Compound Interest

Percentage Accurate: 28.9% → 96.0%

Time: 11.8s

Alternatives: 21

Speedup: 24.3×

• could not determine a ground truth

  1. i = 1.715744732767909e-270
  2. n = -2.8377477399519157e+36

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.9% 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

real(8) function code(i, n)
    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: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\frac{t\_0 \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -1e-84)
     (* (/ (- (* t_0 n) n) i) 100.0)
     (if (<= t_1 INFINITY)
       (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ i n))
       (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-84) {
		tmp = (((t_0 * n) - n) / i) * 100.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) / (i / n);
	} else {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-84)
		tmp = Float64(Float64(Float64(Float64(t_0 * n) - n) / i) * 100.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-84], N[(N[(N[(N[(t$95$0 * n), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e-84

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -1e-84 < (/.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 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 24.5

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 97.1

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

   4. Applied rewrites 97.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 13.3

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

   5. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ t_2 := \frac{t\_0 \cdot n - n}{i} \cdot 100\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.3333333333333333}{n}, \frac{i}{n}, \frac{-0.5}{n}\right) \cdot i, i, i\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n))
        (t_1 (/ (- t_0 1.0) (/ i n)))
        (t_2 (* (/ (- (* t_0 n) n) i) 100.0)))
   (if (<= t_1 -1e-153)
     t_2
     (if (<= t_1 0.0)
       (/
        (*
         (expm1
          (fma (* (fma (/ 0.3333333333333333 n) (/ i n) (/ -0.5 n)) i) i i))
         100.0)
        (/ i n))
       (if (<= t_1 INFINITY)
         t_2
         (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n))))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double t_2 = (((t_0 * n) - n) / i) * 100.0;
	double tmp;
	if (t_1 <= -1e-153) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (expm1(fma((fma((0.3333333333333333 / n), (i / n), (-0.5 / n)) * i), i, i)) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	t_2 = Float64(Float64(Float64(Float64(t_0 * n) - n) / i) * 100.0)
	tmp = 0.0
	if (t_1 <= -1e-153)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(fma(Float64(fma(Float64(0.3333333333333333 / n), Float64(i / n), Float64(-0.5 / n)) * i), i, i)) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$0 * n), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-153], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(N[(N[(N[(0.3333333333333333 / n), $MachinePrecision] * N[(i / n), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] * i + i), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1.00000000000000004e-153 or -0.0 < (/.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 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 92.9

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

   4. Applied rewrites 92.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 93.1

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

   6. Applied rewrites 93.1%

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

   if -1.00000000000000004e-153 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -0.0

   1. Initial program 15.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 14.6

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

      12. lift--.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. lower-expm1.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lower-log1p.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in i around 0

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{i \cdot \left(1 + i \cdot \left(\frac{1}{3} \cdot \frac{i}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right) \cdot 100}{\frac{i}{n}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(i \cdot \color{blue}{\left(i \cdot \left(\frac{1}{3} \cdot \frac{i}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + 1\right)}\right) \cdot 100}{\frac{i}{n}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(i \cdot \left(\frac{1}{3} \cdot \frac{i}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + 1 \cdot i}\right) \cdot 100}{\frac{i}{n}} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\left(i \cdot \left(\frac{1}{3} \cdot \frac{i}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot i + \color{blue}{i}\right) \cdot 100}{\frac{i}{n}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(i \cdot \left(\frac{1}{3} \cdot \frac{i}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right), i, i\right)}\right) \cdot 100}{\frac{i}{n}} \]

   9. Applied rewrites 82.9%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.3333333333333333}{n}, \frac{i}{n}, \frac{-0.5}{n}\right) \cdot i, i, i\right)}\right) \cdot 100}{\frac{i}{n}} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 13.3

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

   5. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.3333333333333333}{n}, \frac{i}{n}, \frac{-0.5}{n}\right) \cdot i, i, i\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot n - n}{i} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ t_2 := \frac{t\_0 \cdot n - n}{i} \cdot 100\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-259}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n))
        (t_1 (/ (- t_0 1.0) (/ i n)))
        (t_2 (* (/ (- (* t_0 n) n) i) 100.0)))
   (if (<= t_1 -5e-259)
     t_2
     (if (<= t_1 0.0)
       (* (* 100.0 n) (/ (expm1 i) i))
       (if (<= t_1 INFINITY)
         t_2
         (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n))))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double t_2 = (((t_0 * n) - n) / i) * 100.0;
	double tmp;
	if (t_1 <= -5e-259) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * n) * (expm1(i) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	t_2 = Float64(Float64(Float64(Float64(t_0 * n) - n) / i) * 100.0)
	tmp = 0.0
	if (t_1 <= -5e-259)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * n) * Float64(expm1(i) / i));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$0 * n), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-259], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(100.0 * n), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -4.99999999999999977e-259 or -0.0 < (/.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 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 93.0

