Compound Interest

Percentage Accurate: 27.9% → 93.8%

Time: 20.1s

Alternatives: 23

Speedup: 8.9×

• could not determine a ground truth

  1. i = 2.39295862586628e-149
  2. n = -2.9010620315617022e+29

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: 27.9% 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: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-239}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{n + i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 1e-239)
     (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
     (if (<= t_0 INFINITY)
       (/ (* 100.0 (- (pow (/ (+ n i) n) n) 1.0)) (/ i n))
       (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-239) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (100.0 * (pow(((n + i) / n), n) - 1.0)) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-239) {
		tmp = (100.0 * Math.expm1((Math.log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (100.0 * (Math.pow(((n + i) / n), n) - 1.0)) / (i / n);
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 1e-239:
		tmp = (100.0 * math.expm1((math.log1p((i / n)) * n))) / (i / n)
	elif t_0 <= math.inf:
		tmp = (100.0 * (math.pow(((n + i) / n), n) - 1.0)) / (i / n)
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 1e-239)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(100.0 * Float64((Float64(Float64(n + i) / n) ^ n) - 1.0)) / Float64(i / n));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-239], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(100.0 * N[(N[Power[N[(N[(n + i), $MachinePrecision] / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 1.0000000000000001e-239

   1. Initial program 26.5%

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

   3. Applied rewrites 35.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 1.0000000000000001e-239 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 97.4%

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

   3. Applied rewrites 57.6%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. div-add N/A

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

      8. *-inverses N/A

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

      9. lower--.f64 N/A

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

      10. *-inverses N/A

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

      11. div-add N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 77.9%

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

Alternative 2: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-239}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;100 \cdot \frac{{\left(\frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n)))))
   (if (<= t_0 1e-239)
     (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
     (if (<= t_0 INFINITY)
       (* 100.0 (/ (- (pow (/ i n) n) 1.0) (/ i n)))
       (* 100.0 n)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-239) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * ((pow((i / n), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
	double tmp;
	if (t_0 <= 1e-239) {
		tmp = (100.0 * Math.expm1((Math.log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((Math.pow((i / n), n) - 1.0) / (i / n));
	} else {
		tmp = 100.0 * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))
	tmp = 0
	if t_0 <= 1e-239:
		tmp = (100.0 * math.expm1((math.log1p((i / n)) * n))) / (i / n)
	elif t_0 <= math.inf:
		tmp = 100.0 * ((math.pow((i / n), n) - 1.0) / (i / n))
	else:
		tmp = 100.0 * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
	tmp = 0.0
	if (t_0 <= 1e-239)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * Float64(Float64((Float64(i / n) ^ n) - 1.0) / Float64(i / n)));
	else
		tmp = Float64(100.0 * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-239], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < 1.0000000000000001e-239

   1. Initial program 26.5%

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

   3. Applied rewrites 35.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 1.0000000000000001e-239 < (*.f64 #s(literal 100 binary64) (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))) < +inf.0

   1. Initial program 97.4%

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

   2. Taylor expanded in i around inf

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

      1. lift-/.f64 92.9

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

   4. Applied rewrites 92.9%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 77.9%

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

Alternative 3: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (/ (* 100.0 (expm1 (* (log (/ (+ n i) n)) n))) (/ i n))
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (100.0 * expm1((log(((n + i) / n)) * n))) / (i / n);
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (100.0 * Math.expm1((Math.log(((n + i) / n)) * n))) / (i / n);
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = (100.0 * math.expm1((math.log(((n + i) / n)) * n))) / (i / n)
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(100.0 * expm1(Float64(log(Float64(Float64(n + i) / n)) * n))) / Float64(i / n));
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(100.0 * N[(Exp[N[(N[Log[N[(N[(n + i), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

