Compound Interest

Percentage Accurate: 28.2% → 98.0%

Time: 12.7s

Alternatives: 16

Speedup: 24.3×

• could not determine a ground truth

  1. i = 1.968378738748379e-273
  2. n = 1.4938603232630504e+39

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: 28.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(\frac{-1}{i} + \frac{t\_0}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 0.0)
     (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) (/ i n))
     (if (<= t_1 INFINITY)
       (* (* (+ (/ -1.0 i) (/ t_0 i)) 100.0) n)
       (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((-1.0 / i) + (t_0 / i)) * 100.0) * n;
	} else {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(-1.0 / i) + Float64(t_0 / i)) * 100.0) * n);
	else
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 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[(N[(N[(-1.0 / i), $MachinePrecision] + N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 28.5

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 98.7

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

   4. Applied rewrites 98.7%

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

   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 96.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. 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 96.8

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

      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 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lower-+.f64 97.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.5

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

   10. Applied rewrites 97.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 0.0

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

      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 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.3

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

   7. Applied rewrites 24.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 24.3

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

   9. Applied rewrites 24.3%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 99.6

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\left(\frac{-1}{i} + \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(\frac{-1}{i} + \frac{t\_0}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* (* (/ 100.0 i) (expm1 (* (log1p (/ i n)) n))) n)
     (if (<= t_1 INFINITY)
       (* (* (+ (/ -1.0 i) (/ t_0 i)) 100.0) n)
       (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((100.0 / i) * expm1((log1p((i / n)) * n))) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((-1.0 / i) + (t_0 / i)) * 100.0) * n;
	} else {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(100.0 / i) * expm1(Float64(log1p(Float64(i / n)) * n))) * n);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(-1.0 / i) + Float64(t_0 / i)) * 100.0) * n);
	else
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 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[(N[(N[(-1.0 / i), $MachinePrecision] + N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 98.3

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

   4. Applied rewrites 98.3%

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

   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 96.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. 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 96.8

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

      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 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lower-+.f64 97.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.5

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

   10. Applied rewrites 97.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 0.0

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

      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 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.3

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

   7. Applied rewrites 24.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 24.3

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

   9. Applied rewrites 24.3%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 99.6

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{100}{i} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\left(\frac{-1}{i} + \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := \frac{t\_0 - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(\frac{-1}{i} + \frac{t\_0}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (/ (- t_0 1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* (* (/ (expm1 i) i) 100.0) n)
     (if (<= t_1 INFINITY)
       (* (* (+ (/ -1.0 i) (/ t_0 i)) 100.0) n)
       (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = (t_0 - 1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((-1.0 / i) + (t_0 / i)) * 100.0) * n;
	} else {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(Float64(t_0 - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(-1.0 / i) + Float64(t_0 / i)) * 100.0) * n);
	else
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 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[(-1.0 / i), $MachinePrecision] + N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   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 96.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. 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 96.8

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

      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 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lower-+.f64 97.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.5

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

   10. Applied rewrites 97.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 0.0

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

      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 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.3

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

   7. Applied rewrites 24.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 24.3

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

   9. Applied rewrites 24.3%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 99.6

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\left(\frac{-1}{i} + \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}\right) \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n} - 1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{t\_0}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \end{array} \]

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 i) 100.0) n)
       (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))))))

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 / i) * 100.0) * n;
	} else {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	}
	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(Float64(t_0 / i) * 100.0) * n);
	else
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	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[(N[(t$95$0 / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   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 96.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. 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 96.8

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

      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 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lower-+.f64 96.9

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

   10. Applied rewrites 96.9%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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 0.0

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

      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 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 24.3

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

   7. Applied rewrites 24.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 24.3

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

   9. Applied rewrites 24.3%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 99.6

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -8 \cdot 10^{-251}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 0.39:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -8e-251)
     t_0
     (if (<= n 0.39)
       (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))
       t_0))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -8e-251) {
		tmp = t_0;
	} else if (n <= 0.39) {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -8e-251)
		tmp = t_0;
	elseif (n <= 0.39)
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -8e-251], t$95$0, If[LessEqual[n, 0.39], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -8.00000000000000012e-251 or 0.39000000000000001 < n