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

   4. Applied rewrites 93.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if -4.99999999999999977e-259 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -0.0

   1. Initial program 14.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 82.8

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 82.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 13.3

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

   5. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -5 \cdot 10^{-259}:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot n - n}{i} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{i} \cdot n, 100, \frac{-1}{\frac{i}{100 \cdot n}}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -10000.0)
     (fma (* (/ t_0 i) n) 100.0 (/ -1.0 (/ i (* 100.0 n))))
     (if (<= t_1 INFINITY)
       (* (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) i) n)
       (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = fma(((t_0 / i) * n), 100.0, (-1.0 / (i / (100.0 * n))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((expm1((log1p((i / n)) * n)) * 100.0) / i) * n;
	} else {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = fma(Float64(Float64(t_0 / i) * n), 100.0, Float64(-1.0 / Float64(i / Float64(100.0 * n))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / i) * n);
	else
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], N[(N[(N[(t$95$0 / i), $MachinePrecision] * n), $MachinePrecision] * 100.0 + N[(-1.0 / N[(i / N[(100.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e4

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -1e4 < (/.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 26.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 25.9

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

      12. lift--.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. lower-expm1.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lower-log1p.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/r/ N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 96.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 13.3

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

   5. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-1}{\frac{i}{100 \cdot n}}\right)\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\frac{t\_0 \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -1e-124)
     (* (/ (- (* t_0 n) n) i) 100.0)
     (if (<= t_1 INFINITY)
       (* (* 100.0 n) (/ (expm1 (* (log1p (/ i n)) n)) i))
       (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-124) {
		tmp = (((t_0 * n) - n) / i) * 100.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (100.0 * n) * (expm1((log1p((i / n)) * n)) / i);
	} else {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-124)
		tmp = Float64(Float64(Float64(Float64(t_0 * n) - n) / i) * 100.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(100.0 * n) * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i));
	else
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-124], N[(N[(N[(N[(t$95$0 * n), $MachinePrecision] - n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(100.0 * n), $MachinePrecision] * N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -9.99999999999999933e-125

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -9.99999999999999933e-125 < (/.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 24.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-to-exp N/A

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

      12. lower-expm1.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 96.6

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

   4. Applied rewrites 96.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 13.3

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

   5. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} \cdot n - n}{i} \cdot 100\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{i} \cdot n, 100, \frac{-1}{\frac{i}{100 \cdot n}}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -10000.0)
     (fma (* (/ t_0 i) n) 100.0 (/ -1.0 (/ i (* 100.0 n))))
     (if (<= t_1 INFINITY)
       (* (* (/ 100.0 i) (expm1 (* (log1p (/ i n)) n))) n)
       (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = fma(((t_0 / i) * n), 100.0, (-1.0 / (i / (100.0 * n))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((100.0 / i) * expm1((log1p((i / n)) * n))) * n;
	} else {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = fma(Float64(Float64(t_0 / i) * n), 100.0, Float64(-1.0 / Float64(i / Float64(100.0 * n))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(100.0 / i) * expm1(Float64(log1p(Float64(i / n)) * n))) * n);
	else
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], N[(N[(N[(t$95$0 / i), $MachinePrecision] * n), $MachinePrecision] * 100.0 + N[(-1.0 / N[(i / N[(100.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e4

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -1e4 < (/.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 26.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 13.3

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

   5. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-1}{\frac{i}{100 \cdot n}}\right)\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{i} \cdot n, 100, \frac{-1}{\frac{i}{100 \cdot n}}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot n\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -10000.0)
     (fma (* (/ t_0 i) n) 100.0 (/ -1.0 (/ i (* 100.0 n))))
     (if (<= t_1 INFINITY)
       (* (* (/ 100.0 i) n) (expm1 (* (log1p (/ i n)) n)))
       (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = fma(((t_0 / i) * n), 100.0, (-1.0 / (i / (100.0 * n))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((100.0 / i) * n) * expm1((log1p((i / n)) * n));
	} else {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = fma(Float64(Float64(t_0 / i) * n), 100.0, Float64(-1.0 / Float64(i / Float64(100.0 * n))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(100.0 / i) * n) * expm1(Float64(log1p(Float64(i / n)) * n)));
	else
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], N[(N[(N[(t$95$0 / i), $MachinePrecision] * n), $MachinePrecision] * 100.0 + N[(-1.0 / N[(i / N[(100.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(100.0 / i), $MachinePrecision] * n), $MachinePrecision] * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e4

