Compound Interest

Percentage Accurate: 28.6% → 98.5%

Time: 17.1s

Alternatives: 14

Speedup: 8.7×

• could not determine a ground truth

  1. i = -2.0000000000000001e231
  2. n = 1.99999999999999997e77

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-74}:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-74)
     (* n (/ (+ (* t_0 100.0) -100.0) i))
     (if (<= t_1 0.0)
       (/ 100.0 (/ (/ i n) (expm1 (* n (log1p (/ i n))))))
       (if (<= t_1 INFINITY)
         (* 100.0 (* (/ n i) (+ -1.0 (pow (/ i n) n))))
         (/
          100.0
          (+
           (/ 1.0 n)
           (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-74) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 / ((i / n) / expm1((n * log1p((i / n)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((n / i) * (-1.0 + pow((i / n), n)));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-74) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 / ((i / n) / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((n / i) * (-1.0 + Math.pow((i / n), n)));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -5e-74:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	elif t_1 <= 0.0:
		tmp = 100.0 / ((i / n) / math.expm1((n * math.log1p((i / n)))))
	elif t_1 <= math.inf:
		tmp = 100.0 * ((n / i) * (-1.0 + math.pow((i / n), n)))
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-74)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 / Float64(Float64(i / n) / expm1(Float64(n * log1p(Float64(i / n))))));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(n / i) * Float64(-1.0 + (Float64(i / n) ^ n))));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-74], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 / N[(N[(i / n), $MachinePrecision] / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(-1.0 + N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -4.99999999999999998e-74

   1. Initial program 99.4%

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

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r/ 99.7%

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

      5. sub-neg 99.7%

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

      6. distribute-lft-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

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

      10. fma-define 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   if -4.99999999999999998e-74 < (/.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 23.9%

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

      1. associate-/r/ 23.5%

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

      2. associate-*r* 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-*r/ 23.4%

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

      5. sub-neg 23.4%

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

      6. distribute-lft-in 23.4%

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

      7. metadata-eval 23.4%

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

      8. metadata-eval 23.4%

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

      9. metadata-eval 23.4%

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

      10. fma-define 23.4%

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

      11. metadata-eval 23.4%

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

   3. Simplified 23.4%

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

      1. *-commutative 23.4%

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

      2. fma-undefine 23.4%

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

      3. *-commutative 23.4%

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

      4. associate-/r/ 23.9%

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

      5. metadata-eval 23.9%

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

      6. metadata-eval 23.9%

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

      7. distribute-rgt-in 23.9%

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

      8. sub-neg 23.9%

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

      9. associate-*r/ 23.9%

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

      10. clear-num 23.9%

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

      11. un-div-inv 23.9%

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

      12. add-exp-log 23.9%

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

      13. expm1-define 23.9%

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

      14. log-pow 33.9%

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

      15. log1p-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   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 99.7%

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

   3. Taylor expanded in i around inf 99.7%

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

   4. Taylor expanded in n around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*l/ 99.9%

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

      4. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   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. Step-by-step derivation

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. sub-neg 0.0%

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

      9. associate-*r/ 0.0%

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

      10. clear-num 0.0%

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

      11. un-div-inv 0.0%

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

      12. add-exp-log 0.0%

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

      13. expm1-define 0.0%

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

      14. log-pow 0.0%

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

      15. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in i around 0 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-74}:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{100}{\frac{\frac{i}{n}}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1e-8)
     (/ (+ (* t_0 100.0) -100.0) (/ i n))
     (if (<= t_1 0.0)
       (* n (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) i)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* (/ n i) (+ -1.0 (pow (/ i n) n))))
         (/
          100.0
          (+
           (/ 1.0 n)
           (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = n * (100.0 * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((n / i) * (-1.0 + pow((i / n), n)));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = n * (100.0 * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((n / i) * (-1.0 + Math.pow((i / n), n)));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -1e-8:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	elif t_1 <= 0.0:
		tmp = n * (100.0 * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = 100.0 * ((n / i) * (-1.0 + math.pow((i / n), n)))
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-8)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(n / i) * Float64(-1.0 + (Float64(i / n) ^ n))));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-8], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(-1.0 + N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e-8

   1. Initial program 99.7%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   if -1e-8 < (/.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 26.4%

