Compound Interest

Percentage Accurate: 28.2% → 94.6%

Time: 20.9s

Alternatives: 15

Speedup: 8.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t\_0}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 10^{-296}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t\_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 1e-296)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 INFINITY) (* 100.0 (* n (/ t_0 i))) (* n 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 1e-296) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= 1e-296) {
		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));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= 1e-296:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * (t_0 / i))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 1e-296)
		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(t_0 / i)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-296], 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[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 1e-296

   1. Initial program 19.1%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   if 1e-296 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.1%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i} \cdot \color{blue}{n}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{i}\right), \color{blue}{n}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right), i\right), n\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n} + \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(1 + \frac{i}{n}\right)}^{n}\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      6. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(e^{\log \left(1 + \frac{i}{n}\right)}\right)}^{n}\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right)}\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      8. rem-exp-log N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{i}{n}\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{i}{n}\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), \left(\mathsf{neg}\left(1\right)\right)\right), i\right), n\right)\right) \]

      11. metadata-eval 99.1%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(i, n\right)\right), n\right), -1\right), i\right), n\right)\right) \]

   4. Applied egg-rr 99.1%

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

   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. Add Preprocessing

   3. Taylor expanded in i around 0

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

      1. *-lowering-*.f64 78.6%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 78.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 10^{-296}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -1.85 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.2:\\ \;\;\;\;\frac{i \cdot \left(0 - \frac{n}{i}\right)}{-0.01}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -1.85e-48)
     t_0
     (if (<= n 2.2) (/ (* i (- 0.0 (/ n i))) -0.01) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -1.85e-48) {
		tmp = t_0;
	} else if (n <= 2.2) {
		tmp = (i * (0.0 - (n / i))) / -0.01;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -1.85e-48) {
		tmp = t_0;
	} else if (n <= 2.2) {
		tmp = (i * (0.0 - (n / i))) / -0.01;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -1.85e-48:
		tmp = t_0
	elif n <= 2.2:
		tmp = (i * (0.0 - (n / i))) / -0.01
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -1.85e-48)
		tmp = t_0;
	elseif (n <= 2.2)
		tmp = Float64(Float64(i * Float64(0.0 - Float64(n / i))) / -0.01);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.85e-48], t$95$0, If[LessEqual[n, 2.2], N[(N[(i * N[(0.0 - N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -0.01), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.8499999999999999e-48 or 2.2000000000000002 < n

   1. Initial program 21.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 91.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 91.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right), \color{blue}{100}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), i\right), 100\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right), 100\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right), 100\right) \]

      7. expm1-lowering-expm1.f64 93.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right), 100\right) \]

   7. Applied egg-rr 93.0%

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

   if -1.8499999999999999e-48 < n < 2.2000000000000002

   1. Initial program 23.3%

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

   3. Taylor expanded in i around 0

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

      1. *-lowering-*.f64 50.4%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 50.4%

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

      1. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(-100\right)\right) \cdot n \]

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

         \[\leadsto \mathsf{neg}\left(-100 \cdot n\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(n \cdot -100\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(1 \cdot \left(n \cdot -100\right)\right) \]

      5. *-inverses N/A

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

      6. associate-/r/ N/A

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

      7. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\frac{i}{\frac{\frac{i}{n}}{-100}}\right) \]

      8. div-inv N/A

         \[\leadsto \mathsf{neg}\left(i \cdot \frac{1}{\frac{\frac{i}{n}}{-100}}\right) \]

      9. clear-num N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. clear-num N/A

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

      13. div-inv N/A

          \[\leadsto \frac{1}{\frac{i}{n} \cdot \frac{1}{-100}} \cdot \left(\mathsf{neg}\left(i\right)\right) \]

      14. associate-/r* N/A

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

      15. clear-num N/A

          \[\leadsto \frac{\frac{n}{i}}{\frac{1}{-100}} \cdot \left(\mathsf{neg}\left(i\right)\right) \]

