Compound Interest

Percentage Accurate: 28.1% → 98.0%

Time: 21.1s

Alternatives: 18

Speedup: 8.1×

• could not determine a ground truth

  1. i = 0.001208785347122663
  2. n = 2.2218209266582675e+38

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: 28.1% 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: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-236}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left(t\_0, 100, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(i, -0.5, 1\right)}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 5e-236)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 INFINITY)
       (* n (/ (fma t_0 100.0 -100.0) i))
       (* 100.0 (/ 1.0 (/ (fma i -0.5 1.0) n)))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-236) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (fma(t_0, 100.0, -100.0) / i);
	} else {
		tmp = 100.0 * (1.0 / (fma(i, -0.5, 1.0) / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 5e-236)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(fma(t_0, 100.0, -100.0) / i));
	else
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(i, -0.5, 1.0) / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $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, 5e-236], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(t$95$0 * 100.0 + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(1.0 / N[(N[(i * -0.5 + 1.0), $MachinePrecision] / 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)) < 4.9999999999999998e-236

   1. Initial program 29.7%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-log1p.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if 4.9999999999999998e-236 < (/.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 95.6%

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

      1. lift-*.f64 N/A

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

      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. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 95.9

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

   4. Applied rewrites 95.9%

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

   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. 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. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 3.0

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

   4. Applied rewrites 3.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 2.7%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in n around inf

       \[\leadsto 100 \cdot \frac{1}{\frac{1 + \frac{-1}{2} \cdot i}{\color{blue}{n}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-236}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{fma}\left({\left(1 + \frac{i}{n}\right)}^{n}, 100, -100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(i, -0.5, 1\right)}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-5}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{i}{n}, \mathsf{fma}\left(i, 0.5, -1\right)\right)}{-n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3e-5)
   (* n (fma (/ (* i (exp i)) n) -50.0 (* 100.0 (/ (expm1 i) i))))
   (if (<= n 1.15e-6)
     (* 100.0 (/ 1.0 (/ (fma -0.5 (/ i n) (fma i 0.5 -1.0)) (- n))))
     (* n (/ (* 100.0 (expm1 i)) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3e-5) {
		tmp = n * fma(((i * exp(i)) / n), -50.0, (100.0 * (expm1(i) / i)));
	} else if (n <= 1.15e-6) {
		tmp = 100.0 * (1.0 / (fma(-0.5, (i / n), fma(i, 0.5, -1.0)) / -n));
	} else {
		tmp = n * ((100.0 * expm1(i)) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3e-5)
		tmp = Float64(n * fma(Float64(Float64(i * exp(i)) / n), -50.0, Float64(100.0 * Float64(expm1(i) / i))));
	elseif (n <= 1.15e-6)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(-0.5, Float64(i / n), fma(i, 0.5, -1.0)) / Float64(-n))));
	else
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3e-5], N[(n * N[(N[(N[(i * N[Exp[i], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * -50.0 + N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-6], N[(100.0 * N[(1.0 / N[(N[(-0.5 * N[(i / n), $MachinePrecision] + N[(i * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.00000000000000008e-5

   1. Initial program 31.0%

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

   3. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-expm1.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -3.00000000000000008e-5 < n < 1.15e-6

   1. Initial program 44.6%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 27.6

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

   4. Applied rewrites 27.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 10.1%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 78.3

