Compound Interest

Percentage Accurate: 29.0% → 99.0%

Time: 21.1s

Alternatives: 13

Speedup: 38.0×

• 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: 29.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-48}:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{t\_0}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

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-48)
     (* n (/ (+ (* (pow (/ i n) n) 100.0) -100.0) i))
     (if (<= t_1 0.0)
       (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (+ (/ t_0 i) (/ -1.0 i))))
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

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-48) {
		tmp = n * (((pow((i / n), n) * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-48) {
		tmp = n * (((Math.pow((i / n), n) * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-48:
		tmp = n * (((math.pow((i / n), n) * 100.0) + -100.0) / i)
	elif t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)))
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	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-48)
		tmp = Float64(n * Float64(Float64(Float64((Float64(i / n) ^ n) * 100.0) + -100.0) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64(t_0 / i) + Float64(-1.0 / i))));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	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-48], N[(n * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $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)) < -9.9999999999999997e-49

   1. Initial program 99.5%

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

      1. associate-/r/ 99.5%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

      10. fma-define 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in i around inf 99.8%

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

   if -9.9999999999999997e-49 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 25.6%

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

      1. *-commutative 25.6%

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

      2. frac-2neg 25.6%

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

      3. associate-*l/ 25.6%

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

      4. add-exp-log 25.6%

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

      5. expm1-define 25.6%

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

      6. log-pow 39.9%

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

      7. log1p-define 99.6%

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

      8. distribute-neg-frac2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. times-frac 99.7%

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

      4. metadata-eval 99.7%

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

      5. add-sqr-sqrt 55.1%

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

      6. sqrt-unprod 51.1%

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

      7. sqr-neg 51.1%

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

      8. sqrt-unprod 7.5%

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

      9. add-sqr-sqrt 19.7%

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

      10. add-sqr-sqrt 12.2%

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

      11. sqrt-unprod 45.8%

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

      12. sqr-neg 45.8%

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

      13. sqrt-unprod 45.4%

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

      14. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

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

   1. Initial program 98.2%

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

      1. associate-/r/ 98.2%

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

      2. add-exp-log 98.2%

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

      3. expm1-define 98.2%

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

      4. log-pow 59.2%

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

      5. log1p-define 59.2%

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

   4. Applied egg-rr 59.2%

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

      1. expm1-undefine 57.7%

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

      2. div-sub 58.0%

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

      3. *-commutative 58.0%

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

      4. log1p-undefine 58.0%

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

      5. exp-to-pow 98.5%

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

      6. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. Taylor expanded in n around inf 1.9%

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

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-define 76.8%

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

   7. Simplified 76.8%

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

      1. clear-num 76.7%

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

      2. un-div-inv 76.6%

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

      3. *-un-lft-identity 76.6%

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

      4. times-frac 76.7%

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

      5. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 76.7%

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

   11. Simplified 76.7%

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

   12. Taylor expanded in i around 0 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-48}:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-100 + t\_0 \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{t\_0}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

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 -2e-81)
     (/ (+ -100.0 (* t_0 100.0)) (/ i n))
     (if (<= t_1 0.0)
       (* 100.0 (* n (/ (expm1 (* n (log1p (/ i n)))) i)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (+ (/ t_0 i) (/ -1.0 i))))
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

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 <= -2e-81) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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 <= -2e-81) {
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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 <= -2e-81:
		tmp = (-100.0 + (t_0 * 100.0)) / (i / n)
	elif t_1 <= 0.0:
		tmp = 100.0 * (n * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)))
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	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 <= -2e-81)
		tmp = Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / Float64(i / n));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64(t_0 / i) + Float64(-1.0 / i))));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	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, -2e-81], N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n * 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[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $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)) < -1.9999999999999999e-81

   1. Initial program 99.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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 -1.9999999999999999e-81 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 24.0%

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

      1. associate-/r/ 24.0%

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

      2. add-exp-log 24.0%

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

      3. expm1-define 24.0%

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

      4. log-pow 38.6%

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

      5. log1p-define 99.1%

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

   4. Applied egg-rr 99.1%

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

   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 98.2%

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

      1. associate-/r/ 98.2%

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

      2. add-exp-log 98.2%

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

      3. expm1-define 98.2%

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

      4. log-pow 59.2%

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

      5. log1p-define 59.2%

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

   4. Applied egg-rr 59.2%

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

      1. expm1-undefine 57.7%

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

      2. div-sub 58.0%

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

      3. *-commutative 58.0%

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

      4. log1p-undefine 58.0%

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

      5. exp-to-pow 98.5%

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

      6. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. Taylor expanded in n around inf 1.9%

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

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-define 76.8%

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

   7. Simplified 76.8%

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

      1. clear-num 76.7%

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

      2. un-div-inv 76.6%

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

      3. *-un-lft-identity 76.6%

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

      4. times-frac 76.7%

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

      5. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 76.7%

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

   11. Simplified 76.7%

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

   12. Taylor expanded in i around 0 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-48}:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{t\_0}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

