Compound Interest

Percentage Accurate: 28.5% → 99.4%

Time: 13.5s

Alternatives: 20

Speedup: 24.3×

• could not determine a ground truth

  1. i = 8.866141953250986e-174
  2. n = -1.4905376104236442e+36

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.5% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-148}:\\ \;\;\;\;\frac{100 \cdot \left(t\_0 \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -2e-148)
     (/ (- (* 100.0 (* t_0 n)) (* 100.0 n)) i)
     (if (<= t_1 0.0)
       (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ i n))
       (if (<= t_1 INFINITY)
         (fma (/ 100.0 i) (- n) (* (* (/ n i) 100.0) t_0))
         (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -2e-148) {
		tmp = ((100.0 * (t_0 * n)) - (100.0 * n)) / i;
	} else if (t_1 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((100.0 / i), -n, (((n / i) * 100.0) * t_0));
	} else {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -2e-148)
		tmp = Float64(Float64(Float64(100.0 * Float64(t_0 * n)) - Float64(100.0 * n)) / i);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(100.0 / i), Float64(-n), Float64(Float64(Float64(n / i) * 100.0) * t_0));
	else
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-148], N[(N[(N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision] - N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(100.0 / i), $MachinePrecision] * (-n) + N[(N[(N[(n / i), $MachinePrecision] * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1.99999999999999987e-148

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.4

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.5

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 72.8

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

   4. Applied rewrites 72.8%

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unsub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 99.7%

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

   if -1.99999999999999987e-148 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -0.0

   1. Initial program 18.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 18.0

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 99.7

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

   4. Applied rewrites 99.7%

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

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

   1. Initial program 97.6%

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

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

      3. lower-*.f64 97.6

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 52.0

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

   4. Applied rewrites 52.0%

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

   6. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 98.0

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

   8. Applied rewrites 98.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -2 \cdot 10^{-148}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-110}:\\ \;\;\;\;\frac{100 \cdot \left(t\_0 \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i} \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -5e-110)
     (/ (- (* 100.0 (* t_0 n)) (* 100.0 n)) i)
     (if (<= t_1 0.0)
       (* (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) i) n)
       (if (<= t_1 INFINITY)
         (fma (/ 100.0 i) (- n) (* (* (/ n i) 100.0) t_0))
         (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-110) {
		tmp = ((100.0 * (t_0 * n)) - (100.0 * n)) / i;
	} else if (t_1 <= 0.0) {
		tmp = ((expm1((log1p((i / n)) * n)) * 100.0) / i) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((100.0 / i), -n, (((n / i) * 100.0) * t_0));
	} else {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-110)
		tmp = Float64(Float64(Float64(100.0 * Float64(t_0 * n)) - Float64(100.0 * n)) / i);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / i) * n);
	elseif (t_1 <= Inf)
		tmp = fma(Float64(100.0 / i), Float64(-n), Float64(Float64(Float64(n / i) * 100.0) * t_0));
	else
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-110], N[(N[(N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision] - N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(100.0 / i), $MachinePrecision] * (-n) + N[(N[(N[(n / i), $MachinePrecision] * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -5e-110

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 68.9

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

   4. Applied rewrites 68.9%

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unsub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if -5e-110 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -0.0

   1. Initial program 18.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 18.9

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 18.9

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-expm1.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log1p.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. associate-*l/ N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