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

   9. Applied rewrites 78.3%

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

   10. Taylor expanded in n around -inf

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

   12. Applied rewrites 83.5%

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

   if 1.15e-6 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-5}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(\frac{i \cdot e^{i}}{n}, -50, 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{i}{n}, \mathsf{fma}\left(i, 0.5, -1\right)\right)}{-n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-5}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{i}{n}, \mathsf{fma}\left(i, 0.5, -1\right)\right)}{-n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3e-5)
   (* 100.0 (/ (* n (expm1 i)) i))
   (if (<= n 1.15e-6)
     (* 100.0 (/ 1.0 (/ (fma -0.5 (/ i n) (fma i 0.5 -1.0)) (- n))))
     (* n (/ (* 100.0 (expm1 i)) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3e-5) {
		tmp = 100.0 * ((n * expm1(i)) / i);
	} else if (n <= 1.15e-6) {
		tmp = 100.0 * (1.0 / (fma(-0.5, (i / n), fma(i, 0.5, -1.0)) / -n));
	} else {
		tmp = n * ((100.0 * expm1(i)) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3e-5)
		tmp = Float64(100.0 * Float64(Float64(n * expm1(i)) / i));
	elseif (n <= 1.15e-6)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(-0.5, Float64(i / n), fma(i, 0.5, -1.0)) / Float64(-n))));
	else
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3e-5], N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-6], N[(100.0 * N[(1.0 / N[(N[(-0.5 * N[(i / n), $MachinePrecision] + N[(i * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.00000000000000008e-5

   1. Initial program 31.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      3. lower-expm1.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -3.00000000000000008e-5 < n < 1.15e-6

   1. Initial program 44.6%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 27.6

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

   4. Applied rewrites 27.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 10.1%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 78.3

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

   9. Applied rewrites 78.3%

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

   10. Taylor expanded in n around -inf

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

   12. Applied rewrites 83.5%

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

   if 1.15e-6 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.7%

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

Alternative 4: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{i}{n}, \mathsf{fma}\left(i, 0.5, -1\right)\right)}{-n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (/ (* 100.0 (expm1 i)) i))))
   (if (<= n -3.4e-5)
     t_0
     (if (<= n 1.15e-6)
       (* 100.0 (/ 1.0 (/ (fma -0.5 (/ i n) (fma i 0.5 -1.0)) (- n))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 * expm1(i)) / i);
	double tmp;
	if (n <= -3.4e-5) {
		tmp = t_0;
	} else if (n <= 1.15e-6) {
		tmp = 100.0 * (1.0 / (fma(-0.5, (i / n), fma(i, 0.5, -1.0)) / -n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
	tmp = 0.0
	if (n <= -3.4e-5)
		tmp = t_0;
	elseif (n <= 1.15e-6)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(-0.5, Float64(i / n), fma(i, 0.5, -1.0)) / Float64(-n))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.4e-5], t$95$0, If[LessEqual[n, 1.15e-6], N[(100.0 * N[(1.0 / N[(N[(-0.5 * N[(i / n), $MachinePrecision] + N[(i * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.4e-5 or 1.15e-6 < n

   1. Initial program 24.6%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 90.4%

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

   if -3.4e-5 < n < 1.15e-6

   1. Initial program 44.6%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 27.6

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

   4. Applied rewrites 27.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 10.1%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 78.3

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

   9. Applied rewrites 78.3%

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

   10. Taylor expanded in n around -inf

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

   12. Applied rewrites 83.5%

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

Alternative 5: 67.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-168}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(i, -0.5, 1\right)}{n}}\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{i \cdot 0.5}{n \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), n \cdot 0.5\right), n\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+142)
   (fma
    n
    (fma 50.0 i 100.0)
    (* (* i i) (* n (fma 4.166666666666667 i 16.666666666666668))))
   (if (<= n -5.5e-168)
     (* 100.0 (/ 1.0 (/ (fma i -0.5 1.0) n)))
     (if (<= n 2.65e-52)
       (* 100.0 (/ 1.0 (/ (* i 0.5) (* n n))))
       (*
        100.0
        (/
         (*
          i
          (fma
           i
           (fma
            i
            (* n (fma 0.041666666666666664 i 0.16666666666666666))
            (* n 0.5))
           n))
         i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+142) {
		tmp = fma(n, fma(50.0, i, 100.0), ((i * i) * (n * fma(4.166666666666667, i, 16.666666666666668))));
	} else if (n <= -5.5e-168) {
		tmp = 100.0 * (1.0 / (fma(i, -0.5, 1.0) / n));
	} else if (n <= 2.65e-52) {
		tmp = 100.0 * (1.0 / ((i * 0.5) / (n * n)));
	} else {
		tmp = 100.0 * ((i * fma(i, fma(i, (n * fma(0.041666666666666664, i, 0.16666666666666666)), (n * 0.5)), n)) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+142)
		tmp = fma(n, fma(50.0, i, 100.0), Float64(Float64(i * i) * Float64(n * fma(4.166666666666667, i, 16.666666666666668))));
	elseif (n <= -5.5e-168)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(i, -0.5, 1.0) / n)));
	elseif (n <= 2.65e-52)
		tmp = Float64(100.0 * Float64(1.0 / Float64(Float64(i * 0.5) / Float64(n * n))));
	else
		tmp = Float64(100.0 * Float64(Float64(i * fma(i, fma(i, Float64(n * fma(0.041666666666666664, i, 0.16666666666666666)), Float64(n * 0.5)), n)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+142], N[(n * N[(50.0 * i + 100.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5.5e-168], N[(100.0 * N[(1.0 / N[(N[(i * -0.5 + 1.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.65e-52], N[(100.0 * N[(1.0 / N[(N[(i * 0.5), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i * N[(i * N[(i * N[(n * N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.8e142