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-48)
     (* n (/ (+ (* (pow (/ i n) n) 100.0) -100.0) i))
     (if (<= t_1 0.0)
       (* 100.0 (/ n (/ i (expm1 i))))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (+ (/ t_0 i) (/ -1.0 i))))
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

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-48) {
		tmp = n * (((pow((i / n), n) * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-48) {
		tmp = n * (((Math.pow((i / n), n) * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-48:
		tmp = n * (((math.pow((i / n), n) * 100.0) + -100.0) / i)
	elif t_1 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * ((t_0 / i) + (-1.0 / i)))
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	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-48)
		tmp = Float64(n * Float64(Float64(Float64((Float64(i / n) ^ n) * 100.0) + -100.0) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(Float64(t_0 / i) + Float64(-1.0 / i))));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	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-48], N[(n * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(N[(t$95$0 / i), $MachinePrecision] + N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $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)) < -9.9999999999999997e-49

   1. Initial program 99.5%

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

      1. associate-/r/ 99.5%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

      10. fma-define 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in i around inf 99.8%

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

   if -9.9999999999999997e-49 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 25.6%

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

      1. associate-/r/ 25.6%

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

      2. associate-*r* 25.6%

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

      3. *-commutative 25.6%

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

      4. associate-*r/ 25.6%

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

      5. sub-neg 25.6%

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

      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 25.6%

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

      11. metadata-eval 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in n around inf 37.2%

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

      1. sub-neg 37.2%

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

      2. metadata-eval 37.2%

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

      3. metadata-eval 37.2%

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

      4. distribute-lft-in 37.2%

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

      5. metadata-eval 37.2%

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

      6. sub-neg 37.2%

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

      7. expm1-define 74.8%

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

   7. Simplified 74.8%

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

      1. clear-num 74.8%

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

      2. un-div-inv 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. times-frac 74.8%

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

      5. metadata-eval 74.8%

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

   9. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 74.7%

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

   11. Simplified 74.7%

       \[\leadsto \color{blue}{\frac{n}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 74.7%

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

       2. associate-/l* 74.8%

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

       3. times-frac 74.9%

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

       4. metadata-eval 74.9%

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

   13. Applied egg-rr 74.9%

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

   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 98.2%

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

      1. associate-/r/ 98.2%

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

      2. add-exp-log 98.2%

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

      3. expm1-define 98.2%

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

      4. log-pow 59.2%

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

      5. log1p-define 59.2%

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

   4. Applied egg-rr 59.2%

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

      1. expm1-undefine 57.7%

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

      2. div-sub 58.0%

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

      3. *-commutative 58.0%

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

      4. log1p-undefine 58.0%

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

      5. exp-to-pow 98.5%

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

      6. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. Taylor expanded in n around inf 1.9%

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

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-define 76.8%

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

   7. Simplified 76.8%

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

      1. clear-num 76.7%

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

      2. un-div-inv 76.6%

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

      3. *-un-lft-identity 76.6%

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

      4. times-frac 76.7%

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

      5. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 76.7%

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

   11. Simplified 76.7%

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

   12. Taylor expanded in i around 0 99.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-48}:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} + \frac{-1}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-48}:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + t\_0 \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

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-48)
     (* n (/ (+ (* (pow (/ i n) n) 100.0) -100.0) i))
     (if (<= t_1 0.0)
       (* 100.0 (/ n (/ i (expm1 i))))
       (if (<= t_1 INFINITY)
         (* n (/ (+ -100.0 (* t_0 100.0)) i))
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

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-48) {
		tmp = n * (((pow((i / n), n) * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-48) {
		tmp = n * (((Math.pow((i / n), n) * 100.0) + -100.0) / i);
	} else if (t_1 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i);
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	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-48:
		tmp = n * (((math.pow((i / n), n) * 100.0) + -100.0) / i)
	elif t_1 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_1 <= math.inf:
		tmp = n * ((-100.0 + (t_0 * 100.0)) / i)
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	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-48)
		tmp = Float64(n * Float64(Float64(Float64((Float64(i / n) ^ n) * 100.0) + -100.0) / i));
	elseif (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(-100.0 + Float64(t_0 * 100.0)) / i));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	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-48], N[(n * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(-100.0 + N[(t$95$0 * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $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)) < -9.9999999999999997e-49

   1. Initial program 99.5%

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

      1. associate-/r/ 99.5%

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

      2. associate-*r* 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*r/ 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

      10. fma-define 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in i around inf 99.8%

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

   if -9.9999999999999997e-49 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 25.6%

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

      1. associate-/r/ 25.6%

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

      2. associate-*r* 25.6%

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

      3. *-commutative 25.6%

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

      4. associate-*r/ 25.6%

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

      5. sub-neg 25.6%

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

      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 25.6%

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

      11. metadata-eval 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in n around inf 37.2%

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

      1. sub-neg 37.2%

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

      2. metadata-eval 37.2%

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

      3. metadata-eval 37.2%

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

      4. distribute-lft-in 37.2%

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

      5. metadata-eval 37.2%

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

      6. sub-neg 37.2%

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

      7. expm1-define 74.8%

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

   7. Simplified 74.8%

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

      1. clear-num 74.8%

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

      2. un-div-inv 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. times-frac 74.8%