      16. associate-*l/ N/A

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

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{n}{i} \cdot \left(\mathsf{neg}\left(i\right)\right)\right), \color{blue}{\left(\frac{1}{-100}\right)}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{n}{i}\right), \left(\mathsf{neg}\left(i\right)\right)\right), \left(\frac{\color{blue}{1}}{-100}\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, i\right), \left(\mathsf{neg}\left(i\right)\right)\right), \left(\frac{1}{-100}\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, i\right), \left(0 - i\right)\right), \left(\frac{1}{-100}\right)\right) \]

      21. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, i\right), \mathsf{\_.f64}\left(0, i\right)\right), \left(\frac{1}{-100}\right)\right) \]

      22. metadata-eval 69.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(n, i\right), \mathsf{\_.f64}\left(0, i\right)\right), \frac{-1}{100}\right) \]

   7. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\frac{\frac{n}{i} \cdot \left(0 - i\right)}{-0.01}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-48}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 2.2:\\ \;\;\;\;\frac{i \cdot \left(0 - \frac{n}{i}\right)}{-0.01}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.00026:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2 \cdot 10^{+196}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} + \frac{i \cdot 0 - i}{i \cdot \frac{\frac{i}{n}}{-100}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.00026)
   (* 100.0 (/ (expm1 i) (/ i n)))
   (if (<= i 2e+196)
     (+
      (* n (+ 100.0 (* i 50.0)))
      (* (* i i) (* n (+ (* i 4.166666666666667) 16.666666666666668))))
     (+ (/ n (/ i -100.0)) (/ (- (* i 0.0) i) (* i (/ (/ i n) -100.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.00026) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (i <= 2e+196) {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	} else {
		tmp = (n / (i / -100.0)) + (((i * 0.0) - i) / (i * ((i / n) / -100.0)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.00026) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (i <= 2e+196) {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	} else {
		tmp = (n / (i / -100.0)) + (((i * 0.0) - i) / (i * ((i / n) / -100.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.00026:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif i <= 2e+196:
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)))
	else:
		tmp = (n / (i / -100.0)) + (((i * 0.0) - i) / (i * ((i / n) / -100.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.00026)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (i <= 2e+196)
		tmp = Float64(Float64(n * Float64(100.0 + Float64(i * 50.0))) + Float64(Float64(i * i) * Float64(n * Float64(Float64(i * 4.166666666666667) + 16.666666666666668))));
	else
		tmp = Float64(Float64(n / Float64(i / -100.0)) + Float64(Float64(Float64(i * 0.0) - i) / Float64(i * Float64(Float64(i / n) / -100.0))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.00026], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2e+196], N[(N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(N[(i * 4.166666666666667), $MachinePrecision] + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(i * 0.0), $MachinePrecision] - i), $MachinePrecision] / N[(i * N[(N[(i / n), $MachinePrecision] / -100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.59999999999999977e-4

   1. Initial program 47.4%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 93.2%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 93.2%

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

   if -2.59999999999999977e-4 < i < 1.9999999999999999e196

   1. Initial program 14.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 75.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 75.9%

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

   6. Taylor expanded in i around 0

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

      1. distribute-lft-in N/A

         \[\leadsto 100 \cdot n + \left(i \cdot \left(50 \cdot n\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(\left(i \cdot i\right) \cdot \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left({i}^{2} \cdot \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left({i}^{2}\right), \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left(i \cdot i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 82.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right) \]

   8. Simplified 82.7%

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

   if 1.9999999999999999e196 < i

   1. Initial program 35.3%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 15.8%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.7%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{i}{n} \cdot \frac{1}{-100}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \frac{i}{n}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(i \cdot \frac{1}{n}\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(\frac{1}{n} \cdot i\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\left(\frac{1}{-100} \cdot \frac{1}{n}\right) \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{\frac{1}{-100}}{n} \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-100}}{n}\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-100}\right), n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      9. metadata-eval 6.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{100}, n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