   1. Initial program 20.5%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 73.3%

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

   if -2.8e142 < n < -5.4999999999999999e-168

   1. Initial program 37.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      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. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 37.7

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

   4. Applied rewrites 37.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 14.9%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in n around inf

       \[\leadsto 100 \cdot \frac{1}{\frac{1 + \frac{-1}{2} \cdot i}{\color{blue}{n}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 65.7%

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

   if -5.4999999999999999e-168 < n < 2.6500000000000002e-52

   1. Initial program 53.8%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 27.1

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

   4. Applied rewrites 27.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 10.3%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 80.9

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

   9. Applied rewrites 80.9%

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

   10. Taylor expanded in n around 0

       \[\leadsto 100 \cdot \frac{1}{\frac{1}{2} \cdot \color{blue}{\frac{i}{{n}^{2}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 72.9%

       \[\leadsto 100 \cdot \frac{1}{\frac{i \cdot 0.5}{\color{blue}{n \cdot n}}} \]

   if 2.6500000000000002e-52 < n

   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. lower-*.f64 N/A

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

      3. lower-expm1.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot \left(n + i \cdot \left(\frac{1}{2} \cdot n + i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right)}{i} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.7%

      \[\leadsto 100 \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), n \cdot 0.5\right), n\right)}{i} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{i}{n}, \mathsf{fma}\left(i, 0.5, -1\right)\right)}{-n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), n \cdot 0.5\right), n\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+142)
   (fma
    n
    (fma 50.0 i 100.0)
    (* (* i i) (* n (fma 4.166666666666667 i 16.666666666666668))))
   (if (<= n 1.15e-6)
     (* 100.0 (/ 1.0 (/ (fma -0.5 (/ i n) (fma i 0.5 -1.0)) (- n))))
     (*
      100.0
      (/
       (*
        i
        (fma
         i
         (fma
          i
          (* n (fma 0.041666666666666664 i 0.16666666666666666))
          (* n 0.5))
         n))
       i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+142) {
		tmp = fma(n, fma(50.0, i, 100.0), ((i * i) * (n * fma(4.166666666666667, i, 16.666666666666668))));
	} else if (n <= 1.15e-6) {
		tmp = 100.0 * (1.0 / (fma(-0.5, (i / n), fma(i, 0.5, -1.0)) / -n));
	} else {
		tmp = 100.0 * ((i * fma(i, fma(i, (n * fma(0.041666666666666664, i, 0.16666666666666666)), (n * 0.5)), n)) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+142)
		tmp = fma(n, fma(50.0, i, 100.0), Float64(Float64(i * i) * Float64(n * fma(4.166666666666667, i, 16.666666666666668))));
	elseif (n <= 1.15e-6)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(-0.5, Float64(i / n), fma(i, 0.5, -1.0)) / Float64(-n))));
	else
		tmp = Float64(100.0 * Float64(Float64(i * fma(i, fma(i, Float64(n * fma(0.041666666666666664, i, 0.16666666666666666)), Float64(n * 0.5)), n)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+142], N[(n * N[(50.0 * i + 100.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-6], N[(100.0 * N[(1.0 / N[(N[(-0.5 * N[(i / n), $MachinePrecision] + N[(i * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / (-n)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i * N[(i * N[(i * N[(n * N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8e142

