Compound Interest

Percentage Accurate: 28.9% → 98.1%

Time: 11.4s

Alternatives: 15

Speedup: 8.1×

• could not determine a ground truth

  1. i = 3.419016440892517e-242
  2. n = -3.0706492627570174e+38

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right) \cdot 100 - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
   (if (<= t_0 1e-302)
     (/ (* 100.0 (expm1 (* (log1p (/ i n)) n))) (/ i n))
     (if (<= t_0 INFINITY)
       (- (* (* (/ (pow (+ (/ i n) 1.0) n) i) n) 100.0) (/ (* n 100.0) i))
       (pow (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)) -1.0)))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) - 1.0) / (i / n);
	double tmp;
	if (t_0 <= 1e-302) {
		tmp = (100.0 * expm1((log1p((i / n)) * n))) / (i / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((pow(((i / n) + 1.0), n) / i) * n) * 100.0) - ((n * 100.0) / i);
	} else {
		tmp = pow(fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n)), -1.0);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 1e-302)
		tmp = Float64(Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))) / Float64(i / n));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) / i) * n) * 100.0) - Float64(Float64(n * 100.0) / i));
	else
		tmp = fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-302], N[(N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] - N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[Power[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], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 9.9999999999999996e-303

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 24.9

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

      6. lift--.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lower-expm1.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 97.9

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

   4. Applied rewrites 97.9%

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

   if 9.9999999999999996e-303 < (/.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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. 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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 \cdot \frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      6. lift-expm1.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

   7. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{100} \cdot \left(i \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \frac{1}{100} \cdot \frac{1}{n}}} \]
   8. 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 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 10^{-302}:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right) \cdot 100 - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right) \cdot 100 - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
   (if (<= t_0 1e-302)
     (* (/ n i) (* 100.0 (expm1 (* (log1p (/ i n)) n))))
     (if (<= t_0 INFINITY)
       (- (* (* (/ (pow (+ (/ i n) 1.0) n) i) n) 100.0) (/ (* n 100.0) i))
       (pow (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)) -1.0)))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) - 1.0) / (i / n);
	double tmp;
	if (t_0 <= 1e-302) {
		tmp = (n / i) * (100.0 * expm1((log1p((i / n)) * n)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((pow(((i / n) + 1.0), n) / i) * n) * 100.0) - ((n * 100.0) / i);
	} else {
		tmp = pow(fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n)), -1.0);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 1e-302)
		tmp = Float64(Float64(n / i) * Float64(100.0 * expm1(Float64(log1p(Float64(i / n)) * n))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) / i) * n) * 100.0) - Float64(Float64(n * 100.0) / i));
	else
		tmp = fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-302], N[(N[(n / i), $MachinePrecision] * N[(100.0 * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] - N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[Power[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], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 9.9999999999999996e-303

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 24.4

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 96.0

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

   4. Applied rewrites 96.0%

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

   if 9.9999999999999996e-303 < (/.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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. 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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 \cdot \frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      6. lift-expm1.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

   7. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{100} \cdot \left(i \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \frac{1}{100} \cdot \frac{1}{n}}} \]
   8. 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 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 10^{-302}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right) \cdot 100 - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 10^{-302}:\\ \;\;\;\;\left(\frac{n}{i} \cdot 100\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right) \cdot 100 - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))
   (if (<= t_0 1e-302)
     (* (* (/ n i) 100.0) (expm1 (* (log1p (/ i n)) n)))
     (if (<= t_0 INFINITY)
       (- (* (* (/ (pow (+ (/ i n) 1.0) n) i) n) 100.0) (/ (* n 100.0) i))
       (pow (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)) -1.0)))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) - 1.0) / (i / n);
	double tmp;
	if (t_0 <= 1e-302) {
		tmp = ((n / i) * 100.0) * expm1((log1p((i / n)) * n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (((pow(((i / n) + 1.0), n) / i) * n) * 100.0) - ((n * 100.0) / i);
	} else {
		tmp = pow(fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n)), -1.0);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 1e-302)
		tmp = Float64(Float64(Float64(n / i) * 100.0) * expm1(Float64(log1p(Float64(i / n)) * n)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) / i) * n) * 100.0) - Float64(Float64(n * 100.0) / i));
	else
		tmp = fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-302], N[(N[(N[(n / i), $MachinePrecision] * 100.0), $MachinePrecision] * N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision] * 100.0), $MachinePrecision] - N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[Power[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], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 9.9999999999999996e-303

   1. Initial program 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 24.4

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

      12. lift--.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. lower-expm1.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lower-log1p.f64 95.6

