Compound Interest

Percentage Accurate: 29.0% → 93.5%

Time: 11.0s

Alternatives: 15

Speedup: 8.1×

• could not determine a ground truth

  1. i = 1.0218393146853892e-250
  2. n = 5.099207290362704e+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: 29.0% 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: 93.5% accurate, 0.3× speedup

Math

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

FPCore

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

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 <= 0.0) {
		tmp = (100.0 * (expm1((log1p((i / n)) * n)) / i)) * n;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 100.0 * fma((-1.0 / i), n, ((pow(((i / n) + 1.0), n) * n) / i));
	} else {
		tmp = 100.0 * n;
	}
	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 <= 0.0)
		tmp = Float64(Float64(100.0 * Float64(expm1(Float64(log1p(Float64(i / n)) * n)) / i)) * n);
	elseif (t_0 <= Inf)
		tmp = Float64(100.0 * fma(Float64(-1.0 / i), n, Float64(Float64((Float64(Float64(i / n) + 1.0) ^ n) * n) / i)));
	else
		tmp = Float64(100.0 * n);
	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, 0.0], N[(N[(100.0 * N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(100.0 * N[(N[(-1.0 / i), $MachinePrecision] * n + N[(N[(N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision] * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * n), $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 25.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 18.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. div-inv N/A

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

      15. neg-mul-1 N/A

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

      16. cancel-sign-sub-inv N/A

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

   6. Applied rewrites 96.9%

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

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

   1. Initial program 96.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 96.4

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

   6. Applied rewrites 96.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

Alternative 2: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.7 \cdot 10^{-302} \lor \neg \left(n \leq 4.2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n 3.7e-302) (not (<= n 4.2e-115)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (/ (* (- (log i) (log n)) n) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= 3.7e-302) || !(n <= 4.2e-115)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * (((log(i) - log(n)) * n) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= 3.7e-302) || !(n <= 4.2e-115)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * (((Math.log(i) - Math.log(n)) * n) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= 3.7e-302) or not (n <= 4.2e-115):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * (((math.log(i) - math.log(n)) * n) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= 3.7e-302) || !(n <= 4.2e-115))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(Float64(log(i) - log(n)) * n) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, 3.7e-302], N[Not[LessEqual[n, 4.2e-115]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.7e-302 or 4.20000000000000003e-115 < n

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 3.7e-302 < n < 4.20000000000000003e-115

   1. Initial program 26.4%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 79.0

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

   5. Applied rewrites 79.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.7 \cdot 10^{-302} \lor \neg \left(n \leq 4.2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\left(\log i - \log n\right) \cdot n}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.7 \cdot 10^{-302} \lor \neg \left(n \leq 4.2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{n}{i} \cdot 100\right) \cdot \left(\left(\log i - \log n\right) \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n 3.7e-302) (not (<= n 4.2e-115)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* (* (/ n i) 100.0) (* (- (log i) (log n)) n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= 3.7e-302) || !(n <= 4.2e-115)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n / i) * 100.0) * ((log(i) - log(n)) * n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= 3.7e-302) || !(n <= 4.2e-115)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n / i) * 100.0) * ((Math.log(i) - Math.log(n)) * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= 3.7e-302) or not (n <= 4.2e-115):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = ((n / i) * 100.0) * ((math.log(i) - math.log(n)) * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= 3.7e-302) || !(n <= 4.2e-115))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(Float64(n / i) * 100.0) * Float64(Float64(log(i) - log(n)) * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, 3.7e-302], N[Not[LessEqual[n, 4.2e-115]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(n / i), $MachinePrecision] * 100.0), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.7e-302 or 4.20000000000000003e-115 < n

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 3.7e-302 < n < 4.20000000000000003e-115

   1. Initial program 26.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. 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 26.5

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

      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 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 79.0