   1. Initial program 20.5%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 73.3%

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

   if -2.8e142 < n < 1.15e-6

   1. Initial program 43.8%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 32.9

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

   4. Applied rewrites 32.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 13.7%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 71.6

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

   9. Applied rewrites 71.6%

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

   10. Taylor expanded in n around -inf

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

   12. Applied rewrites 75.0%

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

   if 1.15e-6 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      3. lower-expm1.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot \left(n + i \cdot \left(\frac{1}{2} \cdot n + i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right)}{i} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.9%

      \[\leadsto 100 \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), n \cdot 0.5\right), n\right)}{i} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(n, \mathsf{fma}\left(i, -0.5, 1\right), i \cdot 0.5\right)}{n \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), n \cdot 0.5\right), n\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.2e+140)
   (fma
    n
    (fma 50.0 i 100.0)
    (* (* i i) (* n (fma 4.166666666666667 i 16.666666666666668))))
   (if (<= n 1.15e-6)
     (* 100.0 (/ 1.0 (/ (fma n (fma i -0.5 1.0) (* i 0.5)) (* n n))))
     (*
      100.0
      (/
       (*
        i
        (fma
         i
         (fma
          i
          (* n (fma 0.041666666666666664 i 0.16666666666666666))
          (* n 0.5))
         n))
       i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.2e+140) {
		tmp = fma(n, fma(50.0, i, 100.0), ((i * i) * (n * fma(4.166666666666667, i, 16.666666666666668))));
	} else if (n <= 1.15e-6) {
		tmp = 100.0 * (1.0 / (fma(n, fma(i, -0.5, 1.0), (i * 0.5)) / (n * n)));
	} else {
		tmp = 100.0 * ((i * fma(i, fma(i, (n * fma(0.041666666666666664, i, 0.16666666666666666)), (n * 0.5)), n)) / i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.2e+140)
		tmp = fma(n, fma(50.0, i, 100.0), Float64(Float64(i * i) * Float64(n * fma(4.166666666666667, i, 16.666666666666668))));
	elseif (n <= 1.15e-6)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(n, fma(i, -0.5, 1.0), Float64(i * 0.5)) / Float64(n * n))));
	else
		tmp = Float64(100.0 * Float64(Float64(i * fma(i, fma(i, Float64(n * fma(0.041666666666666664, i, 0.16666666666666666)), Float64(n * 0.5)), n)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.2e+140], N[(n * N[(50.0 * i + 100.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-6], N[(100.0 * N[(1.0 / N[(N[(n * N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * 0.5), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(i * N[(i * N[(i * N[(n * N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(n * 0.5), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.2000000000000002e140

   1. Initial program 20.5%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 73.3%

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

   if -5.2000000000000002e140 < n < 1.15e-6

   1. Initial program 43.8%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 32.9

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

   4. Applied rewrites 32.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 13.7%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 71.6

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

   9. Applied rewrites 71.6%

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

   10. Taylor expanded in n around 0

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

   12. Applied rewrites 71.1%

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

   if 1.15e-6 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   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. lower-*.f64 N/A

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

      3. lower-expm1.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{i \cdot \left(n + i \cdot \left(\frac{1}{2} \cdot n + i \cdot \left(\frac{1}{24} \cdot \left(i \cdot n\right) + \frac{1}{6} \cdot n\right)\right)\right)}{i} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.9%