   3. Applied rewrites 56.9%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 4: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (* (/ (* 100.0 (expm1 (* (log (/ (+ n i) n)) n))) i) n)
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((100.0 * expm1((log(((n + i) / n)) * n))) / i) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((100.0 * Math.expm1((Math.log(((n + i) / n)) * n))) / i) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = ((100.0 * math.expm1((math.log(((n + i) / n)) * n))) / i) * n
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(Float64(100.0 * expm1(Float64(log(Float64(Float64(n + i) / n)) * n))) / i) * n);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(N[(100.0 * N[(Exp[N[(N[Log[N[(N[(n + i), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 56.5%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 5: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\log \left(\frac{i}{n} + 1\right) \cdot n\right) \cdot 100\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (* (* (expm1 (* (log (+ (/ i n) 1.0)) n)) 100.0) (/ n i))
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (expm1((log(((i / n) + 1.0)) * n)) * 100.0) * (n / i);
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (Math.expm1((Math.log(((i / n) + 1.0)) * n)) * 100.0) * (n / i);
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = (math.expm1((math.log(((i / n) + 1.0)) * n)) * 100.0) * (n / i)
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(expm1(Float64(log(Float64(Float64(i / n) + 1.0)) * n)) * 100.0) * Float64(n / i));
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(N[(Exp[N[(N[Log[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

   3. Applied rewrites 56.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 83.5

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

   5. Applied rewrites 83.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. *-commutative N/A

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

      10. exp-to-pow N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

   7. Applied rewrites 56.3%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 6: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{i} \cdot n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (* (* (/ (expm1 (* (log (/ (+ n i) n)) n)) i) n) 100.0)
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((expm1((log(((n + i) / n)) * n)) / i) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((Math.expm1((Math.log(((n + i) / n)) * n)) / i) * n) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = ((math.expm1((math.log(((n + i) / n)) * n)) / i) * n) * 100.0
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(Float64(expm1(Float64(log(Float64(Float64(n + i) / n)) * n)) / i) * n) * 100.0);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(N[(N[(Exp[N[(N[Log[N[(N[(n + i), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 56.4%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 7: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (/ (* 100.0 (expm1 (* (log (/ i n)) n))) (/ i n))
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (100.0 * expm1((log((i / n)) * n))) / (i / n);
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (100.0 * Math.expm1((Math.log((i / n)) * n))) / (i / n);
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = (100.0 * math.expm1((math.log((i / n)) * n))) / (i / n)
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(100.0 * expm1(Float64(log(Float64(i / n)) * n))) / Float64(i / n));
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(100.0 * N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

   3. Applied rewrites 56.9%

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

   4. Taylor expanded in i around inf

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

   6. Applied rewrites 52.7%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 8: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (* (/ (* (expm1 (* (log (/ i n)) n)) 100.0) i) n)
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((expm1((log((i / n)) * n)) * 100.0) / i) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((Math.expm1((Math.log((i / n)) * n)) * 100.0) / i) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = ((math.expm1((math.log((i / n)) * n)) * 100.0) / i) * n
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) * 100.0) / i) * n);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 27.1

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 52.4%

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

   11. Applied rewrites 52.4%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 9: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (* (* (expm1 (* (log (/ i n)) n)) (/ 100.0 i)) n)
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (expm1((log((i / n)) * n)) * (100.0 / i)) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (Math.expm1((Math.log((i / n)) * n)) * (100.0 / i)) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = (math.expm1((math.log((i / n)) * n)) * (100.0 / i)) * n
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) * Float64(100.0 / i)) * n);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 27.1

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 52.4%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 10: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\left(\mathsf{expm1}\left(\log \left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -5.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (* (* (expm1 (* (log (/ i n)) n)) (/ n i)) 100.0)
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (expm1((log((i / n)) * n)) * (n / i)) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -5.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = (Math.expm1((Math.log((i / n)) * n)) * (n / i)) * 100.0;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -5.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = (math.expm1((math.log((i / n)) * n)) * (n / i)) * 100.0
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -5.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(expm1(Float64(log(Float64(i / n)) * n)) * Float64(n / i)) * 100.0);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -5.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(N[(Exp[N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.1999999999999998e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.2