   1. Initial program 29.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 87.1

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

   5. Applied rewrites 87.1%

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

   if -8.00000000000000012e-251 < n < 0.39000000000000001

   1. Initial program 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 36.6

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

      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 86.0

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

   4. Applied rewrites 86.0%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 49.4

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

   9. Applied rewrites 49.4%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 70.0

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 70.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 70.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{n} - 0.5}{n} + 0.16666666666666666, i, 0.5 - \frac{0.5}{n}\right), i, 1\right) \cdot i\right) \cdot n}{i} \cdot 100\\ \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (*
            (*
             (fma
              (fma
               (+ (/ (- (/ 0.3333333333333333 n) 0.5) n) 0.16666666666666666)
               i
               (- 0.5 (/ 0.5 n)))
              i
              1.0)
             i)
            n)
           i)
          100.0)))
   (if (<= n -1.66e+167)
     t_0
     (if (<= n 4.2e-11)
       (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (((fma(fma(((((0.3333333333333333 / n) - 0.5) / n) + 0.16666666666666666), i, (0.5 - (0.5 / n))), i, 1.0) * i) * n) / i) * 100.0;
	double tmp;
	if (n <= -1.66e+167) {
		tmp = t_0;
	} else if (n <= 4.2e-11) {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(Float64(fma(fma(Float64(Float64(Float64(Float64(0.3333333333333333 / n) - 0.5) / n) + 0.16666666666666666), i, Float64(0.5 - Float64(0.5 / n))), i, 1.0) * i) * n) / i) * 100.0)
	tmp = 0.0
	if (n <= -1.66e+167)
		tmp = t_0;
	elseif (n <= 4.2e-11)
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(0.3333333333333333 / n), $MachinePrecision] - 0.5), $MachinePrecision] / n), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * i + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + 1.0), $MachinePrecision] * i), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -1.66e+167], t$95$0, If[LessEqual[n, 4.2e-11], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.6600000000000001e167 or 4.1999999999999997e-11 < n

   1. Initial program 15.3%

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

   3. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 52.8%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), i, 1\right) \cdot i}}{\frac{i}{n}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 75.9%

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

   if -1.6600000000000001e167 < n < 4.1999999999999997e-11

   1. Initial program 43.8%

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

      1. lift-*.f64 N/A

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

      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 43.8

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

      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 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 49.3

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

   9. Applied rewrites 49.3%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 63.3

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 63.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{n} - 0.5}{n} + 0.16666666666666666, i, 0.5 - \frac{0.5}{n}\right), i, 1\right) \cdot i\right) \cdot n}{i} \cdot 100\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{n} - 0.5}{n} + 0.16666666666666666, i, 0.5 - \frac{0.5}{n}\right), i, 1\right) \cdot i\right) \cdot n}{i} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), i, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.66e+167)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n 4.2e-11)
     (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))
     (*
      (* 100.0 n)
      (fma
       (fma
        (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
        i
        (- 0.5 (/ 0.5 n)))
       i
       1.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.66e+167) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= 4.2e-11) {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	} else {
		tmp = (100.0 * n) * fma(fma((((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), i, (0.5 - (0.5 / n))), i, 1.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.66e+167)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= 4.2e-11)
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	else
		tmp = Float64(Float64(100.0 * n) * fma(fma(Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), i, Float64(0.5 - Float64(0.5 / n))), i, 1.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.66e+167], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 4.2e-11], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * n), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6600000000000001e167

   1. Initial program 11.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.6%

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

   if -1.6600000000000001e167 < n < 4.1999999999999997e-11

   1. Initial program 43.8%

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

      1. lift-*.f64 N/A

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

      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 43.8

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

      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 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 49.3