      \[\leadsto 100 \cdot \frac{i \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), n \cdot 0.5\right), n\right)}{i} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-168}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(i, -0.5, 1\right)}{n}}\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{i \cdot 0.5}{n \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+142)
   (fma
    n
    (fma 50.0 i 100.0)
    (* (* i i) (* n (fma 4.166666666666667 i 16.666666666666668))))
   (if (<= n -5.5e-168)
     (* 100.0 (/ 1.0 (/ (fma i -0.5 1.0) n)))
     (if (<= n 2.65e-52)
       (* 100.0 (/ 1.0 (/ (* i 0.5) (* n n))))
       (*
        n
        (fma
         i
         (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
         100.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+142) {
		tmp = fma(n, fma(50.0, i, 100.0), ((i * i) * (n * fma(4.166666666666667, i, 16.666666666666668))));
	} else if (n <= -5.5e-168) {
		tmp = 100.0 * (1.0 / (fma(i, -0.5, 1.0) / n));
	} else if (n <= 2.65e-52) {
		tmp = 100.0 * (1.0 / ((i * 0.5) / (n * n)));
	} else {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+142)
		tmp = fma(n, fma(50.0, i, 100.0), Float64(Float64(i * i) * Float64(n * fma(4.166666666666667, i, 16.666666666666668))));
	elseif (n <= -5.5e-168)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(i, -0.5, 1.0) / n)));
	elseif (n <= 2.65e-52)
		tmp = Float64(100.0 * Float64(1.0 / Float64(Float64(i * 0.5) / Float64(n * n))));
	else
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+142], N[(n * N[(50.0 * i + 100.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5.5e-168], N[(100.0 * N[(1.0 / N[(N[(i * -0.5 + 1.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.65e-52], N[(100.0 * N[(1.0 / N[(N[(i * 0.5), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.8e142

   1. Initial program 20.5%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 73.3%

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

   if -2.8e142 < n < -5.4999999999999999e-168

   1. Initial program 37.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      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. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 37.7

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

   4. Applied rewrites 37.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 14.9%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in n around inf

       \[\leadsto 100 \cdot \frac{1}{\frac{1 + \frac{-1}{2} \cdot i}{\color{blue}{n}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 65.7%

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

   if -5.4999999999999999e-168 < n < 2.6500000000000002e-52

   1. Initial program 53.8%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 27.1

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

   4. Applied rewrites 27.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 10.3%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 80.9

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

   9. Applied rewrites 80.9%

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

   10. Taylor expanded in n around 0

       \[\leadsto 100 \cdot \frac{1}{\frac{1}{2} \cdot \color{blue}{\frac{i}{{n}^{2}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 72.9%

       \[\leadsto 100 \cdot \frac{1}{\frac{i \cdot 0.5}{\color{blue}{n \cdot n}}} \]

   if 2.6500000000000002e-52 < n

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 92.7

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 84.0%

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

Alternative 9: 66.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-168}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(i, -0.5, 1\right)}{n}}\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{-1 + 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+142)
   (fma
    n
    (fma 50.0 i 100.0)
    (* (* i i) (* n (fma 4.166666666666667 i 16.666666666666668))))
   (if (<= n -5.5e-168)
     (* 100.0 (/ 1.0 (/ (fma i -0.5 1.0) n)))
     (if (<= n 2.65e-52)
       (* 100.0 (/ (+ -1.0 1.0) (/ i n)))
       (*
        n
        (fma
         i
         (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
         100.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+142) {
		tmp = fma(n, fma(50.0, i, 100.0), ((i * i) * (n * fma(4.166666666666667, i, 16.666666666666668))));
	} else if (n <= -5.5e-168) {
		tmp = 100.0 * (1.0 / (fma(i, -0.5, 1.0) / n));
	} else if (n <= 2.65e-52) {
		tmp = 100.0 * ((-1.0 + 1.0) / (i / n));
	} else {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+142)
		tmp = fma(n, fma(50.0, i, 100.0), Float64(Float64(i * i) * Float64(n * fma(4.166666666666667, i, 16.666666666666668))));
	elseif (n <= -5.5e-168)
		tmp = Float64(100.0 * Float64(1.0 / Float64(fma(i, -0.5, 1.0) / n)));
	elseif (n <= 2.65e-52)
		tmp = Float64(100.0 * Float64(Float64(-1.0 + 1.0) / Float64(i / n)));
	else
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+142], N[(n * N[(50.0 * i + 100.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -5.5e-168], N[(100.0 * N[(1.0 / N[(N[(i * -0.5 + 1.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.65e-52], N[(100.0 * N[(N[(-1.0 + 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.8e142