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

      5. metadata-eval 74.8%

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

   9. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 74.7%

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

   11. Simplified 74.7%

       \[\leadsto \color{blue}{\frac{n}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 74.7%

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

       2. associate-/l* 74.8%

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

       3. times-frac 74.9%

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

       4. metadata-eval 74.9%

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

   13. Applied egg-rr 74.9%

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

   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 98.2%

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

      1. associate-/r/ 98.2%

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

      2. associate-*r* 98.1%

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

      3. *-commutative 98.1%

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

      4. associate-*r/ 98.3%

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

      5. sub-neg 98.3%

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

      6. distribute-lft-in 98.2%

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

      7. metadata-eval 98.2%

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

      8. metadata-eval 98.2%

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

      9. metadata-eval 98.2%

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

      10. fma-define 98.3%

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

      11. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. fma-undefine 98.2%

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

      2. *-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. Taylor expanded in n around inf 1.9%

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

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-define 76.8%

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

   7. Simplified 76.8%

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

      1. clear-num 76.7%

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

      2. un-div-inv 76.6%

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

      3. *-un-lft-identity 76.6%

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

      4. times-frac 76.7%

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

      5. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 76.7%

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

   11. Simplified 76.7%

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

   12. Taylor expanded in i around 0 99.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-48}:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{-100 + {\left(1 + \frac{i}{n}\right)}^{n} \cdot 100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ t_1 := n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)))
        (t_1 (* n (/ (+ (* (pow (/ i n) n) 100.0) -100.0) i))))
   (if (<= t_0 -1e-48)
     t_1
     (if (<= t_0 0.0)
       (* 100.0 (/ n (/ i (expm1 i))))
       (if (<= t_0 INFINITY)
         t_1
         (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = n * (((pow((i / n), n) * 100.0) + -100.0) / i);
	double tmp;
	if (t_0 <= -1e-48) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double t_1 = n * (((Math.pow((i / n), n) * 100.0) + -100.0) / i);
	double tmp;
	if (t_0 <= -1e-48) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	t_1 = n * (((math.pow((i / n), n) * 100.0) + -100.0) / i)
	tmp = 0
	if t_0 <= -1e-48:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	elif t_0 <= math.inf:
		tmp = t_1
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	t_1 = Float64(n * Float64(Float64(Float64((Float64(i / n) ^ n) * 100.0) + -100.0) / i))
	tmp = 0.0
	if (t_0 <= -1e-48)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	elseif (t_0 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n * N[(N[(N[(N[Power[N[(i / n), $MachinePrecision], n], $MachinePrecision] * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-48], t$95$1, If[LessEqual[t$95$0, 0.0], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$1, N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -9.9999999999999997e-49 or 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 98.5%

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

      1. associate-/r/ 98.5%

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

      2. associate-*r* 98.4%

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

      3. *-commutative 98.4%

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

      4. associate-*r/ 98.7%

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

      5. sub-neg 98.7%

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

      6. distribute-lft-in 98.6%

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

      7. metadata-eval 98.6%

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

      8. metadata-eval 98.6%

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

      9. metadata-eval 98.6%

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

      10. fma-define 98.7%

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

      11. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

      1. fma-undefine 98.6%

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

      2. *-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in i around inf 98.6%

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

   if -9.9999999999999997e-49 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 25.6%

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

      1. associate-/r/ 25.6%

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

      2. associate-*r* 25.6%

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

      3. *-commutative 25.6%

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

      4. associate-*r/ 25.6%

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

      5. sub-neg 25.6%

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

      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 25.6%

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

      11. metadata-eval 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in n around inf 37.2%

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

      1. sub-neg 37.2%

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

      2. metadata-eval 37.2%

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

      3. metadata-eval 37.2%

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

      4. distribute-lft-in 37.2%

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

      5. metadata-eval 37.2%

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

      6. sub-neg 37.2%

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

      7. expm1-define 74.8%

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

   7. Simplified 74.8%

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

      1. clear-num 74.8%

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

      2. un-div-inv 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. times-frac 74.8%

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

      5. metadata-eval 74.8%

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

   9. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 74.7%

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

   11. Simplified 74.7%

       \[\leadsto \color{blue}{\frac{n}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 74.7%

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

       2. associate-/l* 74.8%

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

       3. times-frac 74.9%

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

       4. metadata-eval 74.9%

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

   13. Applied egg-rr 74.9%

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

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

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. 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. Taylor expanded in n around inf 1.9%

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

      1. sub-neg 1.9%

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

      2. metadata-eval 1.9%

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

      3. metadata-eval 1.9%

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

      4. distribute-lft-in 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. expm1-define 76.8%