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

      1. associate-*r/ 26.4%

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

      2. sub-neg 26.4%

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

      3. distribute-rgt-in 26.4%

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

      4. metadata-eval 26.4%

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

      5. metadata-eval 26.4%

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

   3. Simplified 26.4%

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

      1. metadata-eval 26.4%

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

      2. metadata-eval 26.4%

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

      3. distribute-rgt-in 26.4%

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

      4. sub-neg 26.4%

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

      5. associate-*r/ 26.4%

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

      6. associate-/r/ 26.0%

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

      7. associate-*r* 26.0%

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

      8. add-exp-log 26.0%

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

      9. expm1-define 26.0%

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

      10. log-pow 35.7%

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

      11. log1p-define 98.2%

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

   6. Applied egg-rr 98.2%

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

   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 99.7%

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

   3. Taylor expanded in i around inf 99.7%

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

   4. Taylor expanded in n around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*l/ 99.9%

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

      4. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   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. Step-by-step derivation

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. sub-neg 0.0%

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

      9. associate-*r/ 0.0%

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

      10. clear-num 0.0%

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

      11. un-div-inv 0.0%

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

      12. add-exp-log 0.0%

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

      13. expm1-define 0.0%

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

      14. log-pow 0.0%

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

      15. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in i around 0 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -1e-8)
     (/ (+ (* t_0 100.0) -100.0) (/ i n))
     (if (<= t_1 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* 100.0 (* (/ n i) (+ -1.0 (pow (/ i n) n))))
         (/
          100.0
          (+
           (/ 1.0 n)
           (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((n / i) * (-1.0 + pow((i / n), n)));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -1e-8) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * Math.expm1((n * Math.log1p((i / n))))) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((n / i) * (-1.0 + Math.pow((i / n), n)));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -1e-8:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	elif t_1 <= 0.0:
		tmp = n * ((100.0 * math.expm1((n * math.log1p((i / n))))) / i)
	elif t_1 <= math.inf:
		tmp = 100.0 * ((n / i) * (-1.0 + math.pow((i / n), n)))
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-8)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(n / i) * Float64(-1.0 + (Float64(i / n) ^ n))));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-8], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(-1.0 + N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -1e-8

   1. Initial program 99.7%

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

      1. associate-*r/ 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   if -1e-8 < (/.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 26.4%

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

      1. associate-/r/ 26.0%

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

      2. associate-*r* 26.0%

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

      3. *-commutative 26.0%

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

      4. associate-*r/ 26.0%

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

      5. sub-neg 26.0%

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

      6. distribute-lft-in 26.0%

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

      7. metadata-eval 26.0%

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

      8. metadata-eval 26.0%

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

      9. metadata-eval 26.0%

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

      10. fma-define 26.0%

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

      11. metadata-eval 26.0%

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

   3. Simplified 26.0%

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

      1. fma-undefine 26.0%

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

      2. metadata-eval 26.0%

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

      3. metadata-eval 26.0%

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

      4. distribute-lft-in 26.0%

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

      5. sub-neg 26.0%

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

      6. *-commutative 26.0%

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

      7. add-exp-log 26.0%

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

      8. expm1-define 26.0%

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

      9. log-pow 35.7%

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

      10. log1p-define 98.2%

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

   6. Applied egg-rr 98.2%

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

   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 99.7%

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

   3. Taylor expanded in i around inf 99.7%

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

   4. Taylor expanded in n around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*l/ 99.9%

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

      4. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   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. Step-by-step derivation

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

      1. *-commutative 1.9%

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

      2. fma-undefine 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-/r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. sub-neg 0.0%

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

      9. associate-*r/ 0.0%

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

      10. clear-num 0.0%

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

      11. un-div-inv 0.0%

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

      12. add-exp-log 0.0%

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

      13. expm1-define 0.0%

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

      14. log-pow 0.0%

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

      15. log1p-define 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in i around 0 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \left(-1 + {\left(\frac{i}{n}\right)}^{n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-31} \lor \neg \left(n \leq 1.4\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.7e-31) (not (<= n 1.4)))
   (* 100.0 (* n (/ (expm1 i) i)))
   (/
    100.0
    (+ (/ 1.0 n) (* i (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n))))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.7e-31) || !(n <= 1.4)) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.7e-31) || !(n <= 1.4)) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else {
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.7e-31) or not (n <= 1.4):
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = 100.0 / ((1.0 / n) + (i * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.7e-31) || !(n <= 1.4))
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = Float64(100.0 / Float64(Float64(1.0 / n) + Float64(i * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.7e-31], N[Not[LessEqual[n, 1.4]], $MachinePrecision]], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 / N[(N[(1.0 / n), $MachinePrecision] + N[(i * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.70000000000000014e-31 or 1.3999999999999999 < n