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -1e4 < (/.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 26.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 25.9

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

      12. lift--.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. lower-expm1.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lower-log1p.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.7

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

   6. Applied rewrites 95.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 13.3

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

   5. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 13.3

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -10000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-1}{\frac{i}{100 \cdot n}}\right)\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot n\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -10200000:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot n}{i} \cdot 100\\ \mathbf{elif}\;n \leq 44:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -10200000.0)
   (* (/ (* (expm1 i) n) i) 100.0)
   (if (<= n 44.0)
     (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))
     (* (* 100.0 n) (/ (expm1 i) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -10200000.0) {
		tmp = ((expm1(i) * n) / i) * 100.0;
	} else if (n <= 44.0) {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	} else {
		tmp = (100.0 * n) * (expm1(i) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -10200000.0)
		tmp = Float64(Float64(Float64(expm1(i) * n) / i) * 100.0);
	elseif (n <= 44.0)
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	else
		tmp = Float64(Float64(100.0 * n) * Float64(expm1(i) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -10200000.0], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 44.0], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * n), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.02e7

   1. Initial program 19.9%

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

   3. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if -1.02e7 < n < 44

   1. Initial program 27.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 26.1

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 70.5

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

   10. Applied rewrites 70.5%

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

   if 44 < n

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.4

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 93.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -10200000:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot n}{i} \cdot 100\\ \mathbf{elif}\;n \leq 44:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -10200000:\\ \;\;\;\;\left(t\_0 \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 44:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (expm1 i) i)))
   (if (<= n -10200000.0)
     (* (* t_0 100.0) n)
     (if (<= n 44.0)
       (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))
       (* (* 100.0 n) t_0)))))

C

double code(double i, double n) {
	double t_0 = expm1(i) / i;
	double tmp;
	if (n <= -10200000.0) {
		tmp = (t_0 * 100.0) * n;
	} else if (n <= 44.0) {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	} else {
		tmp = (100.0 * n) * t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(expm1(i) / i)
	tmp = 0.0
	if (n <= -10200000.0)
		tmp = Float64(Float64(t_0 * 100.0) * n);
	elseif (n <= 44.0)
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	else
		tmp = Float64(Float64(100.0 * n) * t_0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -10200000.0], N[(N[(t$95$0 * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 44.0], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * n), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.02e7

   1. Initial program 19.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if -1.02e7 < n < 44

   1. Initial program 27.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 26.1

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 70.5

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

   10. Applied rewrites 70.5%

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

   if 44 < n

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.4

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 93.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -10200000:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 44:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -10200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 44:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -10200000.0)
     t_0
     (if (<= n 44.0)
       (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))
       t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -10200000.0) {
		tmp = t_0;
	} else if (n <= 44.0) {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / 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 <= -10200000.0)
		tmp = t_0;
	elseif (n <= 44.0)
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / 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, -10200000.0], t$95$0, If[LessEqual[n, 44.0], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.02e7 or 44 < n

   1. Initial program 21.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.1

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

   5. Applied rewrites 93.1%

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

   if -1.02e7 < n < 44

   1. Initial program 27.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 26.1

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

   5. Applied rewrites 26.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 26.1

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 70.5

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

   10. Applied rewrites 70.5%

       \[\leadsto \frac{100}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 44:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.3e+138)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n 44.0)
     (/ 100.0 (fma (- (/ 0.5 (* n n)) (/ 0.5 n)) i (/ 1.0 n)))
     (*
      (/
       (*
        (fma
         (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
         i
         1.0)
        i)
       i)
      (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.3e+138) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= 44.0) {
		tmp = 100.0 / fma(((0.5 / (n * n)) - (0.5 / n)), i, (1.0 / n));
	} else {
		tmp = ((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.3e+138)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= 44.0)
		tmp = Float64(100.0 / fma(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), i, Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.3e+138], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 44.0], N[(100.0 / N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.29999999999999978e138

   1. Initial program 11.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 77.0%

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

   if -3.29999999999999978e138 < n < 44

   1. Initial program 28.8%

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

   3. Taylor expanded in i around 0

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

      1. lower-+.f64 22.5

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

   5. Applied rewrites 22.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 22.5

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 69.7

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

   10. Applied rewrites 69.7%

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

   if 44 < n

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.4

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}{i} \cdot \left(n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 44:\\ \;\;\;\;\frac{100}{\mathsf{fma}\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}, i, \frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e-150)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n 1.5e-213)
     (* (/ (- 1.0 1.0) (/ i n)) 100.0)
     (*
      (/
       (*
        (fma
         (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
         i
         1.0)
        i)
       i)
      (* 100.0 n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e-150) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= 1.5e-213) {
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0;
	} else {
		tmp = ((fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e-150)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= 1.5e-213)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / Float64(i / n)) * 100.0);
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * i) / i) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e-150], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 1.5e-213], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.70000000000000001e-150