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

   9. Applied rewrites 49.3%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 63.3

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 63.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]

   if 4.1999999999999997e-11 < n

   1. Initial program 17.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 17.1

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

      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 72.9

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

   4. Applied rewrites 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log1p.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 78.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(100 \cdot n\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), i, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), n\right) \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.66e+167)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n 4.2e-11)
     (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))
     (*
      (fma
       (* n i)
       (fma
        (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
        i
        (- 0.5 (/ 0.5 n)))
       n)
      100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.66e+167) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= 4.2e-11) {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	} else {
		tmp = fma((n * i), fma((((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), i, (0.5 - (0.5 / n))), n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.66e+167)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= 4.2e-11)
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	else
		tmp = Float64(fma(Float64(n * i), fma(Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), i, Float64(0.5 - Float64(0.5 / n))), n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.66e+167], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 4.2e-11], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(n * i), $MachinePrecision] * N[(N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i + N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6600000000000001e167

   1. Initial program 11.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.6%

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

   if -1.6600000000000001e167 < n < 4.1999999999999997e-11

   1. Initial program 43.8%

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

      1. lift-*.f64 N/A

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

      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 43.8

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

      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 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 49.3

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

   9. Applied rewrites 49.3%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 63.3

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 63.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]

   if 4.1999999999999997e-11 < n

   1. Initial program 17.1%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 78.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.66e+167)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n 1.3e+54)
     (/ 1.0 (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)))
     (* (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n) 100.0))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.66e+167) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= 1.3e+54) {
		tmp = 1.0 / fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n));
	} else {
		tmp = (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.66e+167)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= 1.3e+54)
		tmp = Float64(1.0 / fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)));
	else
		tmp = Float64(Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.66e+167], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 1.3e+54], N[(1.0 / N[(N[(0.01 * i), $MachinePrecision] * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + N[(0.01 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.6600000000000001e167

   1. Initial program 11.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 63.6%

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

   if -1.6600000000000001e167 < n < 1.30000000000000003e54

   1. Initial program 40.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 40.7

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

      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 84.8

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.0

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

   7. Applied rewrites 52.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 51.9

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

   9. Applied rewrites 51.9%

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

   10. Taylor expanded in i around 0

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

       1. associate-*r* N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. lower-/.f64 65.0

           \[\leadsto \frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \color{blue}{\frac{0.01}{n}}\right)} \]

   12. Applied rewrites 65.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}} \]

   if 1.30000000000000003e54 < n

   1. Initial program 18.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 78.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.66 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;i \leq 9200000000000:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, i, -0.5\right), i, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{n} \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -7.2e+31)
   (/ 0.0 i)
   (if (<= i 9200000000000.0)
     (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
     (*
      (/
       (fma
        (fma
         (fma -0.5 i -0.5)
         i
         (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n))
        n
        (* (* i i) 0.3333333333333333))
       n)
      100.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -7.2e+31) {
		tmp = 0.0 / i;
	} else if (i <= 9200000000000.0) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else {
		tmp = (fma(fma(fma(-0.5, i, -0.5), i, (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n)), n, ((i * i) * 0.3333333333333333)) / n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -7.2e+31)
		tmp = Float64(0.0 / i);
	elseif (i <= 9200000000000.0)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	else
		tmp = Float64(Float64(fma(fma(fma(-0.5, i, -0.5), i, Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n)), n, Float64(Float64(i * i) * 0.3333333333333333)) / n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -7.2e+31], N[(0.0 / i), $MachinePrecision], If[LessEqual[i, 9200000000000.0], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * i + -0.5), $MachinePrecision] * i + N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] * n + N[(N[(i * i), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * 100.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -7.19999999999999992e31

   1. Initial program 74.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 31.5

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 31.5%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if -7.19999999999999992e31 < i < 9.2e12