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

      9. *-inverses 83.2

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

      10. div-add 83.2

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

      11. exp-to-pow 83.2

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

   6. Applied rewrites 83.2%

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

   if -5.1999999999999998e-113 < n < 3.3e-72

   1. Initial program 39.9%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 52.0%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 11: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.04 \cdot 10^{-138}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-232}:\\ \;\;\;\;\left(\log \left(\frac{i}{n}\right) \cdot \left(n \cdot \frac{n}{i}\right)\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{100 \cdot \left(1 - 1\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -1.04e-138)
     (* (* t_0 n) 100.0)
     (if (<= n -4.2e-232)
       (* (* (log (/ i n)) (* n (/ n i))) 100.0)
       (if (<= n 3.3e-72)
         (* (/ (* 100.0 (- 1.0 1.0)) i) n)
         (* (* t_0 100.0) n))))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -1.04e-138) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= -4.2e-232) {
		tmp = (log((i / n)) * (n * (n / i))) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((100.0 * (1.0 - 1.0)) / i) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -1.04e-138) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= -4.2e-232) {
		tmp = (Math.log((i / n)) * (n * (n / i))) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = ((100.0 * (1.0 - 1.0)) / i) * n;
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -1.04e-138:
		tmp = (t_0 * n) * 100.0
	elif n <= -4.2e-232:
		tmp = (math.log((i / n)) * (n * (n / i))) * 100.0
	elif n <= 3.3e-72:
		tmp = ((100.0 * (1.0 - 1.0)) / i) * n
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -1.04e-138)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= -4.2e-232)
		tmp = Float64(Float64(log(Float64(i / n)) * Float64(n * Float64(n / i))) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(Float64(100.0 * Float64(1.0 - 1.0)) / i) * n);
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -1.04e-138], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, -4.2e-232], N[(N[(N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision] * N[(n * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(N[(N[(100.0 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.0399999999999999e-138

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.8

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

   4. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.8

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.5

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

      9. *-inverses 82.5

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

      10. div-add 82.5

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

      11. exp-to-pow 82.5

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

   6. Applied rewrites 82.5%

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

   if -1.0399999999999999e-138 < n < -4.2000000000000001e-232

   1. Initial program 52.0%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 56.4

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in n around 0

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. *-inverses N/A

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

      5. div-add N/A

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

      6. exp-to-pow N/A

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

      7. distribute-lft-neg-in N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 44.2%

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

   if -4.2000000000000001e-232 < n < 3.3e-72

   1. Initial program 37.2%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 27.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 27.7

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

   6. Applied rewrites 27.7%

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

   7. Taylor expanded in n around 0

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

   9. Applied rewrites 59.6%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 12: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;\left(t\_0 \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -4.2e-113)
     (* (* t_0 n) 100.0)
     (if (<= n 3.3e-72)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (* (* t_0 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -4.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.expm1(i) / i;
	double tmp;
	if (n <= -4.2e-113) {
		tmp = (t_0 * n) * 100.0;
	} else if (n <= 3.3e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * 100.0) * n;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.expm1(i) / i
	tmp = 0
	if n <= -4.2e-113:
		tmp = (t_0 * n) * 100.0
	elif n <= 3.3e-72:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = (t_0 * 100.0) * n
	return tmp

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -4.2e-113)
		tmp = Float64(Float64(t_0 * n) * 100.0);
	elseif (n <= 3.3e-72)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(t_0 * 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -4.2e-113], N[(N[(t$95$0 * n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 3.3e-72], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.2e-113

   1. Initial program 26.7%

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

   2. Taylor expanded in n around inf

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

      1. lower-expm1.f64 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.9

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 83.3

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

      9. *-inverses 83.3

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

      10. div-add 83.3

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

      11. exp-to-pow 83.3

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

   6. Applied rewrites 83.3%

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

   if -4.2e-113 < n < 3.3e-72

   1. Initial program 40.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 56.2%

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

   if 3.3e-72 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 2.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 2.8

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

   6. Applied rewrites 2.8%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 17.8%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 89.2