   1. Initial program 18.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 72.2%

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

   if -3.70000000000000001e-150 < n < 1.49999999999999993e-213

   1. Initial program 59.3%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 66.7%

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

   if 1.49999999999999993e-213 < n

   1. Initial program 20.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in i around 0

      \[\leadsto \frac{i \cdot \left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right)}{i} \cdot \left(n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot \left(n \cdot 100\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot i}{i} \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e-150)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n 1.5e-213)
     (* (/ (- 1.0 1.0) (/ i n)) 100.0)
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e-150) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= 1.5e-213) {
		tmp = ((1.0 - 1.0) / (i / n)) * 100.0;
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e-150)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= 1.5e-213)
		tmp = Float64(Float64(Float64(1.0 - 1.0) / Float64(i / n)) * 100.0);
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e-150], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 1.5e-213], N[(N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.70000000000000001e-150

   1. Initial program 18.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 72.2%

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

   if -3.70000000000000001e-150 < n < 1.49999999999999993e-213

   1. Initial program 59.3%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 66.7%

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

   if 1.49999999999999993e-213 < n

   1. Initial program 20.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 76.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{1 - 1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.7e-150)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n 1.5e-213)
     (/ 0.0 i)
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.7e-150) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= 1.5e-213) {
		tmp = 0.0 / i;
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.7e-150)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= 1.5e-213)
		tmp = Float64(0.0 / i);
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.7e-150], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 1.5e-213], N[(0.0 / i), $MachinePrecision], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.70000000000000001e-150

   1. Initial program 18.4%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 72.2%

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

   if -3.70000000000000001e-150 < n < 1.49999999999999993e-213

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. 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. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \cdot 100 + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \color{blue}{\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100}\right) \]

   4. Applied rewrites 18.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-n}{i} \cdot 100\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{n \cdot \left(-100 + 100\right)}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{n \cdot \color{blue}{0}}{i} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if 1.49999999999999993e-213 < n

   1. Initial program 20.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 83.4

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 76.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)))
   (if (<= n -3.7e-150) t_0 (if (<= n 1.5e-213) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	double tmp;
	if (n <= -3.7e-150) {
		tmp = t_0;
	} else if (n <= 1.5e-213) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0)
	tmp = 0.0
	if (n <= -3.7e-150)
		tmp = t_0;
	elseif (n <= 1.5e-213)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -3.7e-150], t$95$0, If[LessEqual[n, 1.5e-213], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.70000000000000001e-150 or 1.49999999999999993e-213 < n

   1. Initial program 19.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n\right)} \]

   5. Applied rewrites 69.9%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), n\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \mathsf{fma}\left(n \cdot i, \frac{1}{2} + \color{blue}{\frac{1}{6} \cdot i}, n\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.0%

      \[\leadsto 100 \cdot \mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, \color{blue}{i}, 0.5\right), n\right) \]

   if -3.70000000000000001e-150 < n < 1.49999999999999993e-213

   1. Initial program 59.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. 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. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      6. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      7. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \cdot 100 + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \color{blue}{\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100}\right) \]

   4. Applied rewrites 18.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-n}{i} \cdot 100\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{n \cdot \left(-100 + 100\right)}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{n \cdot \color{blue}{0}}{i} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-213}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -3.7e-150) t_0 (if (<= n 1.5e-213) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -3.7e-150) {
		tmp = t_0;
	} else if (n <= 1.5e-213) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -3.7e-150)
		tmp = t_0;
	elseif (n <= 1.5e-213)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -3.7e-150], t$95$0, If[LessEqual[n, 1.5e-213], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.70000000000000001e-150 or 1.49999999999999993e-213 < n

   1. Initial program 19.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 85.9

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 72.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   if -3.70000000000000001e-150 < n < 1.49999999999999993e-213

   1. Initial program 59.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. 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. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      6. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      7. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \cdot 100 + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \color{blue}{\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100}\right) \]