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

   7. Simplified 76.8%

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

      1. clear-num 76.7%

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

      2. un-div-inv 76.6%

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

      3. *-un-lft-identity 76.6%

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

      4. times-frac 76.7%

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

      5. metadata-eval 76.7%

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

   9. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 76.7%

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

   11. Simplified 76.7%

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

   12. Taylor expanded in i around 0 99.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-48}:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(\frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-54} \lor \neg \left(n \leq 6.8 \cdot 10^{-9}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -7e-54) (not (<= n 6.8e-9)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -7e-54) || !(n <= 6.8e-9)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -7e-54) || !(n <= 6.8e-9)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -7e-54) or not (n <= 6.8e-9):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -7e-54) || !(n <= 6.8e-9))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -7e-54], N[Not[LessEqual[n, 6.8e-9]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.99999999999999964e-54 or 6.7999999999999997e-9 < n

   1. Initial program 29.0%

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

      1. associate-/r/ 29.3%

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

      2. associate-*r* 29.2%

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

      3. *-commutative 29.2%

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

      4. associate-*r/ 29.3%

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

      5. sub-neg 29.3%

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

      6. distribute-lft-in 29.3%

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

      7. metadata-eval 29.3%

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

      8. metadata-eval 29.3%

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

      9. metadata-eval 29.3%

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

      10. fma-define 29.3%

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

      11. metadata-eval 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in n around inf 42.9%

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

      1. sub-neg 42.9%

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

      2. metadata-eval 42.9%

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

      3. metadata-eval 42.9%

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

      4. distribute-lft-in 42.9%

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

      5. metadata-eval 42.9%

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

      6. sub-neg 42.9%

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

      7. expm1-define 89.7%

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

   7. Simplified 89.7%

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

      1. clear-num 89.7%

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

      2. un-div-inv 89.6%

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

      3. *-un-lft-identity 89.6%

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

      4. times-frac 89.7%

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

      5. metadata-eval 89.7%

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

   9. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 89.6%

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

   11. Simplified 89.6%

       \[\leadsto \color{blue}{\frac{n}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 89.6%

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

       2. associate-/l* 89.7%

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

       3. times-frac 89.8%

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

       4. metadata-eval 89.8%

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

   13. Applied egg-rr 89.8%

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

   if -6.99999999999999964e-54 < n < 6.7999999999999997e-9

   1. Initial program 38.2%

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

      1. associate-/r/ 38.3%

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

      2. associate-*r* 38.3%

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

      3. *-commutative 38.3%

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

      4. associate-*r/ 38.3%

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

      5. sub-neg 38.3%

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

      6. distribute-lft-in 38.3%

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

      7. metadata-eval 38.3%

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

      8. metadata-eval 38.3%

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

      9. metadata-eval 38.3%

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

      10. fma-define 38.3%

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

      11. metadata-eval 38.3%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in n around inf 22.4%

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

      1. sub-neg 22.4%

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

      2. metadata-eval 22.4%

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

      3. metadata-eval 22.4%

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

      4. distribute-lft-in 22.4%

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

      5. metadata-eval 22.4%

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

      6. sub-neg 22.4%

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

      7. expm1-define 44.0%

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

   7. Simplified 44.0%

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

      1. clear-num 44.0%

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

      2. un-div-inv 44.0%

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

      3. *-un-lft-identity 44.0%

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

      4. times-frac 44.0%

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

      5. metadata-eval 44.0%

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

   9. Applied egg-rr 44.0%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 44.0%

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

   11. Simplified 44.0%

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

   12. Taylor expanded in i around 0 63.1%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-54} \lor \neg \left(n \leq 6.8 \cdot 10^{-9}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.2:\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 65000000:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.75 \cdot 10^{+187}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.2)
   (* -100.0 (/ n i))
   (if (<= i 65000000.0)
     (* n (+ 100.0 (* i 50.0)))
     (if (<= i 3.1e+124)
       (/ 0.0 (/ i n))
       (if (<= i 2.75e+187) (* 50.0 (* i n)) (/ -100.0 (/ i n)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.2) {
		tmp = -100.0 * (n / i);
	} else if (i <= 65000000.0) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (i <= 3.1e+124) {
		tmp = 0.0 / (i / n);
	} else if (i <= 2.75e+187) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.2d0)) then
        tmp = (-100.0d0) * (n / i)
    else if (i <= 65000000.0d0) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (i <= 3.1d+124) then
        tmp = 0.0d0 / (i / n)
    else if (i <= 2.75d+187) then
        tmp = 50.0d0 * (i * n)
    else
        tmp = (-100.0d0) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.2) {
		tmp = -100.0 * (n / i);
	} else if (i <= 65000000.0) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (i <= 3.1e+124) {
		tmp = 0.0 / (i / n);
	} else if (i <= 2.75e+187) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.2:
		tmp = -100.0 * (n / i)
	elif i <= 65000000.0:
		tmp = n * (100.0 + (i * 50.0))
	elif i <= 3.1e+124:
		tmp = 0.0 / (i / n)
	elif i <= 2.75e+187:
		tmp = 50.0 * (i * n)
	else:
		tmp = -100.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.2)
		tmp = Float64(-100.0 * Float64(n / i));
	elseif (i <= 65000000.0)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (i <= 3.1e+124)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (i <= 2.75e+187)
		tmp = Float64(50.0 * Float64(i * n));
	else
		tmp = Float64(-100.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.2)
		tmp = -100.0 * (n / i);
	elseif (i <= 65000000.0)
		tmp = n * (100.0 + (i * 50.0));
	elseif (i <= 3.1e+124)
		tmp = 0.0 / (i / n);
	elseif (i <= 2.75e+187)
		tmp = 50.0 * (i * n);
	else
		tmp = -100.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.2], N[(-100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 65000000.0], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.1e+124], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.75e+187], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision], N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -1.19999999999999996