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

   7. Applied rewrites 79.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.7 \cdot 10^{-302} \lor \neg \left(n \leq 4.2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{n}{i} \cdot 100\right) \cdot \left(\left(\log i - \log n\right) \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.7 \cdot 10^{-302} \lor \neg \left(n \leq 4.2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot n\right) \cdot 100\right) \cdot \frac{\log i - \log n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n 3.7e-302) (not (<= n 4.2e-115)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* (* (* n n) 100.0) (/ (- (log i) (log n)) i))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= 3.7e-302) || !(n <= 4.2e-115)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n * n) * 100.0) * ((log(i) - log(n)) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= 3.7e-302) || !(n <= 4.2e-115)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = ((n * n) * 100.0) * ((Math.log(i) - Math.log(n)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= 3.7e-302) or not (n <= 4.2e-115):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = ((n * n) * 100.0) * ((math.log(i) - math.log(n)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= 3.7e-302) || !(n <= 4.2e-115))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(Float64(Float64(n * n) * 100.0) * Float64(Float64(log(i) - log(n)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, 3.7e-302], N[Not[LessEqual[n, 4.2e-115]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(n * n), $MachinePrecision] * 100.0), $MachinePrecision] * N[(N[(N[Log[i], $MachinePrecision] - N[Log[n], $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 3.7e-302 or 4.20000000000000003e-115 < n

   1. Initial program 25.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 3.7e-302 < n < 4.20000000000000003e-115

   1. Initial program 26.4%

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

   3. Taylor expanded in n around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-log.f64 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.7 \cdot 10^{-302} \lor \neg \left(n \leq 4.2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\left(n \cdot n\right) \cdot 100\right) \cdot \frac{\log i - \log n}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-167} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.4e-167) (not (<= n 3.5e-115)))
   (* (* (/ (expm1 i) i) 100.0) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.4e-167) || !(n <= 3.5e-115)) {
		tmp = ((expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.4e-167) || !(n <= 3.5e-115)) {
		tmp = ((Math.expm1(i) / i) * 100.0) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.4e-167) or not (n <= 3.5e-115):
		tmp = ((math.expm1(i) / i) * 100.0) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.4e-167) || !(n <= 3.5e-115))
		tmp = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.4e-167], N[Not[LessEqual[n, 3.5e-115]], $MachinePrecision]], N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.3999999999999999e-167 or 3.5000000000000002e-115 < n

   1. Initial program 20.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   if -4.3999999999999999e-167 < n < 3.5000000000000002e-115

   1. Initial program 45.1%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 68.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-167} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-167} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.4e-167) (not (<= n 3.5e-115)))
   (* (* (expm1 i) (/ 100.0 i)) n)
   (* 100.0 (/ (- 1.0 1.0) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.4e-167) || !(n <= 3.5e-115)) {
		tmp = (expm1(i) * (100.0 / i)) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.4e-167) || !(n <= 3.5e-115)) {
		tmp = (Math.expm1(i) * (100.0 / i)) * n;
	} else {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.4e-167) or not (n <= 3.5e-115):
		tmp = (math.expm1(i) * (100.0 / i)) * n
	else:
		tmp = 100.0 * ((1.0 - 1.0) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.4e-167) || !(n <= 3.5e-115))
		tmp = Float64(Float64(expm1(i) * Float64(100.0 / i)) * n);
	else
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.4e-167], N[Not[LessEqual[n, 3.5e-115]], $MachinePrecision]], N[(N[(N[(Exp[i] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.3999999999999999e-167 or 3.5000000000000002e-115 < n

   1. Initial program 20.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 87.9%

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

   if -4.3999999999999999e-167 < n < 3.5000000000000002e-115

   1. Initial program 45.1%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 68.3%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-167} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\mathsf{expm1}\left(i\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right)\\ \mathbf{if}\;n \leq -1.26 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(n \cdot t\_0\right) \cdot i\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;100 \cdot \frac{1 - 1}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)))
   (if (<= n -1.26e-151)
     (fma n 100.0 (* (* n t_0) i))
     (if (<= n 3.5e-115)
       (* 100.0 (/ (- 1.0 1.0) (/ i n)))
       (* (fma t_0 i 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0);
	double tmp;
	if (n <= -1.26e-151) {
		tmp = fma(n, 100.0, ((n * t_0) * i));
	} else if (n <= 3.5e-115) {
		tmp = 100.0 * ((1.0 - 1.0) / (i / n));
	} else {
		tmp = fma(t_0, i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0)
	tmp = 0.0
	if (n <= -1.26e-151)
		tmp = fma(n, 100.0, Float64(Float64(n * t_0) * i));
	elseif (n <= 3.5e-115)
		tmp = Float64(100.0 * Float64(Float64(1.0 - 1.0) / Float64(i / n)));
	else
		tmp = Float64(fma(t_0, i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision]}, If[LessEqual[n, -1.26e-151], N[(n * 100.0 + N[(N[(n * t$95$0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-115], N[(100.0 * N[(N[(1.0 - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2600000000000001e-151