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

   12. Applied rewrites 89.2%

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

Alternative 13: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -4.2e-113)
     t_0
     (if (<= n 3.3e-72) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -4.2e-113) {
		tmp = t_0;
	} else if (n <= 3.3e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = ((Math.expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -4.2e-113) {
		tmp = t_0;
	} else if (n <= 3.3e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = ((math.expm1(i) / i) * 100.0) * n
	tmp = 0
	if n <= -4.2e-113:
		tmp = t_0
	elif n <= 3.3e-72:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -4.2e-113)
		tmp = t_0;
	elseif (n <= 3.3e-72)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -4.2e-113], t$95$0, If[LessEqual[n, 3.3e-72], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.2e-113 or 3.3e-72 < n

   1. Initial program 23.8%

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

   2. Taylor expanded in n around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 3.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 3.2

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

   6. Applied rewrites 3.2%

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

   7. Taylor expanded in i around inf

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

   9. Applied rewrites 18.4%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lift-expm1.f64 86.1

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

   12. Applied rewrites 86.1%

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

   if -4.2e-113 < n < 3.3e-72

   1. Initial program 40.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 56.2%

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

Alternative 14: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(100 \cdot n\right) \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -8.7 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(-0.5, i, n\right)\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* (* 100.0 n) (expm1 i)) i)))
   (if (<= n -8.7e-20)
     t_0
     (if (<= n -4.2e-113)
       (* 100.0 (fma -0.5 i n))
       (if (<= n 5e-72) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = ((100.0 * n) * expm1(i)) / i;
	double tmp;
	if (n <= -8.7e-20) {
		tmp = t_0;
	} else if (n <= -4.2e-113) {
		tmp = 100.0 * fma(-0.5, i, n);
	} else if (n <= 5e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(100.0 * n) * expm1(i)) / i)
	tmp = 0.0
	if (n <= -8.7e-20)
		tmp = t_0;
	elseif (n <= -4.2e-113)
		tmp = Float64(100.0 * fma(-0.5, i, n));
	elseif (n <= 5e-72)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(100.0 * n), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -8.7e-20], t$95$0, If[LessEqual[n, -4.2e-113], N[(100.0 * N[(-0.5 * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e-72], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.7e-20 or 4.9999999999999996e-72 < n

   1. Initial program 24.5%

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

   3. Applied rewrites 19.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. exp-to-pow N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 87.0

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

   8. Applied rewrites 87.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 87.2

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

   10. Applied rewrites 87.2%

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

   if -8.7e-20 < n < -4.2e-113

   1. Initial program 16.1%

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

   2. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 59.4

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in n around 0

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

   7. Applied rewrites 59.4%

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

   if -4.2e-113 < n < 4.9999999999999996e-72

   1. Initial program 40.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 56.2%

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

Alternative 15: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}\\ \mathbf{if}\;n \leq -8.7 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(-0.5, i, n\right)\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* 100.0 (* (expm1 i) n)) i)))
   (if (<= n -8.7e-20)
     t_0
     (if (<= n -4.2e-113)
       (* 100.0 (fma -0.5 i n))
       (if (<= n 5e-72) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = (100.0 * (expm1(i) * n)) / i;
	double tmp;
	if (n <= -8.7e-20) {
		tmp = t_0;
	} else if (n <= -4.2e-113) {
		tmp = 100.0 * fma(-0.5, i, n);
	} else if (n <= 5e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * Float64(expm1(i) * n)) / i)
	tmp = 0.0
	if (n <= -8.7e-20)
		tmp = t_0;
	elseif (n <= -4.2e-113)
		tmp = Float64(100.0 * fma(-0.5, i, n));
	elseif (n <= 5e-72)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -8.7e-20], t$95$0, If[LessEqual[n, -4.2e-113], N[(100.0 * N[(-0.5 * i + n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e-72], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -8.7e-20 or 4.9999999999999996e-72 < n

   1. Initial program 24.5%

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

   3. Applied rewrites 19.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. exp-to-pow N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 87.0