   1. Initial program 20.5%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 73.3%

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

   if -2.8e142 < n < -5.4999999999999999e-168

   1. Initial program 37.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      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. sub-neg N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 37.7

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

   4. Applied rewrites 37.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 14.9%

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

   7. Taylor expanded in i around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in n around inf

       \[\leadsto 100 \cdot \frac{1}{\frac{1 + \frac{-1}{2} \cdot i}{\color{blue}{n}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 65.7%

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

   if -5.4999999999999999e-168 < n < 2.6500000000000002e-52

   1. Initial program 53.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 72.6%

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

   if 2.6500000000000002e-52 < n

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 92.7

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 84.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq -5.5 \cdot 10^{-168}:\\ \;\;\;\;100 \cdot \frac{1}{\frac{\mathsf{fma}\left(i, -0.5, 1\right)}{n}}\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{-1 + 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{-1 + 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.15e-109)
   (fma
    n
    (fma 50.0 i 100.0)
    (* (* i i) (* n (fma 4.166666666666667 i 16.666666666666668))))
   (if (<= n 2.65e-52)
     (* 100.0 (/ (+ -1.0 1.0) (/ i n)))
     (*
      n
      (fma
       i
       (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
       100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.15e-109) {
		tmp = fma(n, fma(50.0, i, 100.0), ((i * i) * (n * fma(4.166666666666667, i, 16.666666666666668))));
	} else if (n <= 2.65e-52) {
		tmp = 100.0 * ((-1.0 + 1.0) / (i / n));
	} else {
		tmp = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.15e-109)
		tmp = fma(n, fma(50.0, i, 100.0), Float64(Float64(i * i) * Float64(n * fma(4.166666666666667, i, 16.666666666666668))));
	elseif (n <= 2.65e-52)
		tmp = Float64(100.0 * Float64(Float64(-1.0 + 1.0) / Float64(i / n)));
	else
		tmp = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.15e-109], N[(n * N[(50.0 * i + 100.0), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.65e-52], N[(100.0 * N[(N[(-1.0 + 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1500000000000001e-109

   1. Initial program 29.3%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 63.5%

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

   if -1.1500000000000001e-109 < n < 2.6500000000000002e-52

   1. Initial program 52.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.1%

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

   if 2.6500000000000002e-52 < n

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 92.7

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 84.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(n, \mathsf{fma}\left(50, i, 100\right), \left(i \cdot i\right) \cdot \left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right)\right)\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{-1 + 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{-1 + 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (fma
           i
           (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
           100.0))))
   (if (<= n -1.15e-109)
     t_0
     (if (<= n 2.65e-52) (* 100.0 (/ (+ -1.0 1.0) (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	double tmp;
	if (n <= -1.15e-109) {
		tmp = t_0;
	} else if (n <= 2.65e-52) {
		tmp = 100.0 * ((-1.0 + 1.0) / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0))
	tmp = 0.0
	if (n <= -1.15e-109)
		tmp = t_0;
	elseif (n <= 2.65e-52)
		tmp = Float64(100.0 * Float64(Float64(-1.0 + 1.0) / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.15e-109], t$95$0, If[LessEqual[n, 2.65e-52], N[(100.0 * N[(N[(-1.0 + 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.1500000000000001e-109 or 2.6500000000000002e-52 < n

   1. Initial program 24.6%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 73.7%

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

   if -1.1500000000000001e-109 < n < 2.6500000000000002e-52

   1. Initial program 52.8%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 69.1%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;100 \cdot \frac{-1 + 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, \mathsf{fma}\left(i, 4.166666666666667, 16.666666666666668\right), 50\right), 100\right)\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          n
          (fma
           i
           (fma i (fma i 4.166666666666667 16.666666666666668) 50.0)
           100.0))))
   (if (<= n -1.15e-109) t_0 (if (<= n 2.65e-52) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0);
	double tmp;
	if (n <= -1.15e-109) {
		tmp = t_0;
	} else if (n <= 2.65e-52) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, fma(i, 4.166666666666667, 16.666666666666668), 50.0), 100.0))
	tmp = 0.0
	if (n <= -1.15e-109)
		tmp = t_0;
	elseif (n <= 2.65e-52)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * N[(i * 4.166666666666667 + 16.666666666666668), $MachinePrecision] + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.15e-109], t$95$0, If[LessEqual[n, 2.65e-52], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.1500000000000001e-109 or 2.6500000000000002e-52 < n