   4. Applied rewrites 18.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-n}{i} \cdot 100\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{n \cdot \left(-100 + 100\right)}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{n \cdot \color{blue}{0}}{i} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, i, 1\right) \cdot \left(100 \cdot n\right)\\ \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 0.5 i 1.0) (* 100.0 n))))
   (if (<= n -3.7e-150) t_0 (if (<= n 2.8e-217) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(0.5, i, 1.0) * (100.0 * n);
	double tmp;
	if (n <= -3.7e-150) {
		tmp = t_0;
	} else if (n <= 2.8e-217) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(0.5, i, 1.0) * Float64(100.0 * n))
	tmp = 0.0
	if (n <= -3.7e-150)
		tmp = t_0;
	elseif (n <= 2.8e-217)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(0.5 * i + 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.7e-150], t$95$0, If[LessEqual[n, 2.8e-217], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.70000000000000001e-150 or 2.8e-217 < n

   1. Initial program 19.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 85.9

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.9%

      \[\leadsto \frac{\mathsf{expm1}\left(i\right)}{i} \cdot \color{blue}{\left(n \cdot 100\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \left(1 + \frac{1}{2} \cdot i\right) \cdot \left(\color{blue}{n} \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.7%

       \[\leadsto \mathsf{fma}\left(0.5, i, 1\right) \cdot \left(\color{blue}{n} \cdot 100\right) \]

   if -3.70000000000000001e-150 < n < 2.8e-217

   1. Initial program 59.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. 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. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      6. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      7. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \cdot 100 + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \color{blue}{\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100}\right) \]

   4. Applied rewrites 18.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-n}{i} \cdot 100\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{n \cdot \left(-100 + 100\right)}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{n \cdot \color{blue}{0}}{i} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -3.7 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 50.0 i 100.0) n)))
   (if (<= n -3.7e-150) t_0 (if (<= n 2.8e-217) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(50.0, i, 100.0) * n;
	double tmp;
	if (n <= -3.7e-150) {
		tmp = t_0;
	} else if (n <= 2.8e-217) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(50.0, i, 100.0) * n)
	tmp = 0.0
	if (n <= -3.7e-150)
		tmp = t_0;
	elseif (n <= 2.8e-217)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -3.7e-150], t$95$0, If[LessEqual[n, 2.8e-217], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.70000000000000001e-150 or 2.8e-217 < n

   1. Initial program 19.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 85.9

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 68.7%

      \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]

   if -3.70000000000000001e-150 < n < 2.8e-217

   1. Initial program 59.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. 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. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      6. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      7. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \cdot 100 + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \color{blue}{\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100}\right) \]

   4. Applied rewrites 18.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-n}{i} \cdot 100\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{n \cdot \left(-100 + 100\right)}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{n \cdot \color{blue}{0}}{i} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 54.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 6.3 \cdot 10^{+49}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(50 \cdot i\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 6.3e+49) (* 100.0 n) (* (* 50.0 i) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 6.3e+49) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 6.3d+49) then
        tmp = 100.0d0 * n
    else
        tmp = (50.0d0 * i) * n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 6.3e+49) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 6.3e+49:
		tmp = 100.0 * n
	else:
		tmp = (50.0 * i) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 6.3e+49)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(Float64(50.0 * i) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 6.3e+49)
		tmp = 100.0 * n;
	else
		tmp = (50.0 * i) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 6.3e+49], N[(100.0 * n), $MachinePrecision], N[(N[(50.0 * i), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 6.30000000000000007e49

   1. Initial program 18.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. lower-*.f64 66.9

         \[\leadsto \color{blue}{100 \cdot n} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{100 \cdot n} \]

   if 6.30000000000000007e49 < i

   1. Initial program 45.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 63.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]

   9. Taylor expanded in i around inf

      \[\leadsto \left(50 \cdot i\right) \cdot n \]
   10. Step-by-step derivation

   11. Applied rewrites 42.4%

       \[\leadsto \left(50 \cdot i\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 54.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(50, i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* (fma 50.0 i 100.0) n))

C

double code(double i, double n) {
	return fma(50.0, i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(50.0, i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 23.7%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in n around inf

   \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

   2. *-commutative N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

   8. lower-expm1.f64 80.1

      \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

5. Applied rewrites 80.1%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

6. Taylor expanded in i around 0

   \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
7. Step-by-step derivation

8. Applied rewrites 63.3%

   \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]
9. Add Preprocessing

Alternative 21: 49.4% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot n \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 n))

C

double code(double i, double n) {
	return 100.0 * n;
}

Fortran

real(8) function code(i, n)
    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 23.7%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. lower-*.f64 55.3

      \[\leadsto \color{blue}{100 \cdot n} \]

5. Applied rewrites 55.3%

   \[\leadsto \color{blue}{100 \cdot n} \]
6. Add Preprocessing

Developer Target 1: 33.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    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 2024331 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))