   1. Initial program 12.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 78.0%

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

   if 9.2e12 < i

   1. Initial program 44.7%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in n around 0

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

   8. Applied rewrites 49.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;i \leq 9200000000000:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, i, -0.5\right), i, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{n} \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq -8 \cdot 10^{-251}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.8e+56)
   (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
   (if (<= n -8e-251)
     (/ (* 100.0 i) (/ i n))
     (if (<= n 5.2e-250)
       (/ 0.0 i)
       (if (<= n 1.45e+21)
         (* (* 100.0 i) (/ n i))
         (* (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n) 100.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.8e+56) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= -8e-251) {
		tmp = (100.0 * i) / (i / n);
	} else if (n <= 5.2e-250) {
		tmp = 0.0 / i;
	} else if (n <= 1.45e+21) {
		tmp = (100.0 * i) * (n / i);
	} else {
		tmp = (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.8e+56)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= -8e-251)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	elseif (n <= 5.2e-250)
		tmp = Float64(0.0 / i);
	elseif (n <= 1.45e+21)
		tmp = Float64(Float64(100.0 * i) * Float64(n / i));
	else
		tmp = Float64(Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.8e+56], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, -8e-251], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.2e-250], N[(0.0 / i), $MachinePrecision], If[LessEqual[n, 1.45e+21], N[(N[(100.0 * i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.79999999999999999e56

   1. Initial program 26.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 58.4%

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

   if -1.79999999999999999e56 < n < -8.00000000000000012e-251

   1. Initial program 47.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. 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 47.8

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

      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 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if -8.00000000000000012e-251 < n < 5.20000000000000016e-250

   1. Initial program 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 16.3%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 84.2

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if 5.20000000000000016e-250 < n < 1.45e21

   1. Initial program 16.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 16.0

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

      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 82.0

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 60.0

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

   9. Applied rewrites 60.0%

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

   if 1.45e21 < n

   1. Initial program 17.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 79.1%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq -8 \cdot 10^{-251}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq -8.2 \cdot 10^{-251}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* 100.0 i) (/ n i))))
   (if (<= n -1.4e+56)
     (* (fma (* n i) (fma 0.16666666666666666 i 0.5) n) 100.0)
     (if (<= n -8.2e-251)
       t_0
       (if (<= n 5.2e-250)
         (/ 0.0 i)
         (if (<= n 1.45e+21)
           t_0
           (* (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n) 100.0)))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * i) * (n / i);
	double tmp;
	if (n <= -1.4e+56) {
		tmp = fma((n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0;
	} else if (n <= -8.2e-251) {
		tmp = t_0;
	} else if (n <= 5.2e-250) {
		tmp = 0.0 / i;
	} else if (n <= 1.45e+21) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * i) * Float64(n / i))
	tmp = 0.0
	if (n <= -1.4e+56)
		tmp = Float64(fma(Float64(n * i), fma(0.16666666666666666, i, 0.5), n) * 100.0);
	elseif (n <= -8.2e-251)
		tmp = t_0;
	elseif (n <= 5.2e-250)
		tmp = Float64(0.0 / i);
	elseif (n <= 1.45e+21)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.4e+56], N[(N[(N[(n * i), $MachinePrecision] * N[(0.16666666666666666 * i + 0.5), $MachinePrecision] + n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, -8.2e-251], t$95$0, If[LessEqual[n, 5.2e-250], N[(0.0 / i), $MachinePrecision], If[LessEqual[n, 1.45e+21], t$95$0, N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.40000000000000004e56

   1. Initial program 26.5%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 58.4%

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

   if -1.40000000000000004e56 < n < -8.1999999999999997e-251 or 5.20000000000000016e-250 < n < 1.45e21

   1. Initial program 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 33.7

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

      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 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 58.2

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

   9. Applied rewrites 58.2%

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

   if -8.1999999999999997e-251 < n < 5.20000000000000016e-250

   1. Initial program 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 16.3%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 84.2