   1. Initial program 24.6%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 73.7%

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

   if -1.1500000000000001e-109 < n < 2.6500000000000002e-52

   1. Initial program 52.8%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 29.3

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 69.1

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 69.1%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), n \cdot 100\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.15e-109)
   (* n (fma i (fma i 16.666666666666668 50.0) 100.0))
   (if (<= n 2.65e-52)
     0.0
     (fma i (* n (fma 16.666666666666668 i 50.0)) (* n 100.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.15e-109) {
		tmp = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	} else if (n <= 2.65e-52) {
		tmp = 0.0;
	} else {
		tmp = fma(i, (n * fma(16.666666666666668, i, 50.0)), (n * 100.0));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.15e-109)
		tmp = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0));
	elseif (n <= 2.65e-52)
		tmp = 0.0;
	else
		tmp = fma(i, Float64(n * fma(16.666666666666668, i, 50.0)), Float64(n * 100.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.15e-109], N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.65e-52], 0.0, N[(i * N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] + N[(n * 100.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1500000000000001e-109

   1. Initial program 29.3%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 81.3

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

   5. Applied rewrites 81.3%

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 62.9%

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

   if -1.1500000000000001e-109 < n < 2.6500000000000002e-52

   1. Initial program 52.8%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 29.3

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 69.1

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 69.1%

       \[\leadsto 0 \]

   if 2.6500000000000002e-52 < n

   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-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 80.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), n \cdot 100\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(i, 16.666666666666668, 50\right), 100\right)\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma i (fma i 16.666666666666668 50.0) 100.0))))
   (if (<= n -1.15e-109) t_0 (if (<= n 2.65e-52) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0);
	double tmp;
	if (n <= -1.15e-109) {
		tmp = t_0;
	} else if (n <= 2.65e-52) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(i, fma(i, 16.666666666666668, 50.0), 100.0))
	tmp = 0.0
	if (n <= -1.15e-109)
		tmp = t_0;
	elseif (n <= 2.65e-52)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(i * N[(i * 16.666666666666668 + 50.0), $MachinePrecision] + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.15e-109], t$95$0, If[LessEqual[n, 2.65e-52], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.1500000000000001e-109 or 2.6500000000000002e-52 < n

   1. Initial program 24.6%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in i around 0

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

   10. Applied rewrites 71.6%

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

   if -1.1500000000000001e-109 < n < 2.6500000000000002e-52

   1. Initial program 52.8%

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

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

      7. lift-/.f64 N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      13. lower-neg.f64 29.3

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 29.3%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 69.1

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 69.1%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n, i \cdot 0.5, n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.15e-109)
   (* n (fma 50.0 i 100.0))
   (if (<= n 2.65e-52) 0.0 (* 100.0 (fma n (* i 0.5) n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.15e-109) {
		tmp = n * fma(50.0, i, 100.0);
	} else if (n <= 2.65e-52) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * fma(n, (i * 0.5), n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.15e-109)
		tmp = Float64(n * fma(50.0, i, 100.0));
	elseif (n <= 2.65e-52)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * fma(n, Float64(i * 0.5), n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.15e-109], N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.65e-52], 0.0, N[(100.0 * N[(n * N[(i * 0.5), $MachinePrecision] + n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.1500000000000001e-109

   1. Initial program 29.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right) \cdot 100}}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right) \cdot 100}}{i} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)} \cdot 100}{i} \]

      6. lower-expm1.f64 81.3

         \[\leadsto \frac{\left(n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}\right) \cdot 100}{i} \]

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot 100}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.1%

      \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]

   if -1.1500000000000001e-109 < n < 2.6500000000000002e-52

   1. Initial program 52.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      13. lower-neg.f64 29.3

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 29.3%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 69.1

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 69.1%

       \[\leadsto 0 \]

   if 2.6500000000000002e-52 < n

   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. lower-*.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{n \cdot \left(e^{i} - 1\right)}}{i} \]