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

   8. Applied rewrites 87.0%

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

   if -8.7e-20 < n < -4.2e-113

   1. Initial program 16.1%

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

   2. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 59.4

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in n around 0

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

   7. Applied rewrites 59.4%

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

   if -4.2e-113 < n < 4.9999999999999996e-72

   1. Initial program 40.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 56.2%

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

Alternative 16: 63.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100 \cdot n\right)\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(n \cdot \mathsf{fma}\left(50, i, 100\right)\right) \cdot i}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.2e-113)
   (fma (* n (fma 16.666666666666668 i 50.0)) i (* 100.0 n))
   (if (<= n 5e-72)
     (* 100.0 (/ (- 1.0 1.0) (/ i n)))
     (/ (* (* n (fma 50.0 i 100.0)) i) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.2e-113) {
		tmp = fma((n * fma(16.666666666666668, i, 50.0)), i, (100.0 * n));
	} else if (n <= 5e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = ((n * fma(50.0, i, 100.0)) * i) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.2e-113)
		tmp = fma(Float64(n * fma(16.666666666666668, i, 50.0)), i, Float64(100.0 * n));
	elseif (n <= 5e-72)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(Float64(n * fma(50.0, i, 100.0)) * i) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.2e-113], N[(N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] * i + N[(100.0 * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5e-72], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.2e-113

   1. Initial program 26.7%

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

   3. Applied rewrites 22.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. exp-to-pow N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 80.4

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in i around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-out N/A

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

       6. lower-*.f64 N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 58.4

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

   11. Applied rewrites 58.4%

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

   if -4.2e-113 < n < 4.9999999999999996e-72

   1. Initial program 40.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 56.2%

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

   if 4.9999999999999996e-72 < n

   1. Initial program 20.5%

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

   3. Applied rewrites 20.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. exp-to-pow N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 88.0

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

   8. Applied rewrites 88.0%

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

   9. Taylor expanded in i around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. lower-*.f64 N/A

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

       6. lower-fma.f64 73.8

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

   11. Applied rewrites 73.8%

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

Alternative 17: 62.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot i}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma 50.0 i 100.0))))
   (if (<= n -4.2e-113)
     t_0
     (if (<= n 5e-72) (* 100.0 (/ (- 1.0 1.0) (/ i n))) (/ (* t_0 i) i)))))

C

double code(double i, double n) {
	double t_0 = n * fma(50.0, i, 100.0);
	double tmp;
	if (n <= -4.2e-113) {
		tmp = t_0;
	} else if (n <= 5e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = (t_0 * i) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(50.0, i, 100.0))
	tmp = 0.0
	if (n <= -4.2e-113)
		tmp = t_0;
	elseif (n <= 5e-72)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(Float64(t_0 * i) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.2e-113], t$95$0, If[LessEqual[n, 5e-72], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * i), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.2e-113

   1. Initial program 26.7%

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

   3. Applied rewrites 22.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. exp-to-pow N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 80.4

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-fma.f64 56.0

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

   11. Applied rewrites 56.0%

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

   if -4.2e-113 < n < 4.9999999999999996e-72

   1. Initial program 40.0%

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

   2. Taylor expanded in i around 0

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

   4. Applied rewrites 56.2%

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

   if 4.9999999999999996e-72 < n

   1. Initial program 20.5%

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

   3. Applied rewrites 20.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. exp-to-pow N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 88.0

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

   8. Applied rewrites 88.0%

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

   9. Taylor expanded in i around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

       5. lower-*.f64 N/A

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

       6. lower-fma.f64 73.8

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

   11. Applied rewrites 73.8%

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

Alternative 18: 62.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{100 \cdot \left(1 - 1\right)}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma 50.0 i 100.0))))
   (if (<= n -4.2e-113)
     t_0
     (if (<= n 3.3e-72) (* (/ (* 100.0 (- 1.0 1.0)) i) n) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(50.0, i, 100.0);
	double tmp;
	if (n <= -4.2e-113) {
		tmp = t_0;
	} else if (n <= 3.3e-72) {
		tmp = ((100.0 * (1.0 - 1.0)) / i) * n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(50.0, i, 100.0))
	tmp = 0.0
	if (n <= -4.2e-113)
		tmp = t_0;
	elseif (n <= 3.3e-72)
		tmp = Float64(Float64(Float64(100.0 * Float64(1.0 - 1.0)) / i) * n);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.2e-113], t$95$0, If[LessEqual[n, 3.3e-72], N[(N[(N[(100.0 * N[(1.0 - 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.2e-113 or 3.3e-72 < n