   1. Initial program 20.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 66.9%

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

   10. Applied rewrites 66.9%

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

   if -1.2600000000000001e-151 < n < 3.5000000000000002e-115

   1. Initial program 46.3%

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

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 68.3%

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

   if 3.5000000000000002e-115 < n

   1. Initial program 19.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 75.5%

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

Alternative 8: 65.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.26e-151) (not (<= n 3.5e-115)))
   (*
    (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
    n)
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.26e-151) || !(n <= 3.5e-115)) {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.26e-151) || !(n <= 3.5e-115))
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.26e-151], N[Not[LessEqual[n, 3.5e-115]], $MachinePrecision]], N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.2600000000000001e-151 or 3.5000000000000002e-115 < n

   1. Initial program 19.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 70.9%

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

   if -1.2600000000000001e-151 < n < 3.5000000000000002e-115

   1. Initial program 46.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.3

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

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right)\\ \mathbf{if}\;n \leq -1.26 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(n \cdot t\_0\right) \cdot i\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0)))
   (if (<= n -1.26e-151)
     (fma n 100.0 (* (* n t_0) i))
     (if (<= n 3.5e-115) 0.0 (* (fma t_0 i 100.0) n)))))

C

double code(double i, double n) {
	double t_0 = fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0);
	double tmp;
	if (n <= -1.26e-151) {
		tmp = fma(n, 100.0, ((n * t_0) * i));
	} else if (n <= 3.5e-115) {
		tmp = 0.0;
	} else {
		tmp = fma(t_0, i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0)
	tmp = 0.0
	if (n <= -1.26e-151)
		tmp = fma(n, 100.0, Float64(Float64(n * t_0) * i));
	elseif (n <= 3.5e-115)
		tmp = 0.0;
	else
		tmp = Float64(fma(t_0, i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision]}, If[LessEqual[n, -1.26e-151], N[(n * 100.0 + N[(N[(n * t$95$0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-115], 0.0, N[(N[(t$95$0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2600000000000001e-151

   1. Initial program 20.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 66.9%

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

   10. Applied rewrites 66.9%

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

   if -1.2600000000000001e-151 < n < 3.5000000000000002e-115

   1. Initial program 46.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.3

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

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.3%

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

   if 3.5000000000000002e-115 < n

   1. Initial program 19.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 75.5%

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

Alternative 10: 63.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.26e-151) (not (<= n 3.5e-115)))
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.26e-151) || !(n <= 3.5e-115)) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.26e-151) || !(n <= 3.5e-115))
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.26e-151], N[Not[LessEqual[n, 3.5e-115]], $MachinePrecision]], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.2600000000000001e-151 or 3.5000000000000002e-115 < n

   1. Initial program 19.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 69.4%

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

   if -1.2600000000000001e-151 < n < 3.5000000000000002e-115

   1. Initial program 46.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.3

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

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(n \cdot \mathsf{fma}\left(16.666666666666668, i, 50\right), i, n \cdot 100\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.26e-151)
   (fma (* n (fma 16.666666666666668 i 50.0)) i (* n 100.0))
   (if (<= n 3.5e-115)
     0.0
     (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.26e-151) {
		tmp = fma((n * fma(16.666666666666668, i, 50.0)), i, (n * 100.0));
	} else if (n <= 3.5e-115) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.26e-151)
		tmp = fma(Float64(n * fma(16.666666666666668, i, 50.0)), i, Float64(n * 100.0));
	elseif (n <= 3.5e-115)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.26e-151], N[(N[(n * N[(16.666666666666668 * i + 50.0), $MachinePrecision]), $MachinePrecision] * i + N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-115], 0.0, N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2600000000000001e-151