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

   4. Applied rewrites 95.6%

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

   if 9.9999999999999996e-303 < (/.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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. 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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 \cdot \frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      6. lift-expm1.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

   7. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{100} \cdot \left(i \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \frac{1}{100} \cdot \frac{1}{n}}} \]
   8. 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 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq 10^{-302}:\\ \;\;\;\;\left(\frac{n}{i} \cdot 100\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right) \cdot 100 - \frac{n \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ 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(n / i) * 100.0) * t_0);
	else
		tmp = fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)) ^ -1.0;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.6%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 75.8

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

   5. Applied rewrites 75.8%

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

   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 95.7%

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

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

      4. associate-/r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 95.9

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

      12. lift--.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-to-exp N/A

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

      15. lower-expm1.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lower-log1p.f64 52.5

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

   4. Applied rewrites 52.5%

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

      1. lift-expm1.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. exp-to-pow N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 95.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 95.9

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

   6. Applied rewrites 95.9%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. 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. 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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 \cdot \frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      6. lift-expm1.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

   7. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{100} \cdot \left(i \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \frac{1}{100} \cdot \frac{1}{n}}} \]
   8. 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 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 81.5%

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

Alternative 5: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-209} \lor \neg \left(n \leq 1.6 \cdot 10^{-7}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.6e-209) (not (<= n 1.6e-7)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (pow (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)) -1.0)))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.6e-209) || !(n <= 1.6e-7)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = pow(fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n)), -1.0);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.6e-209) || !(n <= 1.6e-7))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.6e-209], N[Not[LessEqual[n, 1.6e-7]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[Power[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], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.59999999999999984e-209 or 1.6e-7 < n

   1. Initial program 23.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -2.59999999999999984e-209 < n < 1.6e-7

   1. Initial program 31.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 31.1

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

      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 84.1

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

   4. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 \cdot \frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      6. lift-expm1.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

   7. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{100} \cdot \left(i \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \frac{1}{100} \cdot \frac{1}{n}}} \]
   8. 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 64.1

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

   9. Applied rewrites 64.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-209} \lor \neg \left(n \leq 1.6 \cdot 10^{-7}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 - \frac{0.5}{n}\right) \cdot n\\ \mathbf{if}\;n \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot i, \mathsf{fma}\left(i \cdot n, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, t\_0\right), 100 \cdot n\right), i, 0\right)}{i}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(t\_0, i, n\right), i, 0\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (- 0.5 (/ 0.5 n)) n)))
   (if (<= n -1.5e+95)
     (/
      (fma
       (fma
        (* 100.0 i)
        (fma
         (* i n)
         (- (+ (/ 0.3333333333333333 (* n n)) 0.16666666666666666) (/ 0.5 n))
         t_0)
        (* 100.0 n))
       i
       0.0)
      i)
     (if (<= n 1.6e-7)
       (pow (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)) -1.0)
       (/ (fma (* 100.0 (fma t_0 i n)) i 0.0) i)))))

C

double code(double i, double n) {
	double t_0 = (0.5 - (0.5 / n)) * n;
	double tmp;
	if (n <= -1.5e+95) {
		tmp = fma(fma((100.0 * i), fma((i * n), (((0.3333333333333333 / (n * n)) + 0.16666666666666666) - (0.5 / n)), t_0), (100.0 * n)), i, 0.0) / i;
	} else if (n <= 1.6e-7) {
		tmp = pow(fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n)), -1.0);
	} else {
		tmp = fma((100.0 * fma(t_0, i, n)), i, 0.0) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(0.5 - Float64(0.5 / n)) * n)
	tmp = 0.0
	if (n <= -1.5e+95)
		tmp = Float64(fma(fma(Float64(100.0 * i), fma(Float64(i * n), Float64(Float64(Float64(0.3333333333333333 / Float64(n * n)) + 0.16666666666666666) - Float64(0.5 / n)), t_0), Float64(100.0 * n)), i, 0.0) / i);
	elseif (n <= 1.6e-7)
		tmp = fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)) ^ -1.0;
	else
		tmp = Float64(fma(Float64(100.0 * fma(t_0, i, n)), i, 0.0) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -1.5e+95], N[(N[(N[(N[(100.0 * i), $MachinePrecision] * N[(N[(i * n), $MachinePrecision] * N[(N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(100.0 * n), $MachinePrecision]), $MachinePrecision] * i + 0.0), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.6e-7], N[Power[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], -1.0], $MachinePrecision], N[(N[(N[(100.0 * N[(t$95$0 * i + n), $MachinePrecision]), $MachinePrecision] * i + 0.0), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.49999999999999996e95

