[Not[LessEqual[n, 5.6e-149]], $MachinePrecision]], N[(100.0 * n), $MachinePrecision], N[(0.0 * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.74999999999999995e-127 or 5.5999999999999997e-149 < n

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if -1.74999999999999995e-127 < n < 5.5999999999999997e-149

   1. Initial program 55.1%

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

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-*.f64 N/A

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

      16. frac-2neg N/A

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

      17. metadata-eval N/A

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

      18. remove-double-neg N/A

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

      19. lower-/.f64 25.2

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

   4. Applied rewrites 25.2%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{0} \cdot n}{i} \]

      4. lower-*.f64 69.1

         \[\leadsto 100 \cdot \frac{\color{blue}{0 \cdot n}}{i} \]

   7. Applied rewrites 69.1%

      \[\leadsto 100 \cdot \color{blue}{\frac{0 \cdot n}{i}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{0 \cdot n}{i}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

      3. lower-*.f64 69.1

         \[\leadsto \color{blue}{\frac{0 \cdot n}{i} \cdot 100} \]

   9. Applied rewrites 69.1%

      \[\leadsto \color{blue}{0 \cdot 100} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-127} \lor \neg \left(n \leq 5.6 \cdot 10^{-149}\right):\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0 \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.5% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.4%

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

3. Taylor expanded in i around 0

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

   1. lower-*.f64 48.1

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

5. Applied rewrites 48.1%

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

Developer Target 1: 34.0% 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 2024364 
(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))))