   1. Initial program 60.8%

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

      1. associate-*r/ 60.8%

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

      2. sub-neg 60.8%

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

      3. distribute-rgt-in 60.8%

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

      4. metadata-eval 60.8%

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

      5. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in i around 0 23.2%

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

      1. *-commutative 23.2%

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

   7. Simplified 23.2%

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

      1. flip-+ 6.6%

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

      2. clear-num 6.6%

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

      3. *-commutative 6.6%

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

      4. cancel-sign-sub-inv 6.6%

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

      5. metadata-eval 6.6%

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

      6. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 79.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 80.0%

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

   if -1.19999999999999996 < i < 6.5e7

   1. Initial program 7.6%

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

      1. associate-/r/ 8.0%

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

      2. associate-*r* 8.0%

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

      3. *-commutative 8.0%

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

      4. associate-*r/ 8.0%

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

      5. sub-neg 8.0%

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

      6. distribute-lft-in 8.0%

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

      7. metadata-eval 8.0%

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

      8. metadata-eval 8.0%

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

      9. metadata-eval 8.0%

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

      10. fma-define 8.0%

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

      11. metadata-eval 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in n around inf 11.2%

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

      1. sub-neg 11.2%

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

      2. metadata-eval 11.2%

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

      3. metadata-eval 11.2%

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

      4. distribute-lft-in 11.2%

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

      5. metadata-eval 11.2%

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

      6. sub-neg 11.2%

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

      7. expm1-define 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in i around 0 82.4%

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

      1. *-commutative 82.4%

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

   10. Simplified 82.4%

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

   if 6.5e7 < i < 3.1000000000000002e124

   1. Initial program 54.9%

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

      1. associate-*r/ 54.9%

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

      2. sub-neg 54.9%

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

      3. distribute-rgt-in 54.8%

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

      4. metadata-eval 54.8%

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

      5. metadata-eval 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in i around 0 5.1%

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

      1. *-commutative 5.1%

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

   7. Simplified 5.1%

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

   8. Taylor expanded in i around 0 30.4%

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

   if 3.1000000000000002e124 < i < 2.74999999999999999e187

   1. Initial program 56.3%

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

      1. associate-/r/ 56.3%

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

      2. associate-*r* 56.3%

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

      3. *-commutative 56.3%

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

      4. associate-*r/ 56.3%

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

      5. sub-neg 56.3%

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

      6. distribute-lft-in 56.3%

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

      7. metadata-eval 56.3%

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

      8. metadata-eval 56.3%

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

      9. metadata-eval 56.3%

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

      10. fma-define 56.3%

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

      11. metadata-eval 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in n around inf 77.9%

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

      1. sub-neg 77.9%

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

      2. metadata-eval 77.9%

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

      3. metadata-eval 77.9%

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

      4. distribute-lft-in 77.9%

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

      5. metadata-eval 77.9%

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

      6. sub-neg 77.9%

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

      7. expm1-define 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in i around 0 57.1%

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

      1. *-commutative 57.1%

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

   10. Simplified 57.1%

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

   11. Taylor expanded in i around inf 57.1%

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

   if 2.74999999999999999e187 < i

   1. Initial program 59.5%

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

      1. associate-*r/ 59.5%

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

      2. sub-neg 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. metadata-eval 59.5%

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

      5. metadata-eval 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in i around 0 35.6%

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

      1. *-commutative 35.6%

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

   7. Simplified 35.6%

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

      1. flip-+ 33.7%

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

      2. clear-num 33.7%

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

      3. *-commutative 33.7%

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

      4. cancel-sign-sub-inv 33.7%

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

      5. metadata-eval 33.7%

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

      6. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 33.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 53.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 53.0%

       \[\leadsto \frac{\color{blue}{-100}}{\frac{i}{n}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.2:\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 65000000:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;i \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.75 \cdot 10^{+187}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.82:\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+189}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.82)
   (* -100.0 (/ n i))
   (if (<= i 3.8e+189)
     (*
      n
      (+
       100.0
       (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))
     (/ -100.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.82) {
		tmp = -100.0 * (n / i);
	} else if (i <= 3.8e+189) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.82d0)) then
        tmp = (-100.0d0) * (n / i)
    else if (i <= 3.8d+189) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    else
        tmp = (-100.0d0) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.82) {
		tmp = -100.0 * (n / i);
	} else if (i <= 3.8e+189) {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.82:
		tmp = -100.0 * (n / i)
	elif i <= 3.8e+189:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	else:
		tmp = -100.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.82)
		tmp = Float64(-100.0 * Float64(n / i));
	elseif (i <= 3.8e+189)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	else
		tmp = Float64(-100.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.82)
		tmp = -100.0 * (n / i);
	elseif (i <= 3.8e+189)
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	else
		tmp = -100.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.82], N[(-100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.8e+189], 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], N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.82000000000000006