   1. Initial program 17.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 17.4%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 2.1

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 2.1%

       \[\leadsto 0 \]

   11. Taylor expanded in i around 0

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

       1. lower-/.f64 N/A

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

   13. Applied rewrites 61.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot i, \mathsf{fma}\left(i \cdot n, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), 100 \cdot n\right), i, 0\right)}{i}} \]

   if -1.49999999999999996e95 < n < 1.6e-7

   1. Initial program 31.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 31.8

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

      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 85.4

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

   4. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 \cdot \frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      6. lift-expm1.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

   7. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{100} \cdot \left(i \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \frac{1}{100} \cdot \frac{1}{n}}} \]
   8. 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 64.6

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

   9. Applied rewrites 64.6%

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

   if 1.6e-7 < n

   1. Initial program 18.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(100 \cdot i, \mathsf{fma}\left(i \cdot n, \left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, \left(0.5 - \frac{0.5}{n}\right) \cdot n\right), 100 \cdot n\right), i, 0\right)}{i}\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.1e+97)
   (* (fma (+ 50.0 (* i 16.666666666666668)) i 100.0) n)
   (if (<= n 1.6e-7)
     (pow (fma (* 0.01 i) (- (/ 0.5 (* n n)) (/ 0.5 n)) (/ 0.01 n)) -1.0)
     (/ (fma (* 100.0 (fma (* (- 0.5 (/ 0.5 n)) n) i n)) i 0.0) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.1e+97) {
		tmp = fma((50.0 + (i * 16.666666666666668)), i, 100.0) * n;
	} else if (n <= 1.6e-7) {
		tmp = pow(fma((0.01 * i), ((0.5 / (n * n)) - (0.5 / n)), (0.01 / n)), -1.0);
	} else {
		tmp = fma((100.0 * fma(((0.5 - (0.5 / n)) * n), i, n)), i, 0.0) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.1e+97)
		tmp = Float64(fma(Float64(50.0 + Float64(i * 16.666666666666668)), i, 100.0) * n);
	elseif (n <= 1.6e-7)
		tmp = fma(Float64(0.01 * i), Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)), Float64(0.01 / n)) ^ -1.0;
	else
		tmp = Float64(fma(Float64(100.0 * fma(Float64(Float64(0.5 - Float64(0.5 / n)) * n), i, n)), i, 0.0) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.1e+97], N[(N[(N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.6e-7], N[Power[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], -1.0], $MachinePrecision], N[(N[(N[(100.0 * N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision] * i + 0.0), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.10000000000000012e97

   1. Initial program 17.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 17.3

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

      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 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 59.7%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 59.7%

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

   if -2.10000000000000012e97 < n < 1.6e-7

   1. Initial program 31.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 31.8

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

      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 85.4

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

   4. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}{\frac{i}{n}}} \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{n}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \frac{i}{n}}}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1 \cdot \frac{i}{n}}{\color{blue}{100 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

      6. lift-expm1.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log1p.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-/.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{1}{0.01 \cdot \frac{\frac{i}{n}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)}}} \]

   7. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{100} \cdot \left(i \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + \frac{1}{100} \cdot \frac{1}{n}}} \]
   8. 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 64.6

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

   9. Applied rewrites 64.6%

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

   if 1.6e-7 < n

   1. Initial program 18.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.01 \cdot i, \frac{0.5}{n \cdot n} - \frac{0.5}{n}, \frac{0.01}{n}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.5e+66)
   (* (fma (+ 50.0 (* i 16.666666666666668)) i 100.0) n)
   (if (<= n 1.5e-7)
     (/ (* 100.0 i) (/ i n))
     (/ (fma (* 100.0 (fma (* (- 0.5 (/ 0.5 n)) n) i n)) i 0.0) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.5e+66) {
		tmp = fma((50.0 + (i * 16.666666666666668)), i, 100.0) * n;
	} else if (n <= 1.5e-7) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = fma((100.0 * fma(((0.5 - (0.5 / n)) * n), i, n)), i, 0.0) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.5e+66)
		tmp = Float64(fma(Float64(50.0 + Float64(i * 16.666666666666668)), i, 100.0) * n);
	elseif (n <= 1.5e-7)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(fma(Float64(100.0 * fma(Float64(Float64(0.5 - Float64(0.5 / n)) * n), i, n)), i, 0.0) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.5e+66], N[(N[(N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.5e-7], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(100.0 * N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] * i + n), $MachinePrecision]), $MachinePrecision] * i + 0.0), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.50000000000000024e66