   6. Applied rewrites 98.9%

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

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

   1. Initial program 97.6%

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

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

      3. lower-*.f64 97.6

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 52.0

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

   4. Applied rewrites 52.0%

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

   6. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 98.0

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

   8. Applied rewrites 98.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -5 \cdot 10^{-110}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i} \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-110}:\\ \;\;\;\;\frac{100 \cdot \left(t\_0 \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 -5e-110)
     (/ (- (* 100.0 (* t_0 n)) (* 100.0 n)) i)
     (if (<= t_1 0.0)
       (* (* (/ 100.0 i) (expm1 (* (log1p (/ i n)) n))) n)
       (if (<= t_1 INFINITY)
         (fma (/ 100.0 i) (- n) (* (* (/ n i) 100.0) t_0))
         (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-110) {
		tmp = ((100.0 * (t_0 * n)) - (100.0 * n)) / i;
	} else if (t_1 <= 0.0) {
		tmp = ((100.0 / i) * expm1((log1p((i / n)) * n))) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((100.0 / i), -n, (((n / i) * 100.0) * t_0));
	} else {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-110)
		tmp = Float64(Float64(Float64(100.0 * Float64(t_0 * n)) - Float64(100.0 * n)) / i);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(100.0 / i) * expm1(Float64(log1p(Float64(i / n)) * n))) * n);
	elseif (t_1 <= Inf)
		tmp = fma(Float64(100.0 / i), Float64(-n), Float64(Float64(Float64(n / i) * 100.0) * t_0));
	else
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-110], N[(N[(N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision] - N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(100.0 / i), $MachinePrecision] * (-n) + N[(N[(N[(n / i), $MachinePrecision] * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -5e-110

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 68.9

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

   4. Applied rewrites 68.9%

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unsub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if -5e-110 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -0.0

   1. Initial program 18.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-/.f64 98.2

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

   4. Applied rewrites 98.2%

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

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

   1. Initial program 97.6%

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

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

      3. lower-*.f64 97.6

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 52.0

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

   4. Applied rewrites 52.0%

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

   6. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 98.0

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

   8. Applied rewrites 98.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -5 \cdot 10^{-110}:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* (* (/ (expm1 i) i) 100.0) n)
     (if (<= t_1 INFINITY)
       (fma (/ 100.0 i) (- n) (* (* (/ n i) 100.0) t_0))
       (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((100.0 / i), -n, (((n / i) * 100.0) * t_0));
	} else {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (t_1 <= Inf)
		tmp = fma(Float64(100.0 / i), Float64(-n), Float64(Float64(Float64(n / i) * 100.0) * t_0));
	else
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(100.0 / i), $MachinePrecision] * (-n) + N[(N[(N[(n / i), $MachinePrecision] * 100.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $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)) < -0.0

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 78.7

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

   5. Applied rewrites 78.7%

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

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

   1. Initial program 97.6%

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

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

      3. lower-*.f64 97.6

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 52.0

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

   4. Applied rewrites 52.0%

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

   6. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-pow.f64 98.0

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

   8. Applied rewrites 98.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{100}{i}, -n, \left(\frac{n}{i} \cdot 100\right) \cdot {\left(\frac{i}{n} + 1\right)}^{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{n}{i} \cdot t\_0 - \frac{n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* (* (/ (expm1 i) i) 100.0) n)
     (if (<= t_1 INFINITY)
       (* (- (* (/ n i) t_0) (/ n i)) 100.0)
       (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((n / i) * t_0) - (n / i)) * 100.0;
	} else {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(n / i) * t_0) - Float64(n / i)) * 100.0);
	else
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(n / i), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(n / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $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)) < -0.0

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 78.7

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

   5. Applied rewrites 78.7%

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

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

   1. Initial program 97.6%

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

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

      3. lower-*.f64 97.6

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 52.0

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

   4. Applied rewrites 52.0%

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

   6. Applied rewrites 97.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{n}{i} \cdot {\left(\frac{i}{n} + 1\right)}^{n} - \frac{n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{100 \cdot \left(t\_0 \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* (* (/ (expm1 i) i) 100.0) n)
     (if (<= t_1 INFINITY)
       (/ (- (* 100.0 (* t_0 n)) (* 100.0 n)) i)
       (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((100.0 * (t_0 * n)) - (100.0 * n)) / i;
	} else {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(100.0 * Float64(t_0 * n)) - Float64(100.0 * n)) / i);
	else
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(100.0 * N[(t$95$0 * n), $MachinePrecision]), $MachinePrecision] - N[(100.0 * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $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)) < -0.0