   9. Applied egg-rr 6.7%

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

   11. Applied egg-rr 63.5%

       \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{i + i \cdot 0}{i \cdot \frac{\frac{i}{n}}{-100}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00026:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2 \cdot 10^{+196}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} + \frac{i \cdot 0 - i}{i \cdot \frac{\frac{i}{n}}{-100}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{if}\;i \leq -3.3 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-162}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (/ n (/ i -100.0)) (* i (/ -100.0 (/ i (/ n i)))))))
   (if (<= i -3.3e+25)
     t_0
     (if (<= i 9.5e-162)
       (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
       (if (<= i 2.7e+196)
         (/
          (* 100.0 (* n (* i (+ 1.0 (* i (+ 0.5 (* i 0.16666666666666666)))))))
          i)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -3.3e+25) {
		tmp = t_0;
	} else if (i <= 9.5e-162) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (i <= 2.7e+196) {
		tmp = (100.0 * (n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))))) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (n / (i / (-100.0d0))) - (i * ((-100.0d0) / (i / (n / i))))
    if (i <= (-3.3d+25)) then
        tmp = t_0
    else if (i <= 9.5d-162) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (i <= 2.7d+196) then
        tmp = (100.0d0 * (n * (i * (1.0d0 + (i * (0.5d0 + (i * 0.16666666666666666d0))))))) / i
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -3.3e+25) {
		tmp = t_0;
	} else if (i <= 9.5e-162) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (i <= 2.7e+196) {
		tmp = (100.0 * (n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))))) / i;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))))
	tmp = 0
	if i <= -3.3e+25:
		tmp = t_0
	elif i <= 9.5e-162:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif i <= 2.7e+196:
		tmp = (100.0 * (n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))))) / i
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n / Float64(i / -100.0)) - Float64(i * Float64(-100.0 / Float64(i / Float64(n / i)))))
	tmp = 0.0
	if (i <= -3.3e+25)
		tmp = t_0;
	elseif (i <= 9.5e-162)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (i <= 2.7e+196)
		tmp = Float64(Float64(100.0 * Float64(n * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(i * 0.16666666666666666))))))) / i);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	tmp = 0.0;
	if (i <= -3.3e+25)
		tmp = t_0;
	elseif (i <= 9.5e-162)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (i <= 2.7e+196)
		tmp = (100.0 * (n * (i * (1.0 + (i * (0.5 + (i * 0.16666666666666666))))))) / i;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision] - N[(i * N[(-100.0 / N[(i / N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -3.3e+25], t$95$0, If[LessEqual[i, 9.5e-162], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.7e+196], N[(N[(100.0 * N[(n * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(i * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -3.3000000000000001e25 or 2.69999999999999995e196 < i

   1. Initial program 46.9%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 22.7%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.4%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{i}{n} \cdot \frac{1}{-100}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \frac{i}{n}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(i \cdot \frac{1}{n}\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(\frac{1}{n} \cdot i\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\left(\frac{1}{-100} \cdot \frac{1}{n}\right) \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{\frac{1}{-100}}{n} \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-100}}{n}\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-100}\right), n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      9. metadata-eval 6.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{100}, n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

   9. Applied egg-rr 6.6%

      \[\leadsto \frac{\left(n \cdot -100\right) \cdot \color{blue}{\left(\frac{-0.01}{n} \cdot i\right)} - i \cdot 1}{i \cdot \frac{\frac{i}{n}}{-100}} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-rgt-identity N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

   11. Applied egg-rr 44.1%

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

   if -3.3000000000000001e25 < i < 9.5000000000000004e-162

   1. Initial program 6.3%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 6.2%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 9.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 9.9%

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

   8. Taylor expanded in i around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot 100\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 87.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   10. Simplified 87.8%

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

   if 9.5000000000000004e-162 < i < 2.69999999999999995e196

   1. Initial program 30.8%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 74.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 74.2%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}\right)\right), i\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)\right)\right), i\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \left(i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)\right)\right)\right), i\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)\right)\right)\right), i\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot i\right)\right)\right)\right)\right)\right)\right), i\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(i \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), i\right) \]