   1. Initial program 17.4%

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

      1. lift-*.f64 N/A

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

      2. 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 17.3

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

      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 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 62.3%

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

   if -7.50000000000000024e66 < n < 1.4999999999999999e-7

   1. Initial program 32.4%

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

      1. lift-*.f64 N/A

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

      2. 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 32.4

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

      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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if 1.4999999999999999e-7 < n

   1. Initial program 18.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(100 \cdot \mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot n, i, n\right), i, 0\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot i, 100, 100\right) \cdot i\right) \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.5e+66)
   (* (fma (+ 50.0 (* i 16.666666666666668)) i 100.0) n)
   (if (<= n 1.5e-7)
     (/ (* 100.0 i) (/ i n))
     (/ (* (* (fma (* (- 0.5 (/ 0.5 n)) i) 100.0 100.0) i) n) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.5e+66) {
		tmp = fma((50.0 + (i * 16.666666666666668)), i, 100.0) * n;
	} else if (n <= 1.5e-7) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = ((fma(((0.5 - (0.5 / n)) * i), 100.0, 100.0) * i) * n) / i;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.5e+66)
		tmp = Float64(fma(Float64(50.0 + Float64(i * 16.666666666666668)), i, 100.0) * n);
	elseif (n <= 1.5e-7)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 - Float64(0.5 / n)) * i), 100.0, 100.0) * i) * n) / i);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.5e+66], N[(N[(N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.5e-7], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * i), $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.50000000000000024e66

   1. Initial program 17.4%

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

      1. lift-*.f64 N/A

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

      2. 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 17.3

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

      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 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 62.3%

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

   if -7.50000000000000024e66 < n < 1.4999999999999999e-7

   1. Initial program 32.4%

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

      1. lift-*.f64 N/A

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

      2. 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 32.4

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

      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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if 1.4999999999999999e-7 < n

   1. Initial program 18.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 18.2

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

      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 64.3

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 52.5

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

   7. Applied rewrites 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.9

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

   9. Applied rewrites 83.9%

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

Alternative 10: 63.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \frac{0.5}{n}\right) \cdot i, 100, 100\right) \cdot i}{i} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.5e+66)
   (* (fma (+ 50.0 (* i 16.666666666666668)) i 100.0) n)
   (if (<= n 1.5e-7)
     (/ (* 100.0 i) (/ i n))
     (* (/ (* (fma (* (- 0.5 (/ 0.5 n)) i) 100.0 100.0) i) i) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.5e+66) {
		tmp = fma((50.0 + (i * 16.666666666666668)), i, 100.0) * n;
	} else if (n <= 1.5e-7) {
		tmp = (100.0 * i) / (i / n);
	} else {
		tmp = ((fma(((0.5 - (0.5 / n)) * i), 100.0, 100.0) * i) / i) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.5e+66)
		tmp = Float64(fma(Float64(50.0 + Float64(i * 16.666666666666668)), i, 100.0) * n);
	elseif (n <= 1.5e-7)
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.5 - Float64(0.5 / n)) * i), 100.0, 100.0) * i) / i) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.5e+66], N[(N[(N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 1.5e-7], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] * 100.0 + 100.0), $MachinePrecision] * i), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.50000000000000024e66

   1. Initial program 17.4%

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

      1. lift-*.f64 N/A

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

      2. 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 17.3

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

      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 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 62.3%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 62.3%

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

   if -7.50000000000000024e66 < n < 1.4999999999999999e-7

   1. Initial program 32.4%

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

      1. lift-*.f64 N/A

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

      2. 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 32.4

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

      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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if 1.4999999999999999e-7 < n

   1. Initial program 18.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 18.2

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

      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 64.3

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in i around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 52.5

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

   7. Applied rewrites 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 81.2

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

   9. Applied rewrites 81.2%

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

Alternative 11: 63.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+66} \lor \neg \left(n \leq 7 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -7.5e+66) (not (<= n 7e-8)))
   (* (fma (+ 50.0 (* i 16.666666666666668)) i 100.0) n)
   (/ (* 100.0 i) (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -7.5e+66) || !(n <= 7e-8)) {
		tmp = fma((50.0 + (i * 16.666666666666668)), i, 100.0) * n;
	} else {
		tmp = (100.0 * i) / (i / n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -7.5e+66) || !(n <= 7e-8))
		tmp = Float64(fma(Float64(50.0 + Float64(i * 16.666666666666668)), i, 100.0) * n);
	else
		tmp = Float64(Float64(100.0 * i) / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -7.5e+66], N[Not[LessEqual[n, 7e-8]], $MachinePrecision]], N[(N[(N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(100.0 * i), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.50000000000000024e66 or 7.00000000000000048e-8 < n