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 78.7

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

   5. Applied rewrites 78.7%

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

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

   1. Initial program 97.6%

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

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

      3. lower-*.f64 97.6

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 52.0

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

   4. Applied rewrites 52.0%

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

   6. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-lft-neg-out N/A

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

      5. unsub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{100 \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} \cdot n\right) - 100 \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n} - 1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot \frac{n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (pow (+ (/ i n) 1.0) n) 1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 0.0)
     (* (* (/ (expm1 i) i) 100.0) n)
     (if (<= t_1 INFINITY)
       (* (* t_0 (/ n i)) 100.0)
       (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n) - 1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * (n / i)) * 100.0;
	} else {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64((Float64(Float64(i / n) + 1.0) ^ n) - 1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * Float64(n / i)) * 100.0);
	else
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * N[(n / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $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)) < -0.0

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 78.7

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

   5. Applied rewrites 78.7%

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

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

   1. Initial program 97.6%

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

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

      3. lower-*.f64 97.6

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.6

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 52.0

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

   4. Applied rewrites 52.0%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log1p.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lift-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 97.6

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

   6. Applied rewrites 97.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.7%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right) \cdot \frac{n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-56}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7e-56)
   (* (* (/ (expm1 i) i) 100.0) n)
   (if (<= n 2.8e-75)
     (/
      n
      (fma
       (fma (fma (* i i) -1.388888888888889e-5 0.0008333333333333334) i -0.005)
       i
       0.01))
     (* (/ (* (expm1 i) 100.0) i) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7e-56) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (n <= 2.8e-75) {
		tmp = n / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01);
	} else {
		tmp = ((expm1(i) * 100.0) / i) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7e-56)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (n <= 2.8e-75)
		tmp = Float64(n / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01));
	else
		tmp = Float64(Float64(Float64(expm1(i) * 100.0) / i) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7e-56], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.8e-75], N[(n / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.9999999999999996e-56

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 85.1

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

   5. Applied rewrites 85.1%

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

   if -6.9999999999999996e-56 < n < 2.79999999999999998e-75

   1. Initial program 35.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 42.0

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

   5. Applied rewrites 42.0%

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

   7. Applied rewrites 41.9%

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

   9. Applied rewrites 42.0%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.5%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), \color{blue}{i}, 0.01\right)} \]

   if 2.79999999999999998e-75 < n

   1. Initial program 19.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-56}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(i\right) \cdot 100}{i} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -7 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -7e-56)
     t_0
     (if (<= n 2.8e-75)
       (/
        n
        (fma
         (fma
          (fma (* i i) -1.388888888888889e-5 0.0008333333333333334)
          i
          -0.005)
         i
         0.01))
       t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -7e-56) {
		tmp = t_0;
	} else if (n <= 2.8e-75) {
		tmp = n / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -7e-56)
		tmp = t_0;
	elseif (n <= 2.8e-75)
		tmp = Float64(n / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -7e-56], t$95$0, If[LessEqual[n, 2.8e-75], N[(n / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.9999999999999996e-56 or 2.79999999999999998e-75 < n

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -6.9999999999999996e-56 < n < 2.79999999999999998e-75

   1. Initial program 35.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 42.0

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

   5. Applied rewrites 42.0%

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

   7. Applied rewrites 41.9%

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

   9. Applied rewrites 42.0%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.5%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), \color{blue}{i}, 0.01\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-56}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7e-56)
   (/ n (fma -0.005 i 0.01))
   (if (<= n 2.8e-75)
     (/
      n
      (fma
       (fma (fma (* i i) -1.388888888888889e-5 0.0008333333333333334) i -0.005)
       i
       0.01))
     (fma
      n
      100.0
      (* (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7e-56) {
		tmp = n / fma(-0.005, i, 0.01);
	} else if (n <= 2.8e-75) {
		tmp = n / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01);
	} else {
		tmp = fma(n, 100.0, ((fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7e-56)
		tmp = Float64(n / fma(-0.005, i, 0.01));
	elseif (n <= 2.8e-75)
		tmp = Float64(n / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01));
	else
		tmp = fma(n, 100.0, Float64(Float64(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7e-56], N[(n / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8e-75], N[(n / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision], N[(n * 100.0 + N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.9999999999999996e-56