   1. Initial program 60.8%

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

      1. associate-*r/ 60.8%

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

      2. sub-neg 60.8%

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

      3. distribute-rgt-in 60.8%

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

      4. metadata-eval 60.8%

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

      5. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in i around 0 23.2%

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

      1. *-commutative 23.2%

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

   7. Simplified 23.2%

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

      1. flip-+ 6.6%

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

      2. clear-num 6.6%

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

      3. *-commutative 6.6%

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

      4. cancel-sign-sub-inv 6.6%

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

      5. metadata-eval 6.6%

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

      6. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 79.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 80.0%

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

   if -1.82000000000000006 < i < 3.7999999999999998e189

   1. Initial program 17.6%

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

      1. associate-/r/ 17.9%

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

      2. associate-*r* 17.9%

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

      3. *-commutative 17.9%

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

      4. associate-*r/ 17.9%

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

      5. sub-neg 17.9%

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

      6. distribute-lft-in 17.9%

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

      7. metadata-eval 17.9%

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

      8. metadata-eval 17.9%

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

      9. metadata-eval 17.9%

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

      10. fma-define 17.9%

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

      11. metadata-eval 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in n around inf 18.0%

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

      1. sub-neg 18.0%

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

      2. metadata-eval 18.0%

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

      3. metadata-eval 18.0%

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

      4. distribute-lft-in 18.1%

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

      5. metadata-eval 18.1%

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

      6. sub-neg 18.1%

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

      7. expm1-define 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in i around 0 72.8%

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

      1. *-commutative 72.8%

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

   10. Simplified 72.8%

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

   if 3.7999999999999998e189 < i

   1. Initial program 59.5%

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

      1. associate-*r/ 59.5%

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

      2. sub-neg 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. metadata-eval 59.5%

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

      5. metadata-eval 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in i around 0 35.6%

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

      1. *-commutative 35.6%

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

   7. Simplified 35.6%

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

      1. flip-+ 33.7%

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

      2. clear-num 33.7%

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

      3. *-commutative 33.7%

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

      4. cancel-sign-sub-inv 33.7%

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

      5. metadata-eval 33.7%

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

      6. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 33.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 53.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 53.0%

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

Alternative 9: 72.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.56:\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{+187}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.56)
   (* -100.0 (/ n i))
   (if (<= i 8.5e+187)
     (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))
     (/ -100.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.56) {
		tmp = -100.0 * (n / i);
	} else if (i <= 8.5e+187) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.56d0)) then
        tmp = (-100.0d0) * (n / i)
    else if (i <= 8.5d+187) then
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    else
        tmp = (-100.0d0) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.56) {
		tmp = -100.0 * (n / i);
	} else if (i <= 8.5e+187) {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.56:
		tmp = -100.0 * (n / i)
	elif i <= 8.5e+187:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	else:
		tmp = -100.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.56)
		tmp = Float64(-100.0 * Float64(n / i));
	elseif (i <= 8.5e+187)
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	else
		tmp = Float64(-100.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.56)
		tmp = -100.0 * (n / i);
	elseif (i <= 8.5e+187)
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	else
		tmp = -100.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.56], N[(-100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 8.5e+187], N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.5600000000000001

   1. Initial program 60.8%

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

      1. associate-*r/ 60.8%

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

      2. sub-neg 60.8%

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

      3. distribute-rgt-in 60.8%

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

      4. metadata-eval 60.8%

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

      5. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in i around 0 23.2%

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

      1. *-commutative 23.2%

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

   7. Simplified 23.2%

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

      1. flip-+ 6.6%

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

      2. clear-num 6.6%

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

      3. *-commutative 6.6%

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

      4. cancel-sign-sub-inv 6.6%

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

      5. metadata-eval 6.6%

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

      6. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 79.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 80.0%

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

   if -1.5600000000000001 < i < 8.49999999999999989e187

   1. Initial program 17.6%

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

      1. associate-/r/ 17.9%

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

      2. associate-*r* 17.9%

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

      3. *-commutative 17.9%

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

      4. associate-*r/ 17.9%

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

      5. sub-neg 17.9%

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

      6. distribute-lft-in 17.9%

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

      7. metadata-eval 17.9%

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

      8. metadata-eval 17.9%

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

      9. metadata-eval 17.9%

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

      10. fma-define 17.9%

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

      11. metadata-eval 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in n around inf 18.0%

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

      1. sub-neg 18.0%

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

      2. metadata-eval 18.0%

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

      3. metadata-eval 18.0%

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

      4. distribute-lft-in 18.1%

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

      5. metadata-eval 18.1%

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

      6. sub-neg 18.1%

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

      7. expm1-define 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in i around 0 72.1%