   1. Initial program 25.6%

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

   3. Taylor expanded in n around inf 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-/l* 42.1%

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

      3. expm1-define 91.6%

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

   5. Simplified 91.6%

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

   if -2.70000000000000014e-31 < n < 1.3999999999999999

   1. Initial program 35.6%

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

      1. associate-/r/ 34.8%

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

      2. associate-*r* 34.8%

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

      3. *-commutative 34.8%

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

      4. associate-*r/ 34.8%

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

      5. sub-neg 34.8%

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

      6. distribute-lft-in 34.8%

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

      7. metadata-eval 34.8%

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

      8. metadata-eval 34.8%

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

      9. metadata-eval 34.8%

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

      10. fma-define 34.8%

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

      11. metadata-eval 34.8%

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

   3. Simplified 34.8%

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

      1. *-commutative 34.8%

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

      2. fma-undefine 34.8%

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

      3. *-commutative 34.8%

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

      4. associate-/r/ 35.6%

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

      5. metadata-eval 35.6%

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

      6. metadata-eval 35.6%

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

      7. distribute-rgt-in 35.6%

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

      8. sub-neg 35.6%

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

      9. associate-*r/ 35.6%

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

      10. clear-num 35.6%

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

      11. un-div-inv 35.6%

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

      12. add-exp-log 35.6%

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

      13. expm1-define 35.6%

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

      14. log-pow 55.1%

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

      15. log1p-define 87.4%

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

   6. Applied egg-rr 87.4%

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

   7. Taylor expanded in i around 0 75.0%

      \[\leadsto \frac{100}{\color{blue}{i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-31} \lor \neg \left(n \leq 1.4\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100}{\frac{1}{n} + i \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{-259} \lor \neg \left(n \leq 1.4 \cdot 10^{-111}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.55e-259) (not (<= n 1.4e-111)))
   (* 100.0 (* n (/ (expm1 i) i)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.55e-259) || !(n <= 1.4e-111)) {
		tmp = 100.0 * (n * (expm1(i) / i));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.55e-259) || !(n <= 1.4e-111)) {
		tmp = 100.0 * (n * (Math.expm1(i) / i));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.55e-259) or not (n <= 1.4e-111):
		tmp = 100.0 * (n * (math.expm1(i) / i))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.55e-259) || !(n <= 1.4e-111))
		tmp = Float64(100.0 * Float64(n * Float64(expm1(i) / i)));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.55e-259], N[Not[LessEqual[n, 1.4e-111]], $MachinePrecision]], N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.5499999999999999e-259 or 1.39999999999999998e-111 < n

   1. Initial program 25.7%

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

   3. Taylor expanded in n around inf 35.7%

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

      1. *-commutative 35.7%

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

      2. associate-/l* 35.7%

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

      3. expm1-define 85.7%

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

   5. Simplified 85.7%

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

   if -1.5499999999999999e-259 < n < 1.39999999999999998e-111

   1. Initial program 48.1%

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

      1. associate-*r/ 48.1%

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

      2. sub-neg 48.1%

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

      3. distribute-rgt-in 48.1%

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

      4. metadata-eval 48.1%

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

      5. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in i around 0 76.3%

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

   6. Taylor expanded in i around 0 76.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{-259} \lor \neg \left(n \leq 1.4 \cdot 10^{-111}\right):\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-259} \lor \neg \left(n \leq 1.55 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.4e-259) (not (<= n 1.55e-111)))
   (* n (/ (* 100.0 (expm1 i)) i))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e-259) || !(n <= 1.55e-111)) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e-259) || !(n <= 1.55e-111)) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.4e-259) or not (n <= 1.55e-111):
		tmp = n * ((100.0 * math.expm1(i)) / i)
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.4e-259) || !(n <= 1.55e-111))
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.4e-259], N[Not[LessEqual[n, 1.55e-111]], $MachinePrecision]], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e-259 or 1.55000000000000007e-111 < n