      3. lower-expm1.f64 93.0

         \[\leadsto 100 \cdot \frac{n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}}{i} \]

   5. Applied rewrites 93.0%

      \[\leadsto 100 \cdot \color{blue}{\frac{n \cdot \mathsf{expm1}\left(i\right)}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \left(n + \color{blue}{\frac{1}{2} \cdot \left(i \cdot n\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 73.3%

      \[\leadsto 100 \cdot \mathsf{fma}\left(n, \color{blue}{0.5 \cdot i}, n\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \mathsf{fma}\left(n, i \cdot 0.5, n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \mathsf{fma}\left(50, i, 100\right)\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (fma 50.0 i 100.0))))
   (if (<= n -1.15e-109) t_0 (if (<= n 2.65e-52) 0.0 t_0))))

C

double code(double i, double n) {
	double t_0 = n * fma(50.0, i, 100.0);
	double tmp;
	if (n <= -1.15e-109) {
		tmp = t_0;
	} else if (n <= 2.65e-52) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(n * fma(50.0, i, 100.0))
	tmp = 0.0
	if (n <= -1.15e-109)
		tmp = t_0;
	elseif (n <= 2.65e-52)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(50.0 * i + 100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.15e-109], t$95$0, If[LessEqual[n, 2.65e-52], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.1500000000000001e-109 or 2.6500000000000002e-52 < n

   1. Initial program 24.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)}{i}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right) \cdot 100}}{i} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right) \cdot 100}}{i} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(n \cdot \left(e^{i} - 1\right)\right)} \cdot 100}{i} \]

      6. lower-expm1.f64 87.0

         \[\leadsto \frac{\left(n \cdot \color{blue}{\mathsf{expm1}\left(i\right)}\right) \cdot 100}{i} \]

   5. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\frac{\left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot 100}{i}} \]

   6. Taylor expanded in i around 0

      \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{100 \cdot n} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.6%

      \[\leadsto n \cdot \color{blue}{\mathsf{fma}\left(50, i, 100\right)} \]

   if -1.1500000000000001e-109 < n < 2.6500000000000002e-52

   1. Initial program 52.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      13. lower-neg.f64 29.3

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 29.3%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 69.1

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 69.1%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.6 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+65}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.6e+84) 0.0 (if (<= i 7e+65) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.6e+84) {
		tmp = 0.0;
	} else if (i <= 7e+65) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.6d+84)) then
        tmp = 0.0d0
    else if (i <= 7d+65) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.6e+84) {
		tmp = 0.0;
	} else if (i <= 7e+65) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.6e+84:
		tmp = 0.0
	elif i <= 7e+65:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.6e+84)
		tmp = 0.0;
	elseif (i <= 7e+65)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.6e+84)
		tmp = 0.0;
	elseif (i <= 7e+65)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.6e+84], 0.0, If[LessEqual[i, 7e+65], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.60000000000000005e84 or 7.0000000000000002e65 < i

   1. Initial program 71.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      13. lower-neg.f64 68.1

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 68.1%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 36.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 36.7%

       \[\leadsto 0 \]

   if -1.60000000000000005e84 < i < 7.0000000000000002e65

   1. Initial program 9.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{n \cdot 100} \]

      2. lower-*.f64 77.9

         \[\leadsto \color{blue}{n \cdot 100} \]

   5. Applied rewrites 77.9%

      \[\leadsto \color{blue}{n \cdot 100} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 17.7% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 30.7%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
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. sub-neg N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

   8. associate-/r/ N/A

      \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

   13. lower-neg.f64 26.6

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

4. Applied rewrites 26.6%

   \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

5. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

   3. metadata-eval N/A

      \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

   4. mul0-lft N/A

      \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{i} \]

   6. lower-/.f64 18.0

      \[\leadsto \color{blue}{\frac{0}{i}} \]

7. Applied rewrites 18.0%

   \[\leadsto \color{blue}{\frac{0}{i}} \]

8. Taylor expanded in i around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 18.0%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 34.7% 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 2024221 
(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))))