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

      1. *-commutative 72.1%

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

   10. Simplified 72.1%

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

   if 8.49999999999999989e187 < i

   1. Initial program 59.5%

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

      1. associate-*r/ 59.5%

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

      2. sub-neg 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. metadata-eval 59.5%

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

      5. metadata-eval 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in i around 0 35.6%

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

      1. *-commutative 35.6%

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

   7. Simplified 35.6%

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

      1. flip-+ 33.7%

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

      2. clear-num 33.7%

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

      3. *-commutative 33.7%

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

      4. cancel-sign-sub-inv 33.7%

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

      5. metadata-eval 33.7%

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

      6. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 33.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 53.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 53.0%

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

Alternative 10: 70.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.2:\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 5.5 \cdot 10^{+187}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.2)
   (* -100.0 (/ n i))
   (if (<= i 5.5e+187) (* n (+ 100.0 (* i 50.0))) (/ -100.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.2) {
		tmp = -100.0 * (n / i);
	} else if (i <= 5.5e+187) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.2d0)) then
        tmp = (-100.0d0) * (n / i)
    else if (i <= 5.5d+187) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = (-100.0d0) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.2) {
		tmp = -100.0 * (n / i);
	} else if (i <= 5.5e+187) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.2:
		tmp = -100.0 * (n / i)
	elif i <= 5.5e+187:
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = -100.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.2)
		tmp = Float64(-100.0 * Float64(n / i));
	elseif (i <= 5.5e+187)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(-100.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.2)
		tmp = -100.0 * (n / i);
	elseif (i <= 5.5e+187)
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = -100.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.2], N[(-100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.5e+187], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.19999999999999996

   1. Initial program 60.8%

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

      1. associate-*r/ 60.8%

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

      2. sub-neg 60.8%

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

      3. distribute-rgt-in 60.8%

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

      4. metadata-eval 60.8%

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

      5. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in i around 0 23.2%

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

      1. *-commutative 23.2%

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

   7. Simplified 23.2%

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

      1. flip-+ 6.6%

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

      2. clear-num 6.6%

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

      3. *-commutative 6.6%

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

      4. cancel-sign-sub-inv 6.6%

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

      5. metadata-eval 6.6%

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

      6. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 79.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 80.0%

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

   if -1.19999999999999996 < i < 5.49999999999999997e187

   1. Initial program 17.6%

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

      1. associate-/r/ 17.9%

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

      2. associate-*r* 17.9%

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

      3. *-commutative 17.9%

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

      4. associate-*r/ 17.9%

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

      5. sub-neg 17.9%

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

      6. distribute-lft-in 17.9%

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

      7. metadata-eval 17.9%

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

      8. metadata-eval 17.9%

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

      9. metadata-eval 17.9%

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

      10. fma-define 17.9%

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

      11. metadata-eval 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in n around inf 18.0%

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

      1. sub-neg 18.0%

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

      2. metadata-eval 18.0%

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

      3. metadata-eval 18.0%

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

      4. distribute-lft-in 18.1%

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

      5. metadata-eval 18.1%

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

      6. sub-neg 18.1%

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

      7. expm1-define 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in i around 0 70.0%

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

      1. *-commutative 70.0%

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

   10. Simplified 70.0%

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

   if 5.49999999999999997e187 < i

   1. Initial program 59.5%

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

      1. associate-*r/ 59.5%

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

      2. sub-neg 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. metadata-eval 59.5%

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

      5. metadata-eval 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in i around 0 35.6%

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

      1. *-commutative 35.6%

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

   7. Simplified 35.6%

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

      1. flip-+ 33.7%

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

      2. clear-num 33.7%

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

      3. *-commutative 33.7%

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

      4. cancel-sign-sub-inv 33.7%

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

      5. metadata-eval 33.7%

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

      6. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 33.7%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 33.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 53.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 53.0%

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

Alternative 11: 69.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1 \lor \neg \left(i \leq 3.5 \cdot 10^{+93}\right):\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.0) (not (<= i 3.5e+93))) (* -100.0 (/ n i)) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.0) || !(i <= 3.5e+93)) {
		tmp = -100.0 * (n / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-1.0d0)) .or. (.not. (i <= 3.5d+93))) then
        tmp = (-100.0d0) * (n / i)
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.0) || !(i <= 3.5e+93)) {
		tmp = -100.0 * (n / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.0) or not (i <= 3.5e+93):
		tmp = -100.0 * (n / i)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.0) || !(i <= 3.5e+93))
		tmp = Float64(-100.0 * Float64(n / i));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -1.0) || ~((i <= 3.5e+93)))
		tmp = -100.0 * (n / i);
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.0], N[Not[LessEqual[i, 3.5e+93]], $MachinePrecision]], N[(-100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1 or 3.49999999999999998e93 < i