   1. Initial program 25.7%

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

      1. associate-/r/ 25.7%

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

      2. associate-*r* 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*r/ 25.7%

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

      5. sub-neg 25.7%

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

      6. distribute-lft-in 25.6%

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

      7. metadata-eval 25.6%

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

      8. metadata-eval 25.6%

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

      9. metadata-eval 25.6%

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

      10. fma-define 25.7%

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

      11. metadata-eval 25.7%

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

   3. Simplified 25.7%

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

   5. Taylor expanded in n around inf 35.7%

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

      1. sub-neg 35.7%

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

      2. metadata-eval 35.7%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 35.7%

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

      4. distribute-lft-in 35.7%

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

      5. metadata-eval 35.7%

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

      6. sub-neg 35.7%

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

      7. expm1-define 85.6%

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

   7. Simplified 85.6%

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

   if -1.4e-259 < n < 1.55000000000000007e-111

   1. Initial program 48.1%

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

      1. associate-*r/ 48.1%

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

      2. sub-neg 48.1%

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

      3. distribute-rgt-in 48.1%

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

      4. metadata-eval 48.1%

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

      5. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in i around 0 76.3%

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

   6. Taylor expanded in i around 0 76.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-259} \lor \neg \left(n \leq 1.55 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.4 \cdot 10^{-25} \lor \neg \left(i \leq 3.2 \cdot 10^{-62}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.4e-25) (not (<= i 3.2e-62)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (* 100.0 (+ n (* (* i n) (- 0.5 (/ 0.5 n)))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.4e-25) || !(i <= 3.2e-62)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.4e-25) || !(i <= 3.2e-62)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.4e-25) or not (i <= 3.2e-62):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.4e-25) || !(i <= 3.2e-62))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(Float64(i * n) * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.4e-25], N[Not[LessEqual[i, 3.2e-62]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(N[(i * n), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.39999999999999994e-25 or 3.20000000000000021e-62 < i

   1. Initial program 47.6%

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

   3. Taylor expanded in n around inf 56.9%

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

      1. expm1-define 64.3%

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

   5. Simplified 64.3%

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

   if -1.39999999999999994e-25 < i < 3.20000000000000021e-62

   1. Initial program 6.7%

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

   3. Taylor expanded in i around 0 89.9%

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(n \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 90.1%

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

      2. *-commutative 90.1%

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

      3. associate-*r/ 90.1%

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

      4. metadata-eval 90.1%

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

   5. Simplified 90.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(n \cdot i\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.4 \cdot 10^{-25} \lor \neg \left(i \leq 3.2 \cdot 10^{-62}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-139}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.5e-139)
   (*
    n
    (+
     100.0
     (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
   (if (<= n 1.4e-111)
     0.0
     (*
      100.0
      (*
       n
       (/
        (*
         i
         (+
          1.0
          (*
           i
           (+ 0.5 (* i (+ 0.16666666666666666 (* i 0.041666666666666664)))))))
        i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-139) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else if (n <= 1.4e-111) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.5d-139)) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    else if (n <= 1.4d-111) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n * ((i * (1.0d0 + (i * (0.5d0 + (i * (0.16666666666666666d0 + (i * 0.041666666666666664d0))))))) / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-139) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else if (n <= 1.4e-111) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.5e-139:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	elif n <= 1.4e-111:
		tmp = 0.0
	else:
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.5e-139)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	elseif (n <= 1.4e-111)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n * Float64(Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * Float64(0.16666666666666666 + Float64(i * 0.041666666666666664))))))) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.5e-139)
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	elseif (n <= 1.4e-111)
		tmp = 0.0;
	else
		tmp = 100.0 * (n * ((i * (1.0 + (i * (0.5 + (i * (0.16666666666666666 + (i * 0.041666666666666664))))))) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.5e-139], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4e-111], 0.0, N[(100.0 * N[(n * N[(N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * N[(0.16666666666666666 + N[(i * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.5000000000000006e-139

   1. Initial program 29.2%

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

      1. associate-/r/ 28.9%

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

      2. associate-*r* 28.8%

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

      3. *-commutative 28.8%

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

      4. associate-*r/ 28.8%

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

      5. sub-neg 28.8%

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

      6. distribute-lft-in 28.8%

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

      7. metadata-eval 28.8%

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

      8. metadata-eval 28.8%

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

      9. metadata-eval 28.8%

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

      10. fma-define 28.8%

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

      11. metadata-eval 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in n around inf 35.2%