   1. Initial program 20.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 65.4%

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

   if -1.2600000000000001e-151 < n < 3.5000000000000002e-115

   1. Initial program 46.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.3

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

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.3%

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

   if 3.5000000000000002e-115 < n

   1. Initial program 19.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 74.0%

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

Alternative 12: 61.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.26e-151) (not (<= n 3.5e-115)))
   (* (fma 50.0 i 100.0) n)
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.26e-151) || !(n <= 3.5e-115)) {
		tmp = fma(50.0, i, 100.0) * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.26e-151) || !(n <= 3.5e-115))
		tmp = Float64(fma(50.0, i, 100.0) * n);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.26e-151], N[Not[LessEqual[n, 3.5e-115]], $MachinePrecision]], N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.2600000000000001e-151 or 3.5000000000000002e-115 < n

   1. Initial program 19.9%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 65.4%

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

   if -1.2600000000000001e-151 < n < 3.5000000000000002e-115

   1. Initial program 46.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.3

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

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151} \lor \neg \left(n \leq 3.5 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(n, 100, \left(50 \cdot i\right) \cdot n\right)\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.26e-151)
   (fma n 100.0 (* (* 50.0 i) n))
   (if (<= n 3.5e-115) 0.0 (* (fma 50.0 i 100.0) n))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.26e-151) {
		tmp = fma(n, 100.0, ((50.0 * i) * n));
	} else if (n <= 3.5e-115) {
		tmp = 0.0;
	} else {
		tmp = fma(50.0, i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.26e-151)
		tmp = fma(n, 100.0, Float64(Float64(50.0 * i) * n));
	elseif (n <= 3.5e-115)
		tmp = 0.0;
	else
		tmp = Float64(fma(50.0, i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.26e-151], N[(n * 100.0 + N[(N[(50.0 * i), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.5e-115], 0.0, N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2600000000000001e-151

   1. Initial program 20.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 66.9%

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

   10. Applied rewrites 66.9%

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

   11. Taylor expanded in i around 0

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

   13. Applied rewrites 62.9%

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

   if -1.2600000000000001e-151 < n < 3.5000000000000002e-115

   1. Initial program 46.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.3

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

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.3%

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

   if 3.5000000000000002e-115 < n

   1. Initial program 19.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-expm1.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in i around 0

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

   8. Applied rewrites 68.4%

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

Alternative 14: 58.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 8 \cdot 10^{-12}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.15e-9) 0.0 (if (<= i 8e-12) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.15e-9) {
		tmp = 0.0;
	} else if (i <= 8e-12) {
		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 <= (-1.15d-9)) then
        tmp = 0.0d0
    else if (i <= 8d-12) 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 <= -1.15e-9) {
		tmp = 0.0;
	} else if (i <= 8e-12) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.15e-9:
		tmp = 0.0
	elif i <= 8e-12:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.15e-9)
		tmp = 0.0;
	elseif (i <= 8e-12)
		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 <= -1.15e-9)
		tmp = 0.0;
	elseif (i <= 8e-12)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.15e-9], 0.0, If[LessEqual[i, 8e-12], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.15e-9 or 7.99999999999999984e-12 < i

   1. Initial program 50.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

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

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

      11. distribute-frac-neg2 N/A

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

      12. lower-fma.f64 N/A

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

      13. frac-2neg N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lift-/.f64 N/A

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

      18. associate-/r/ N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in i around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 32.9

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

   7. Applied rewrites 32.9%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 32.9%

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

   if -1.15e-9 < i < 7.99999999999999984e-12

   1. Initial program 7.8%

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

   3. Taylor expanded in i around 0

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

      1. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

Alternative 15: 18.0% 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 26.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

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

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

   11. distribute-frac-neg2 N/A

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

   12. lower-fma.f64 N/A

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

   13. frac-2neg N/A

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

   14. remove-double-neg N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-/.f64 N/A

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

   18. associate-/r/ N/A

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

   19. lower-*.f64 N/A

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

4. Applied rewrites 21.4%

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

5. Taylor expanded in i around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 19.0

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

7. Applied rewrites 19.0%

   \[\leadsto \color{blue}{\frac{0}{i}} \]
8. Step-by-step derivation

9. Applied rewrites 19.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 35.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024317 
(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))))