   1. Initial program 61.3%

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

      1. associate-*r/ 61.3%

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

      2. sub-neg 61.3%

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

      3. distribute-rgt-in 61.3%

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

      4. metadata-eval 61.3%

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

      5. metadata-eval 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in i around 0 23.5%

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

      1. *-commutative 23.5%

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

   7. Simplified 23.5%

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

      1. flip-+ 15.0%

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

      2. clear-num 15.0%

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

      3. *-commutative 15.0%

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

      4. cancel-sign-sub-inv 15.0%

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

      5. metadata-eval 15.0%

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

      6. metadata-eval 15.0%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 15.0%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 15.0%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 15.0%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 15.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 65.7%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 65.2%

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

   if -1 < i < 3.49999999999999998e93

   1. Initial program 12.4%

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

   3. Taylor expanded in i around 0 71.6%

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

      1. *-commutative 71.6%

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

   5. Simplified 71.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \lor \neg \left(i \leq 3.5 \cdot 10^{+93}\right):\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1:\\ \;\;\;\;-100 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+93}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{-100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.0)
   (* -100.0 (/ n i))
   (if (<= i 3.5e+93) (* n 100.0) (/ -100.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.0) {
		tmp = -100.0 * (n / i);
	} else if (i <= 3.5e+93) {
		tmp = n * 100.0;
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.0d0)) then
        tmp = (-100.0d0) * (n / i)
    else if (i <= 3.5d+93) then
        tmp = n * 100.0d0
    else
        tmp = (-100.0d0) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.0) {
		tmp = -100.0 * (n / i);
	} else if (i <= 3.5e+93) {
		tmp = n * 100.0;
	} else {
		tmp = -100.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.0:
		tmp = -100.0 * (n / i)
	elif i <= 3.5e+93:
		tmp = n * 100.0
	else:
		tmp = -100.0 / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.0)
		tmp = Float64(-100.0 * Float64(n / i));
	elseif (i <= 3.5e+93)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(-100.0 / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.0)
		tmp = -100.0 * (n / i);
	elseif (i <= 3.5e+93)
		tmp = n * 100.0;
	else
		tmp = -100.0 / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.0], N[(-100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.5e+93], N[(n * 100.0), $MachinePrecision], N[(-100.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1

   1. Initial program 60.8%

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

      1. associate-*r/ 60.8%

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

      2. sub-neg 60.8%

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

      3. distribute-rgt-in 60.8%

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

      4. metadata-eval 60.8%

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

      5. metadata-eval 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in i around 0 23.2%

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

      1. *-commutative 23.2%

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

   7. Simplified 23.2%

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

      1. flip-+ 6.6%

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

      2. clear-num 6.6%

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

      3. *-commutative 6.6%

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

      4. cancel-sign-sub-inv 6.6%

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

      5. metadata-eval 6.6%

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

      6. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 6.6%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 79.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 80.0%

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

   if -1 < i < 3.49999999999999998e93

   1. Initial program 12.4%

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

   3. Taylor expanded in i around 0 71.6%

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

      1. *-commutative 71.6%

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

   5. Simplified 71.6%

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

   if 3.49999999999999998e93 < i

   1. Initial program 62.1%

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

      1. associate-*r/ 62.1%

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

      2. sub-neg 62.1%

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

      3. distribute-rgt-in 62.1%

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

      4. metadata-eval 62.1%

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

      5. metadata-eval 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in i around 0 24.0%

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

      1. *-commutative 24.0%

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

   7. Simplified 24.0%

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

      1. flip-+ 27.2%

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

      2. clear-num 27.2%

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

      3. *-commutative 27.2%

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

      4. cancel-sign-sub-inv 27.2%

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

      5. metadata-eval 27.2%

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

      6. metadata-eval 27.2%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{\color{blue}{10000} - \left(i \cdot 100\right) \cdot \left(i \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      7. swap-sqr 27.2%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{\left(i \cdot i\right) \cdot \left(100 \cdot 100\right)}}} + -100}{\frac{i}{n}} \]

      8. pow2 27.2%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - \color{blue}{{i}^{2}} \cdot \left(100 \cdot 100\right)}} + -100}{\frac{i}{n}} \]

      9. metadata-eval 27.2%

         \[\leadsto \frac{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot \color{blue}{10000}}} + -100}{\frac{i}{n}} \]

   9. Applied egg-rr 27.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{100 + -100 \cdot i}{10000 - {i}^{2} \cdot 10000}}} + -100}{\frac{i}{n}} \]

   10. Taylor expanded in i around 0 45.4%

       \[\leadsto \frac{\frac{1}{\color{blue}{0.01 + i \cdot \left(i \cdot \left(0.01 + -0.01 \cdot i\right) - 0.01\right)}} + -100}{\frac{i}{n}} \]

   11. Taylor expanded in i around inf 45.4%

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

Alternative 13: 48.7% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

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

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 32.5%

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

3. Taylor expanded in i around 0 44.2%

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

   1. *-commutative 44.2%

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

5. Simplified 44.2%

   \[\leadsto \color{blue}{n \cdot 100} \]
6. Add Preprocessing

Developer target: 34.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))