   1. Initial program 23.8%

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

   3. Applied rewrites 21.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add N/A

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

      5. *-inverses N/A

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

      6. lower-log1p.f64 N/A

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

      7. lift-/.f64 73.3

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

   5. Applied rewrites 73.3%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{100} \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      2. exp-to-pow N/A

         \[\leadsto 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      8. lift-expm1.f64 84.0

         \[\leadsto \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i} \]

   8. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(50 \cdot i\right) \cdot n + 100 \cdot n \]

       2. distribute-rgt-out N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       4. lower-fma.f64 61.7

          \[\leadsto n \cdot \mathsf{fma}\left(50, i, 100\right) \]

   11. Applied rewrites 61.7%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]

   if -4.2e-113 < n < 3.3e-72

   1. Initial program 40.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{\left(1 + n \cdot \left(\log i + -1 \cdot \log n\right)\right)} - 1}{\frac{i}{n}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \frac{\left(n \cdot \left(\log i + -1 \cdot \log n\right) + \color{blue}{1}\right) - 1}{\frac{i}{n}} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\left(\left(\log i + -1 \cdot \log n\right) \cdot n + 1\right) - 1}{\frac{i}{n}} \]

      3. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\log i + -1 \cdot \log n, \color{blue}{n}, 1\right) - 1}{\frac{i}{n}} \]

   4. Applied rewrites 26.7%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right)} - 1}{\frac{i}{n}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1}{\frac{i}{n}}} \]

      3. lift-/.f64 N/A

         \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1}{\color{blue}{\frac{i}{n}}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1\right)}{\frac{i}{n}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1\right)}{i} \cdot n} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1\right)}{i} \cdot n} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1\right)}{i}} \cdot n \]

      8. lower-*.f64 27.1

         \[\leadsto \frac{\color{blue}{100 \cdot \left(\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1\right)}}{i} \cdot n \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{fma}\left(\log \left(\frac{i}{n}\right), n, 1\right) - 1\right)}{i} \cdot n} \]

   7. Taylor expanded in n around 0

      \[\leadsto \frac{100 \cdot \left(1 - 1\right)}{i} \cdot n \]
   8. Step-by-step derivation

   9. Applied rewrites 56.2%

      \[\leadsto \frac{100 \cdot \left(1 - 1\right)}{i} \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 61.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.3 \cdot 10^{-72}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma 50.0 i 100.0))))
   (if (<= n -4.2e-113)
     t_0
     (if (<= n 3.3e-72) (* 100.0 (/ (- 1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(50.0, i, 100.0);
	double tmp;
	if (n <= -4.2e-113) {
		tmp = t_0;
	} else if (n <= 3.3e-72) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(50.0, i, 100.0))
	tmp = 0.0
	if (n <= -4.2e-113)
		tmp = t_0;
	elseif (n <= 3.3e-72)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.2e-113], t$95$0, If[LessEqual[n, 3.3e-72], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.2e-113 or 3.3e-72 < n

   1. Initial program 23.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   3. Applied rewrites 21.5%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{\color{blue}{n + i}}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      4. div-add N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n}{n} + \frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      5. *-inverses N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\color{blue}{1} + \frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      6. lower-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      7. lift-/.f64 73.3

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

   5. Applied rewrites 73.3%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{100} \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      2. exp-to-pow N/A

         \[\leadsto 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      8. lift-expm1.f64 84.0

         \[\leadsto \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i} \]

   8. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(50 \cdot i\right) \cdot n + 100 \cdot n \]