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

      1. sub-neg 35.2%

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

      2. metadata-eval 35.2%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 35.2%

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

      4. distribute-lft-in 35.3%

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

      5. metadata-eval 35.3%

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

      6. sub-neg 35.3%

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

      7. expm1-define 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in i around 0 56.7%

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

      1. *-commutative 56.7%

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

   10. Simplified 56.7%

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

   if -9.5000000000000006e-139 < n < 1.39999999999999998e-111

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in i around 0 69.0%

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

   6. Taylor expanded in i around 0 69.0%

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

   if 1.39999999999999998e-111 < n

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-/l* 36.9%

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

      3. expm1-define 91.0%

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

   5. Simplified 91.0%

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

   6. Taylor expanded in i around 0 79.8%

      \[\leadsto \left(n \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot i\right)\right)\right)}}{i}\right) \cdot 100 \]
   7. Step-by-step derivation

      1. *-commutative 79.8%

         \[\leadsto \left(n \cdot \frac{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + \color{blue}{i \cdot 0.041666666666666664}\right)\right)\right)}{i}\right) \cdot 100 \]

   8. Simplified 79.8%

      \[\leadsto \left(n \cdot \frac{\color{blue}{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)}}{i}\right) \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-139}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{i \cdot \left(1 + i \cdot \left(0.5 + i \cdot \left(0.16666666666666666 + i \cdot 0.041666666666666664\right)\right)\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\\ \mathbf{if}\;n \leq -1.32 \cdot 10^{-138}:\\ \;\;\;\;n \cdot t\_0\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-111}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot t\_0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (+
          100.0
          (*
           i
           (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667))))))))
   (if (<= n -1.32e-138)
     (* n t_0)
     (if (<= n 1.4e-111) 0.0 (* n (/ (* i t_0) i))))))

C

double code(double i, double n) {
	double t_0 = 100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))));
	double tmp;
	if (n <= -1.32e-138) {
		tmp = n * t_0;
	} else if (n <= 1.4e-111) {
		tmp = 0.0;
	} else {
		tmp = n * ((i * t_0) / i);
	}
	return tmp;
}

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 = 100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0)))))
    if (n <= (-1.32d-138)) then
        tmp = n * t_0
    else if (n <= 1.4d-111) then
        tmp = 0.0d0
    else
        tmp = n * ((i * t_0) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))));
	double tmp;
	if (n <= -1.32e-138) {
		tmp = n * t_0;
	} else if (n <= 1.4e-111) {
		tmp = 0.0;
	} else {
		tmp = n * ((i * t_0) / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))))
	tmp = 0
	if n <= -1.32e-138:
		tmp = n * t_0
	elif n <= 1.4e-111:
		tmp = 0.0
	else:
		tmp = n * ((i * t_0) / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667))))))
	tmp = 0.0
	if (n <= -1.32e-138)
		tmp = Float64(n * t_0);
	elseif (n <= 1.4e-111)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(Float64(i * t_0) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667)))));
	tmp = 0.0;
	if (n <= -1.32e-138)
		tmp = n * t_0;
	elseif (n <= 1.4e-111)
		tmp = 0.0;
	else
		tmp = n * ((i * t_0) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.32e-138], N[(n * t$95$0), $MachinePrecision], If[LessEqual[n, 1.4e-111], 0.0, N[(n * N[(N[(i * t$95$0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.32e-138

   1. Initial program 29.2%

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

      1. associate-/r/ 28.9%

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

      2. associate-*r* 28.8%

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

      3. *-commutative 28.8%

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

      4. associate-*r/ 28.8%

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

      5. sub-neg 28.8%

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

      6. distribute-lft-in 28.8%

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

      7. metadata-eval 28.8%

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

      8. metadata-eval 28.8%

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

      9. metadata-eval 28.8%

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

      10. fma-define 28.8%

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

      11. metadata-eval 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in n around inf 35.2%

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

      1. sub-neg 35.2%

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

      2. metadata-eval 35.2%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 35.2%

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

      4. distribute-lft-in 35.3%

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

      5. metadata-eval 35.3%

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

      6. sub-neg 35.3%

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

      7. expm1-define 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in i around 0 56.7%