      6. *-lowering-*.f64 70.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(i, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), i\right) \]

   8. Simplified 70.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-162}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + i \cdot 0.16666666666666666\right)\right)\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\frac{i}{-100}}\\ \mathbf{if}\;i \leq -2.7:\\ \;\;\;\;t\_0 - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{i \cdot 0 - i}{i \cdot \frac{\frac{i}{n}}{-100}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ n (/ i -100.0))))
   (if (<= i -2.7)
     (- t_0 (* i (/ -100.0 (/ i (/ n i)))))
     (if (<= i 2.7e+196)
       (+
        (* n (+ 100.0 (* i 50.0)))
        (* (* i i) (* n (+ (* i 4.166666666666667) 16.666666666666668))))
       (+ t_0 (/ (- (* i 0.0) i) (* i (/ (/ i n) -100.0))))))))

C

double code(double i, double n) {
	double t_0 = n / (i / -100.0);
	double tmp;
	if (i <= -2.7) {
		tmp = t_0 - (i * (-100.0 / (i / (n / i))));
	} else if (i <= 2.7e+196) {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	} else {
		tmp = t_0 + (((i * 0.0) - i) / (i * ((i / n) / -100.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n / (i / (-100.0d0))
    if (i <= (-2.7d0)) then
        tmp = t_0 - (i * ((-100.0d0) / (i / (n / i))))
    else if (i <= 2.7d+196) then
        tmp = (n * (100.0d0 + (i * 50.0d0))) + ((i * i) * (n * ((i * 4.166666666666667d0) + 16.666666666666668d0)))
    else
        tmp = t_0 + (((i * 0.0d0) - i) / (i * ((i / n) / (-100.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n / (i / -100.0);
	double tmp;
	if (i <= -2.7) {
		tmp = t_0 - (i * (-100.0 / (i / (n / i))));
	} else if (i <= 2.7e+196) {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	} else {
		tmp = t_0 + (((i * 0.0) - i) / (i * ((i / n) / -100.0)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n / (i / -100.0)
	tmp = 0
	if i <= -2.7:
		tmp = t_0 - (i * (-100.0 / (i / (n / i))))
	elif i <= 2.7e+196:
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)))
	else:
		tmp = t_0 + (((i * 0.0) - i) / (i * ((i / n) / -100.0)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n / Float64(i / -100.0))
	tmp = 0.0
	if (i <= -2.7)
		tmp = Float64(t_0 - Float64(i * Float64(-100.0 / Float64(i / Float64(n / i)))));
	elseif (i <= 2.7e+196)
		tmp = Float64(Float64(n * Float64(100.0 + Float64(i * 50.0))) + Float64(Float64(i * i) * Float64(n * Float64(Float64(i * 4.166666666666667) + 16.666666666666668))));
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(i * 0.0) - i) / Float64(i * Float64(Float64(i / n) / -100.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n / (i / -100.0);
	tmp = 0.0;
	if (i <= -2.7)
		tmp = t_0 - (i * (-100.0 / (i / (n / i))));
	elseif (i <= 2.7e+196)
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	else
		tmp = t_0 + (((i * 0.0) - i) / (i * ((i / n) / -100.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.7], N[(t$95$0 - N[(i * N[(-100.0 / N[(i / N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.7e+196], N[(N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(N[(i * 4.166666666666667), $MachinePrecision] + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(N[(i * 0.0), $MachinePrecision] - i), $MachinePrecision] / N[(i * N[(N[(i / n), $MachinePrecision] / -100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.7000000000000002

   1. Initial program 48.4%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 24.3%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.0%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{i}{n} \cdot \frac{1}{-100}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \frac{i}{n}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(i \cdot \frac{1}{n}\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(\frac{1}{n} \cdot i\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\left(\frac{1}{-100} \cdot \frac{1}{n}\right) \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{\frac{1}{-100}}{n} \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-100}}{n}\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-100}\right), n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      9. metadata-eval 6.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{100}, n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