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.0%

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

   9. Applied rewrites 84.8%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{\frac{-1}{200} \cdot i}} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(-0.005, \color{blue}{i}, 0.01\right)} \]

   if -6.9999999999999996e-56 < n < 2.79999999999999998e-75

   1. Initial program 35.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 42.0

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

   5. Applied rewrites 42.0%

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

   7. Applied rewrites 41.9%

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

   9. Applied rewrites 42.0%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.5%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), \color{blue}{i}, 0.01\right)} \]

   if 2.79999999999999998e-75 < n

   1. Initial program 19.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 91.0

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

   5. Applied rewrites 91.0%

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

   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 79.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-56}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-55}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2e-55)
   (/ n (fma -0.005 i 0.01))
   (if (<= n 7.2e-8)
     (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01))
     (fma
      n
      100.0
      (* (* (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2e-55) {
		tmp = n / fma(-0.005, i, 0.01);
	} else if (n <= 7.2e-8) {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	} else {
		tmp = fma(n, 100.0, ((fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2e-55)
		tmp = Float64(n / fma(-0.005, i, 0.01));
	elseif (n <= 7.2e-8)
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	else
		tmp = fma(n, 100.0, Float64(Float64(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0) * n) * i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2e-55], N[(n / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.2e-8], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision], N[(n * 100.0 + N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * n), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.99999999999999999e-55

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.0%

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

   9. Applied rewrites 84.8%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{\frac{-1}{200} \cdot i}} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(-0.005, \color{blue}{i}, 0.01\right)} \]

   if -1.99999999999999999e-55 < n < 7.19999999999999962e-8

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 46.6%

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

   9. Applied rewrites 46.6%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 66.2%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]

   if 7.19999999999999962e-8 < n

   1. Initial program 21.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 94.8

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

   5. Applied rewrites 94.8%

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

   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 81.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(n \cdot \mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50 \cdot n\right), \color{blue}{i}, n \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 81.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-55}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right) \cdot n\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-55}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2e-55)
   (/ n (fma -0.005 i 0.01))
   (if (<= n 7.2e-8)
     (/ n (fma (fma 0.0008333333333333334 i -0.005) i 0.01))
     (*
      (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
      n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2e-55) {
		tmp = n / fma(-0.005, i, 0.01);
	} else if (n <= 7.2e-8) {
		tmp = n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01);
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2e-55)
		tmp = Float64(n / fma(-0.005, i, 0.01));
	elseif (n <= 7.2e-8)
		tmp = Float64(n / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01));
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2e-55], N[(n / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.2e-8], N[(n / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.99999999999999999e-55

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.0%

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

   9. Applied rewrites 84.8%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{\frac{-1}{200} \cdot i}} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.9%

       \[\leadsto \frac{n}{\mathsf{fma}\left(-0.005, \color{blue}{i}, 0.01\right)} \]

   if -1.99999999999999999e-55 < n < 7.19999999999999962e-8

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 46.7

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

   5. Applied rewrites 46.7%

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

   7. Applied rewrites 46.6%

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

   9. Applied rewrites 46.6%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 66.2%

       \[\leadsto \frac{n}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), \color{blue}{i}, 0.01\right)} \]

   if 7.19999999999999962e-8 < n

   1. Initial program 21.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 94.8

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 81.0%

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

Alternative 13: 63.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -6.2e-177) t_0 (if (<= n 5.2e-129) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -6.2e-177) {
		tmp = t_0;
	} else if (n <= 5.2e-129) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -6.2e-177)
		tmp = t_0;
	elseif (n <= 5.2e-129)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -6.2e-177], t$95$0, If[LessEqual[n, 5.2e-129], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.20000000000000036e-177 or 5.2000000000000001e-129 < n