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

      1. *-commutative 56.7%

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

   10. Simplified 56.7%

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

   if -1.32e-138 < n < 1.39999999999999998e-111

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in i around 0 69.0%

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

   6. Taylor expanded in i around 0 69.0%

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

   if 1.39999999999999998e-111 < n

   1. Initial program 16.6%

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

      1. associate-/r/ 17.0%

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

      2. associate-*r* 17.0%

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

      3. *-commutative 17.0%

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

      4. associate-*r/ 17.0%

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

      5. sub-neg 17.0%

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

      6. distribute-lft-in 17.0%

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

      7. metadata-eval 17.0%

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

      8. metadata-eval 17.0%

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

      9. metadata-eval 17.0%

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

      10. fma-define 17.0%

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

      11. metadata-eval 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in n around inf 36.8%

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

      1. sub-neg 36.8%

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

      2. metadata-eval 36.8%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 36.8%

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

      4. distribute-lft-in 36.9%

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

      5. metadata-eval 36.9%

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

      6. sub-neg 36.9%

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

      7. expm1-define 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in i around 0 79.7%

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

      1. *-commutative 79.7%

         \[\leadsto n \cdot \frac{i \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right)}{i} \]

   10. Simplified 79.7%

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

Alternative 10: 66.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-138} \lor \neg \left(n \leq 1.4 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.85e-138) (not (<= n 1.4e-111)))
   (*
    n
    (+
     100.0
     (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-138) || !(n <= 1.4e-111)) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} 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 ((n <= (-1.85d-138)) .or. (.not. (n <= 1.4d-111))) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-138) || !(n <= 1.4e-111)) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.85e-138) or not (n <= 1.4e-111):
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.85e-138) || !(n <= 1.4e-111))
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.85e-138) || ~((n <= 1.4e-111)))
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.85e-138], N[Not[LessEqual[n, 1.4e-111]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.84999999999999995e-138 or 1.39999999999999998e-111 < n

   1. Initial program 23.2%

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

      1. associate-/r/ 23.2%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.2%

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

      5. sub-neg 23.2%

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

      6. distribute-lft-in 23.2%

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

      7. metadata-eval 23.2%

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

      8. metadata-eval 23.2%

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

      9. metadata-eval 23.2%

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

      10. fma-define 23.2%

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

      11. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in n around inf 36.0%

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

      1. sub-neg 36.0%

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

      2. metadata-eval 36.0%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 36.0%

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

      4. distribute-lft-in 36.0%

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

      5. metadata-eval 36.0%

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

      6. sub-neg 36.0%

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

      7. expm1-define 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in i around 0 67.3%

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

      1. *-commutative 67.3%

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

   10. Simplified 67.3%

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

   if -1.84999999999999995e-138 < n < 1.39999999999999998e-111

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in i around 0 69.0%

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

   6. Taylor expanded in i around 0 69.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-138} \lor \neg \left(n \leq 1.4 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-139} \lor \neg \left(n \leq 1.4 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9.2e-139) (not (<= n 1.4e-111)))
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9.2e-139) || !(n <= 1.4e-111)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} 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 ((n <= (-9.2d-139)) .or. (.not. (n <= 1.4d-111))) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9.2e-139) || !(n <= 1.4e-111)) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9.2e-139) or not (n <= 1.4e-111):
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9.2e-139) || !(n <= 1.4e-111))
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -9.2e-139) || ~((n <= 1.4e-111)))
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9.2e-139], N[Not[LessEqual[n, 1.4e-111]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -9.2000000000000005e-139 or 1.39999999999999998e-111 < n

   1. Initial program 23.2%

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

      1. associate-/r/ 23.2%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.2%

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

      5. sub-neg 23.2%

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

      6. distribute-lft-in 23.2%

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

      7. metadata-eval 23.2%

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

      8. metadata-eval 23.2%

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

      9. metadata-eval 23.2%

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

      10. fma-define 23.2%

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

      11. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in n around inf 36.0%

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

      1. sub-neg 36.0%

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

      2. metadata-eval 36.0%

         \[\leadsto n \cdot \frac{100 \cdot e^{i} + \color{blue}{-100}}{i} \]

      3. metadata-eval 36.0%

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

      4. distribute-lft-in 36.0%

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

      5. metadata-eval 36.0%

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

      6. sub-neg 36.0%

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

      7. expm1-define 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in i around 0 65.4%