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if 1.45e21 < n

   1. Initial program 17.0%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 79.1%

      \[\leadsto 100 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot \color{blue}{n}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot i, \mathsf{fma}\left(0.16666666666666666, i, 0.5\right), n\right) \cdot 100\\ \mathbf{elif}\;n \leq -8.2 \cdot 10^{-251}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(100 \cdot i\right) \cdot \frac{n}{i}\\ t_1 := \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -8.2 \cdot 10^{-251}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* 100.0 i) (/ n i)))
        (t_1 (* (* (fma (fma 0.16666666666666666 i 0.5) i 1.0) n) 100.0)))
   (if (<= n -1.4e+56)
     t_1
     (if (<= n -8.2e-251)
       t_0
       (if (<= n 5.2e-250) (/ 0.0 i) (if (<= n 1.45e+21) t_0 t_1))))))

C

double code(double i, double n) {
	double t_0 = (100.0 * i) * (n / i);
	double t_1 = (fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0;
	double tmp;
	if (n <= -1.4e+56) {
		tmp = t_1;
	} else if (n <= -8.2e-251) {
		tmp = t_0;
	} else if (n <= 5.2e-250) {
		tmp = 0.0 / i;
	} else if (n <= 1.45e+21) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(100.0 * i) * Float64(n / i))
	t_1 = Float64(Float64(fma(fma(0.16666666666666666, i, 0.5), i, 1.0) * n) * 100.0)
	tmp = 0.0
	if (n <= -1.4e+56)
		tmp = t_1;
	elseif (n <= -8.2e-251)
		tmp = t_0;
	elseif (n <= 5.2e-250)
		tmp = Float64(0.0 / i);
	elseif (n <= 1.45e+21)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(100.0 * i), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.16666666666666666 * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision]}, If[LessEqual[n, -1.4e+56], t$95$1, If[LessEqual[n, -8.2e-251], t$95$0, If[LessEqual[n, 5.2e-250], N[(0.0 / i), $MachinePrecision], If[LessEqual[n, 1.45e+21], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.40000000000000004e56 or 1.45e21 < n

   1. Initial program 21.6%

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

   3. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in n around inf

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

   8. Applied rewrites 68.9%

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

   if -1.40000000000000004e56 < n < -8.1999999999999997e-251 or 5.20000000000000016e-250 < n < 1.45e21

   1. Initial program 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 33.7

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

      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 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 58.2

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

   9. Applied rewrites 58.2%

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

   if -8.1999999999999997e-251 < n < 5.20000000000000016e-250

   1. Initial program 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 16.3%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 84.2

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{+56}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \mathbf{elif}\;n \leq -8.2 \cdot 10^{-251}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;\left(100 \cdot i\right) \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right) \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{-201}:\\ \;\;\;\;100 \cdot n\\ \mathbf{elif}\;i \leq 7.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{\left(100 \cdot i\right) \cdot n}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -7.2e+31)
   (/ 0.0 i)
   (if (<= i 3.1e-201)
     (* 100.0 n)
     (if (<= i 7.4e+171) (/ (* (* 100.0 i) n) i) (/ 0.0 i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -7.2e+31) {
		tmp = 0.0 / i;
	} else if (i <= 3.1e-201) {
		tmp = 100.0 * n;
	} else if (i <= 7.4e+171) {
		tmp = ((100.0 * i) * n) / i;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-7.2d+31)) then
        tmp = 0.0d0 / i
    else if (i <= 3.1d-201) then
        tmp = 100.0d0 * n
    else if (i <= 7.4d+171) then
        tmp = ((100.0d0 * i) * n) / i
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -7.2e+31) {
		tmp = 0.0 / i;
	} else if (i <= 3.1e-201) {
		tmp = 100.0 * n;
	} else if (i <= 7.4e+171) {
		tmp = ((100.0 * i) * n) / i;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -7.2e+31:
		tmp = 0.0 / i
	elif i <= 3.1e-201:
		tmp = 100.0 * n
	elif i <= 7.4e+171:
		tmp = ((100.0 * i) * n) / i
	else:
		tmp = 0.0 / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -7.2e+31)
		tmp = Float64(0.0 / i);
	elseif (i <= 3.1e-201)
		tmp = Float64(100.0 * n);
	elseif (i <= 7.4e+171)
		tmp = Float64(Float64(Float64(100.0 * i) * n) / i);
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -7.2e+31)
		tmp = 0.0 / i;
	elseif (i <= 3.1e-201)
		tmp = 100.0 * n;
	elseif (i <= 7.4e+171)
		tmp = ((100.0 * i) * n) / i;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -7.2e+31], N[(0.0 / i), $MachinePrecision], If[LessEqual[i, 3.1e-201], N[(100.0 * n), $MachinePrecision], If[LessEqual[i, 7.4e+171], N[(N[(N[(100.0 * i), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision], N[(0.0 / i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if i < -7.19999999999999992e31 or 7.39999999999999996e171 < i