   1. Initial program 22.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 64.2%

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

   if -6.20000000000000036e-177 < n < 5.2000000000000001e-129

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-*.f64 N/A

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

   4. Applied rewrites 20.6%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 67.3

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 64.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.8e-75)
   (/ n (fma -0.005 i 0.01))
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.8e-75) {
		tmp = n / fma(-0.005, i, 0.01);
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.8e-75)
		tmp = Float64(n / fma(-0.005, i, 0.01));
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.8e-75], N[(n / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.79999999999999998e-75

   1. Initial program 31.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.9%

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

   9. Applied rewrites 66.8%

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

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{\frac{-1}{200} \cdot i}} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.4%

       \[\leadsto \frac{n}{\mathsf{fma}\left(-0.005, \color{blue}{i}, 0.01\right)} \]

   if 2.79999999999999998e-75 < n

   1. Initial program 19.5%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 79.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, i, 1\right) \cdot \left(100 \cdot n\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 0.5 i 1.0) (* 100.0 n))))
   (if (<= n -6.2e-177) t_0 (if (<= n 5.2e-129) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(0.5, i, 1.0) * (100.0 * n);
	double tmp;
	if (n <= -6.2e-177) {
		tmp = t_0;
	} else if (n <= 5.2e-129) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(0.5, i, 1.0) * Float64(100.0 * n))
	tmp = 0.0
	if (n <= -6.2e-177)
		tmp = t_0;
	elseif (n <= 5.2e-129)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(0.5 * i + 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-177], t$95$0, If[LessEqual[n, 5.2e-129], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.20000000000000036e-177 or 5.2000000000000001e-129 < n

   1. Initial program 22.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 82.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.4%

      \[\leadsto \frac{\mathsf{expm1}\left(i\right)}{i} \cdot \color{blue}{\left(n \cdot 100\right)} \]

   8. Taylor expanded in i around 0

      \[\leadsto \left(1 + \frac{1}{2} \cdot i\right) \cdot \left(\color{blue}{n} \cdot 100\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 63.7%

       \[\leadsto \mathsf{fma}\left(0.5, i, 1\right) \cdot \left(\color{blue}{n} \cdot 100\right) \]

   if -6.20000000000000036e-177 < n < 5.2000000000000001e-129

   1. Initial program 48.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      4. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      6. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      7. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \cdot 100 + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \color{blue}{\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100}\right) \]

   4. Applied rewrites 20.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-n}{i} \cdot 100\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{n \cdot \left(-100 + 100\right)}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{n \cdot \color{blue}{0}}{i} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 67.3

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(100 \cdot n\right)\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma 50.0 i 100.0) n)))
   (if (<= n -6.2e-177) t_0 (if (<= n 5.2e-129) (/ 0.0 i) t_0))))

C

double code(double i, double n) {
	double t_0 = fma(50.0, i, 100.0) * n;
	double tmp;
	if (n <= -6.2e-177) {
		tmp = t_0;
	} else if (n <= 5.2e-129) {
		tmp = 0.0 / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(50.0, i, 100.0) * n)
	tmp = 0.0
	if (n <= -6.2e-177)
		tmp = t_0;
	elseif (n <= 5.2e-129)
		tmp = Float64(0.0 / i);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -6.2e-177], t$95$0, If[LessEqual[n, 5.2e-129], N[(0.0 / i), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -6.20000000000000036e-177 or 5.2000000000000001e-129 < n

   1. Initial program 22.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 82.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 63.7%

      \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]

   if -6.20000000000000036e-177 < n < 5.2000000000000001e-129

   1. Initial program 48.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      4. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      6. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      7. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} \cdot 100 + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i} \cdot n, 100, \left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \color{blue}{\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) \cdot 100}\right) \]

   4. Applied rewrites 20.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n, 100, \frac{-n}{i} \cdot 100\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{-100 \cdot n + 100 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{n \cdot \left(-100 + 100\right)}}{i} \]