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

      1. *-commutative 65.4%

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

   10. Simplified 65.4%

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

   if -9.2000000000000005e-139 < n < 1.39999999999999998e-111

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in i around 0 69.0%

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

   6. Taylor expanded in i around 0 69.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-139} \lor \neg \left(n \leq 1.4 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-139} \lor \neg \left(n \leq 3 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9.2e-139) (not (<= n 3e-111))) (* n (+ 100.0 (* i 50.0))) 0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9.2e-139) || !(n <= 3e-111)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-9.2d-139)) .or. (.not. (n <= 3d-111))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9.2e-139) || !(n <= 3e-111)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9.2e-139) or not (n <= 3e-111):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9.2e-139) || !(n <= 3e-111))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -9.2e-139) || ~((n <= 3e-111)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9.2e-139], N[Not[LessEqual[n, 3e-111]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -9.2000000000000005e-139 or 3.00000000000000008e-111 < n

   1. Initial program 23.2%

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

      1. associate-/r/ 23.2%

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

      2. associate-*r* 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-*r/ 23.2%

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

      5. sub-neg 23.2%

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

      6. distribute-lft-in 23.2%

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

      7. metadata-eval 23.2%

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

      8. metadata-eval 23.2%

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

      9. metadata-eval 23.2%

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

      10. fma-define 23.2%

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

      11. metadata-eval 23.2%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in i around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-*r/ 63.4%

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

      3. metadata-eval 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in n around inf 62.9%

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

      1. *-commutative 62.9%

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

   10. Simplified 62.9%

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

   if -9.2000000000000005e-139 < n < 3.00000000000000008e-111

   1. Initial program 49.7%

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

      1. associate-*r/ 49.7%

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

      2. sub-neg 49.7%

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

      3. distribute-rgt-in 49.7%

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

      4. metadata-eval 49.7%

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

      5. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in i around 0 69.0%

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

   6. Taylor expanded in i around 0 69.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-139} \lor \neg \left(n \leq 3 \cdot 10^{-111}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.2:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.35e+17) 0.0 (if (<= i 1.2) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.35e+17) {
		tmp = 0.0;
	} else if (i <= 1.2) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.35d+17)) then
        tmp = 0.0d0
    else if (i <= 1.2d0) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.35e+17) {
		tmp = 0.0;
	} else if (i <= 1.2) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.35e+17:
		tmp = 0.0
	elif i <= 1.2:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.35e+17)
		tmp = 0.0;
	elseif (i <= 1.2)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.35e+17)
		tmp = 0.0;
	elseif (i <= 1.2)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.35e+17], 0.0, If[LessEqual[i, 1.2], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.35e17 or 1.19999999999999996 < i

   1. Initial program 54.2%

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

      1. associate-*r/ 54.3%

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

      2. sub-neg 54.3%

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

      3. distribute-rgt-in 54.2%

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

      4. metadata-eval 54.2%

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

      5. metadata-eval 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in i around 0 29.8%

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

   6. Taylor expanded in i around 0 29.8%

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

   if -1.35e17 < i < 1.19999999999999996

   1. Initial program 8.2%

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

      1. associate-/r/ 8.8%

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

      2. associate-*r* 8.8%

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

      3. *-commutative 8.8%

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

      4. associate-*r/ 8.8%

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

      5. sub-neg 8.8%

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

      6. distribute-lft-in 8.8%

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

      7. metadata-eval 8.8%

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

      8. metadata-eval 8.8%

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

      9. metadata-eval 8.8%

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

      10. fma-define 8.8%

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

      11. metadata-eval 8.8%

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

   3. Simplified 8.8%

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

   5. Taylor expanded in i around 0 84.8%

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

      1. *-commutative 84.8%

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

   7. Simplified 84.8%

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

Alternative 14: 18.3% accurate, 114.0× 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 29.1%

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

   1. associate-*r/ 29.1%

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

   2. sub-neg 29.1%

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

   3. distribute-rgt-in 29.1%

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

   4. metadata-eval 29.1%

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

   5. metadata-eval 29.1%

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

3. Simplified 29.1%

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

5. Taylor expanded in i around 0 18.0%

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

6. Taylor expanded in i around 0 18.3%

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

Developer Target 1: 34.4% 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 2024160 
(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))))