   1. Initial program 17.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 17.8

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

      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 65.6

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 72.0%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 72.0%

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

   if -7.50000000000000024e66 < n < 7.00000000000000048e-8

   1. Initial program 32.4%

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

      1. lift-*.f64 N/A

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

      2. 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 32.4

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

      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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 60.2

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

   7. Applied rewrites 60.2%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{+66} \lor \neg \left(n \leq 7 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-10} \lor \neg \left(n \leq 1.45 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.2e-10) (not (<= n 1.45e-7)))
   (* (fma (+ 50.0 (* i 16.666666666666668)) i 100.0) n)
   (* (/ n i) (* 100.0 i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.2e-10) || !(n <= 1.45e-7)) {
		tmp = fma((50.0 + (i * 16.666666666666668)), i, 100.0) * n;
	} else {
		tmp = (n / i) * (100.0 * i);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.2e-10) || !(n <= 1.45e-7))
		tmp = Float64(fma(Float64(50.0 + Float64(i * 16.666666666666668)), i, 100.0) * n);
	else
		tmp = Float64(Float64(n / i) * Float64(100.0 * i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.2e-10], N[Not[LessEqual[n, 1.45e-7]], $MachinePrecision]], N[(N[(N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * N[(100.0 * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.2e-10 or 1.4499999999999999e-7 < n

   1. Initial program 22.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 22.3

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

      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 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 69.0%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 69.0%

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

   if -4.2e-10 < n < 1.4499999999999999e-7

   1. Initial program 29.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 28.5

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

      11. lift--.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow-to-exp N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-log1p.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in i around 0

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

      1. lower-*.f64 60.5

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

   7. Applied rewrites 60.5%

      \[\leadsto \frac{n}{i} \cdot \color{blue}{\left(100 \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-10} \lor \neg \left(n \leq 1.45 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot \left(100 \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -7.2) 0.0 (* (fma (+ 50.0 (* i 16.666666666666668)) i 100.0) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -7.2) {
		tmp = 0.0;
	} else {
		tmp = fma((50.0 + (i * 16.666666666666668)), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -7.2)
		tmp = 0.0;
	else
		tmp = Float64(fma(Float64(50.0 + Float64(i * 16.666666666666668)), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -7.2], 0.0, N[(N[(N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -7.20000000000000018

   1. Initial program 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 36.0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 36.0%

       \[\leadsto 0 \]

   if -7.20000000000000018 < i

   1. Initial program 17.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. 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 17.4

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

      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 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 60.9%

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

   8. Taylor expanded in n around inf

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

   10. Applied rewrites 69.5%

       \[\leadsto \mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot \color{blue}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50 + i \cdot 16.666666666666668, i, 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.2:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.8 \cdot 10^{+20}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -7.2) 0.0 (if (<= i 1.8e+20) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -7.2) {
		tmp = 0.0;
	} else if (i <= 1.8e+20) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-7.2d0)) then
        tmp = 0.0d0
    else if (i <= 1.8d+20) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -7.2) {
		tmp = 0.0;
	} else if (i <= 1.8e+20) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -7.2:
		tmp = 0.0
	elif i <= 1.8e+20:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -7.2)
		tmp = 0.0;
	elseif (i <= 1.8e+20)
		tmp = Float64(100.0 * n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -7.2)
		tmp = 0.0;
	elseif (i <= 1.8e+20)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -7.2], 0.0, If[LessEqual[i, 1.8e+20], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -7.20000000000000018 or 1.8e20 < i

   1. Initial program 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sub-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in i around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. mul0-rgt N/A

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

      4. lower-/.f64 33.1

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

   7. Applied rewrites 33.1%

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

   8. Taylor expanded in i around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 33.1%

       \[\leadsto 0 \]

   if -7.20000000000000018 < i < 1.8e20

   1. Initial program 7.4%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.2:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.8 \cdot 10^{+20}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 18.1% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 25.5%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
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. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. sub-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-+.f64 N/A

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

4. Applied rewrites 24.4%

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

5. Taylor expanded in i around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt N/A

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

   4. lower-/.f64 18.5

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

7. Applied rewrites 18.5%

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

8. Taylor expanded in i around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 18.5%

    \[\leadsto 0 \]

11. Final simplification 18.5%

    \[\leadsto 0 \]
12. Add Preprocessing

Developer Target 1: 34.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(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))))