       2. distribute-rgt-out N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       4. lower-fma.f64 61.7

          \[\leadsto n \cdot \mathsf{fma}\left(50, i, 100\right) \]

   11. Applied rewrites 61.7%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]

   if -4.2e-113 < n < 3.3e-72

   1. Initial program 40.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
   3. Step-by-step derivation

   4. Applied rewrites 56.2%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -24000000:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -24000000.0)
   (/ (* 100.0 (* i n)) i)
   (if (<= n 1.05e-16) (* 100.0 (/ i (/ i n))) (* n (fma 50.0 i 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -24000000.0) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 1.05e-16) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * fma(50.0, i, 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -24000000.0)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= 1.05e-16)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * fma(50.0, i, 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -24000000.0], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.05e-16], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.4e7

   1. Initial program 28.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   3. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{\color{blue}{n + i}}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      4. div-add N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n}{n} + \frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      5. *-inverses N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\color{blue}{1} + \frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      6. lower-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      7. lift-/.f64 64.0

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

   5. Applied rewrites 64.0%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{100} \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      2. exp-to-pow N/A

         \[\leadsto 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      8. lift-expm1.f64 86.8

         \[\leadsto \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i} \]

   8. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}} \]

   9. Taylor expanded in i around 0

      \[\leadsto \frac{100 \cdot \left(i \cdot n\right)}{i} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.4%

       \[\leadsto \frac{100 \cdot \left(i \cdot n\right)}{i} \]

   if -2.4e7 < n < 1.0500000000000001e-16

   1. Initial program 32.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.1%

      \[\leadsto 100 \cdot \frac{\color{blue}{i}}{\frac{i}{n}} \]

   if 1.0500000000000001e-16 < n

   1. Initial program 22.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   3. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{\color{blue}{n + i}}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      4. div-add N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n}{n} + \frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      5. *-inverses N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\color{blue}{1} + \frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      6. lower-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      7. lift-/.f64 72.6

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

   5. Applied rewrites 72.6%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{100} \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      2. exp-to-pow N/A

         \[\leadsto 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      8. lift-expm1.f64 92.6

         \[\leadsto \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i} \]

   8. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(50 \cdot i\right) \cdot n + 100 \cdot n \]

       2. distribute-rgt-out N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       4. lower-fma.f64 69.1

          \[\leadsto n \cdot \mathsf{fma}\left(50, i, 100\right) \]

   11. Applied rewrites 69.1%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -24000000:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -24000000.0)
   (/ (* 100.0 (* i n)) i)
   (if (<= n 1.02e-16) (* 100.0 (* i (/ n i))) (* n (fma 50.0 i 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -24000000.0) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 1.02e-16) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = n * fma(50.0, i, 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -24000000.0)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= 1.02e-16)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	else
		tmp = Float64(n * fma(50.0, i, 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -24000000.0], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.02e-16], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.4e7

   1. Initial program 28.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   3. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{\color{blue}{n + i}}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      4. div-add N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n}{n} + \frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      5. *-inverses N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\color{blue}{1} + \frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      6. lower-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      7. lift-/.f64 64.0

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

   5. Applied rewrites 64.0%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{100} \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      2. exp-to-pow N/A

         \[\leadsto 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      8. lift-expm1.f64 86.8

         \[\leadsto \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i} \]

   8. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}} \]

   9. Taylor expanded in i around 0

      \[\leadsto \frac{100 \cdot \left(i \cdot n\right)}{i} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.4%

       \[\leadsto \frac{100 \cdot \left(i \cdot n\right)}{i} \]

   if -2.4e7 < n < 1.0200000000000001e-16

   1. Initial program 32.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   2. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 100 \cdot \frac{\left(e^{i} - 1\right) \cdot n}{i} \]

      2. associate-/l* N/A

         \[\leadsto 100 \cdot \left(\left(e^{i} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 100 \cdot \left(\left(e^{i} - 1\right) \cdot \color{blue}{\frac{n}{i}}\right) \]

      4. lower-expm1.f64 N/A

         \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{\color{blue}{n}}{i}\right) \]