   1. Initial program 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 36.6

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 36.6%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if -7.19999999999999992e31 < i < 3.0999999999999999e-201

   1. Initial program 8.8%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 3.0999999999999999e-201 < i < 7.39999999999999996e171

   1. Initial program 26.2%

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

      1. lift-*.f64 N/A

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

      2. 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 26.2

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

      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 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.9

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

   7. Applied rewrites 31.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 54.7

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

   9. Applied rewrites 54.7%

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

Alternative 15: 59.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{0}{i}\\ \mathbf{elif}\;i \leq 2 \cdot 10^{-16}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -7.2e+31) (/ 0.0 i) (if (<= i 2e-16) (* 100.0 n) (/ 0.0 i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -7.2e+31) {
		tmp = 0.0 / i;
	} else if (i <= 2e-16) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-7.2d+31)) then
        tmp = 0.0d0 / i
    else if (i <= 2d-16) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -7.2e+31) {
		tmp = 0.0 / i;
	} else if (i <= 2e-16) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -7.2e+31:
		tmp = 0.0 / i
	elif i <= 2e-16:
		tmp = 100.0 * n
	else:
		tmp = 0.0 / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -7.2e+31)
		tmp = Float64(0.0 / i);
	elseif (i <= 2e-16)
		tmp = Float64(100.0 * n);
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -7.2e+31)
		tmp = 0.0 / i;
	elseif (i <= 2e-16)
		tmp = 100.0 * n;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -7.2e+31], N[(0.0 / i), $MachinePrecision], If[LessEqual[i, 2e-16], N[(100.0 * n), $MachinePrecision], N[(0.0 / i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < -7.19999999999999992e31 or 2e-16 < i

   1. Initial program 56.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      13. frac-2neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

      18. associate-/r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   4. Applied rewrites 48.4%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 27.5

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 27.5%

      \[\leadsto \color{blue}{\frac{0}{i}} \]

   if -7.19999999999999992e31 < i < 2e-16

   1. Initial program 10.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. lower-*.f64 79.8

         \[\leadsto \color{blue}{100 \cdot n} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.4% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ 100 \cdot n \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* 100.0 n))

C

double code(double i, double n) {
	return 100.0 * n;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * n
end function

Java

public static double code(double i, double n) {
	return 100.0 * n;
}

Python

def code(i, n):
	return 100.0 * n

Julia

function code(i, n)
	return Float64(100.0 * n)
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * n;
end

Wolfram

code[i_, n_] := N[(100.0 * n), $MachinePrecision]

Derivation

1. Initial program 30.7%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot n} \]
4. Step-by-step derivation

   1. lower-*.f64 47.3

      \[\leadsto \color{blue}{100 \cdot n} \]

5. Applied rewrites 47.3%

   \[\leadsto \color{blue}{100 \cdot n} \]
6. Add Preprocessing

Developer Target 1: 34.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

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 2024268 
(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))))