      2. metadata-eval N/A

         \[\leadsto \frac{n \cdot \color{blue}{0}}{i} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      4. lower-/.f64 67.3

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 63.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{n}{\mathsf{fma}\left(-0.005, i, 0.01\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.8e-75)
   (/ n (fma -0.005 i 0.01))
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.8e-75) {
		tmp = n / fma(-0.005, i, 0.01);
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.8e-75)
		tmp = Float64(n / fma(-0.005, i, 0.01));
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.8e-75], N[(n / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.79999999999999998e-75

   1. Initial program 31.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 67.0

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i} \cdot n \]
   8. Step-by-step derivation

   9. Applied rewrites 66.8%

      \[\leadsto \frac{n}{\color{blue}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}}} \]

   10. Taylor expanded in i around 0

       \[\leadsto \frac{n}{\frac{1}{100} + \color{blue}{\frac{-1}{200} \cdot i}} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.4%

       \[\leadsto \frac{n}{\mathsf{fma}\left(-0.005, \color{blue}{i}, 0.01\right)} \]

   if 2.79999999999999998e-75 < n

   1. Initial program 19.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 91.0

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 74.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 54.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 3.7 \cdot 10^{+42}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(50 \cdot i\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 3.7e+42) (* 100.0 n) (* (* 50.0 i) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= 3.7e+42) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 3.7d+42) then
        tmp = 100.0d0 * n
    else
        tmp = (50.0d0 * i) * n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 3.7e+42) {
		tmp = 100.0 * n;
	} else {
		tmp = (50.0 * i) * n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 3.7e+42:
		tmp = 100.0 * n
	else:
		tmp = (50.0 * i) * n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 3.7e+42)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(Float64(50.0 * i) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 3.7e+42)
		tmp = 100.0 * n;
	else
		tmp = (50.0 * i) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 3.7e+42], N[(100.0 * n), $MachinePrecision], N[(N[(50.0 * i), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 3.69999999999999996e42

   1. Initial program 20.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. lower-*.f64 62.4

         \[\leadsto \color{blue}{100 \cdot n} \]

   5. Applied rewrites 62.4%

      \[\leadsto \color{blue}{100 \cdot n} \]

   if 3.69999999999999996e42 < i

   1. Initial program 55.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 41.2

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 29.9%

      \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]

   9. Taylor expanded in i around inf

      \[\leadsto \left(50 \cdot i\right) \cdot n \]
   10. Step-by-step derivation

   11. Applied rewrites 29.9%

       \[\leadsto \left(50 \cdot i\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 54.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(50, i, 100\right) \cdot n \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* (fma 50.0 i 100.0) n))

C

double code(double i, double n) {
	return fma(50.0, i, 100.0) * n;
}

Julia

function code(i, n)
	return Float64(fma(50.0, i, 100.0) * n)
end

Wolfram

code[i_, n_] := N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]

Derivation

1. Initial program 27.1%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in n around inf

   \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

   2. *-commutative N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

   8. lower-expm1.f64 75.6

      \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

5. Applied rewrites 75.6%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

6. Taylor expanded in i around 0

   \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
7. Step-by-step derivation

8. Applied rewrites 57.1%

   \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]
9. Add Preprocessing

Alternative 20: 49.0% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot n \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 n))

C

double code(double i, double n) {
	return 100.0 * n;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * n
end function

Java

public static double code(double i, double n) {
	return 100.0 * n;
}

Python

def code(i, n):
	return 100.0 * n

Julia

function code(i, n)
	return Float64(100.0 * n)
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * n;
end

Wolfram

code[i_, n_] := N[(100.0 * n), $MachinePrecision]

Derivation

1. Initial program 27.1%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. lower-*.f64 51.8

      \[\leadsto \color{blue}{100 \cdot n} \]

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{100 \cdot n} \]
6. Add Preprocessing

Developer Target 1: 34.3% 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 2024278 
(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))))