      5. lower-/.f64 50.6

         \[\leadsto 100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{\color{blue}{i}}\right) \]

   4. Applied rewrites 50.6%

      \[\leadsto 100 \cdot \color{blue}{\left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \left(i \cdot \frac{\color{blue}{n}}{i}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 60.0%

      \[\leadsto 100 \cdot \left(i \cdot \frac{\color{blue}{n}}{i}\right) \]

   if 1.0200000000000001e-16 < n

   1. Initial program 22.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

   3. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   4. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{\color{blue}{n + i}}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      4. div-add N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n}{n} + \frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      5. *-inverses N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\color{blue}{1} + \frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

      6. lower-log1p.f64 N/A

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

      7. lift-/.f64 72.6

         \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

   5. Applied rewrites 72.6%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   6. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{100} \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      2. exp-to-pow N/A

         \[\leadsto 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i} \]

      6. *-commutative N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

      8. lift-expm1.f64 92.6

         \[\leadsto \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i} \]

   8. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(50 \cdot i\right) \cdot n + 100 \cdot n \]

       2. distribute-rgt-out N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

       4. lower-fma.f64 69.1

          \[\leadsto n \cdot \mathsf{fma}\left(50, i, 100\right) \]

   11. Applied rewrites 69.1%

       \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 54.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ n \cdot \mathsf{fma}\left(50, i, 100\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n (fma 50.0 i 100.0)))

C

double code(double i, double n) {
	return n * fma(50.0, i, 100.0);
}

Julia

function code(i, n)
	return Float64(n * fma(50.0, i, 100.0))
end

Wolfram

code[i_, n_] := N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.9%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

3. Applied rewrites 30.6%

   \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{n + i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
4. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\frac{\color{blue}{n + i}}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n + i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   4. div-add N/A

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{n}{n} + \frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   5. *-inverses N/A

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\log \left(\color{blue}{1} + \frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}} \]

   6. lower-log1p.f64 N/A

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

   7. lift-/.f64 75.9

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{i}{n}}\right) \cdot n\right)}{\frac{i}{n}} \]

5. Applied rewrites 75.9%

   \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right)}{\frac{i}{n}} \]

6. Taylor expanded in n around inf

   \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
7. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{100} \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

   2. exp-to-pow N/A

      \[\leadsto 100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{\color{blue}{i}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i} \]

   6. *-commutative N/A

      \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{100 \cdot \left(\left(e^{i} - 1\right) \cdot n\right)}{i} \]

   8. lift-expm1.f64 70.1

      \[\leadsto \frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i} \]

8. Applied rewrites 70.1%

   \[\leadsto \color{blue}{\frac{100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot n\right)}{i}} \]

9. Taylor expanded in i around 0

   \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(50 \cdot i\right) \cdot n + 100 \cdot n \]

    2. distribute-rgt-out N/A

       \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto n \cdot \left(50 \cdot i + \color{blue}{100}\right) \]

    4. lower-fma.f64 54.3

       \[\leadsto n \cdot \mathsf{fma}\left(50, i, 100\right) \]

11. Applied rewrites 54.3%

    \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]
12. Add Preprocessing

Alternative 23: 48.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ 100 \cdot n \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 n))

C

double code(double i, double n) {
	return 100.0 * n;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * n
end function

Java

public static double code(double i, double n) {
	return 100.0 * n;
}

Python

def code(i, n):
	return 100.0 * n

Julia

function code(i, n)
	return Float64(100.0 * n)
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * n;
end

Wolfram

code[i_, n_] := N[(100.0 * n), $MachinePrecision]

Derivation

1. Initial program 27.9%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]

2. Taylor expanded in i around 0

   \[\leadsto 100 \cdot \color{blue}{n} \]
3. Step-by-step derivation

4. Applied rewrites 48.8%

   \[\leadsto 100 \cdot \color{blue}{n} \]
5. Add Preprocessing

Developer Target 1: 34.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(i, n)
use fmin_fmax_functions
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025130 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform c (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))