   9. Applied egg-rr 6.4%

      \[\leadsto \frac{\left(n \cdot -100\right) \cdot \color{blue}{\left(\frac{-0.01}{n} \cdot i\right)} - i \cdot 1}{i \cdot \frac{\frac{i}{n}}{-100}} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-rgt-identity N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

   11. Applied egg-rr 30.9%

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

   if -2.7000000000000002 < i < 2.69999999999999995e196

   1. Initial program 14.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 76.0%

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

   6. Taylor expanded in i around 0

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

      1. distribute-lft-in N/A

         \[\leadsto 100 \cdot n + \left(i \cdot \left(50 \cdot n\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(\left(i \cdot i\right) \cdot \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left({i}^{2} \cdot \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left({i}^{2}\right), \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left(i \cdot i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 82.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right) \]

   8. Simplified 82.7%

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

   if 2.69999999999999995e196 < i

   1. Initial program 35.3%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 15.8%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.7%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{i}{n} \cdot \frac{1}{-100}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \frac{i}{n}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(i \cdot \frac{1}{n}\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(\frac{1}{n} \cdot i\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\left(\frac{1}{-100} \cdot \frac{1}{n}\right) \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{\frac{1}{-100}}{n} \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-100}}{n}\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-100}\right), n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      9. metadata-eval 6.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{100}, n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

   9. Applied egg-rr 6.7%

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

   11. Applied egg-rr 63.5%

       \[\leadsto \color{blue}{\frac{n}{\frac{i}{-100}} - \frac{i + i \cdot 0}{i \cdot \frac{\frac{i}{n}}{-100}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.7:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} + \frac{i \cdot 0 - i}{i \cdot \frac{\frac{i}{n}}{-100}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{if}\;i \leq -2.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (/ n (/ i -100.0)) (* i (/ -100.0 (/ i (/ n i)))))))
   (if (<= i -2.35)
     t_0
     (if (<= i 2.7e+196)
       (+
        (* n (+ 100.0 (* i 50.0)))
        (* (* i i) (* n (+ (* i 4.166666666666667) 16.666666666666668))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -2.35) {
		tmp = t_0;
	} else if (i <= 2.7e+196) {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (n / (i / (-100.0d0))) - (i * ((-100.0d0) / (i / (n / i))))
    if (i <= (-2.35d0)) then
        tmp = t_0
    else if (i <= 2.7d+196) then
        tmp = (n * (100.0d0 + (i * 50.0d0))) + ((i * i) * (n * ((i * 4.166666666666667d0) + 16.666666666666668d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -2.35) {
		tmp = t_0;
	} else if (i <= 2.7e+196) {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))))
	tmp = 0
	if i <= -2.35:
		tmp = t_0
	elif i <= 2.7e+196:
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n / Float64(i / -100.0)) - Float64(i * Float64(-100.0 / Float64(i / Float64(n / i)))))
	tmp = 0.0
	if (i <= -2.35)
		tmp = t_0;
	elseif (i <= 2.7e+196)
		tmp = Float64(Float64(n * Float64(100.0 + Float64(i * 50.0))) + Float64(Float64(i * i) * Float64(n * Float64(Float64(i * 4.166666666666667) + 16.666666666666668))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	tmp = 0.0;
	if (i <= -2.35)
		tmp = t_0;
	elseif (i <= 2.7e+196)
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * ((i * 4.166666666666667) + 16.666666666666668)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision] - N[(i * N[(-100.0 / N[(i / N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.35], t$95$0, If[LessEqual[i, 2.7e+196], N[(N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(N[(i * 4.166666666666667), $MachinePrecision] + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -2.35000000000000009 or 2.69999999999999995e196 < i

   1. Initial program 44.1%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 21.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.2%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{i}{n} \cdot \frac{1}{-100}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \frac{i}{n}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(i \cdot \frac{1}{n}\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(\frac{1}{n} \cdot i\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\left(\frac{1}{-100} \cdot \frac{1}{n}\right) \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{\frac{1}{-100}}{n} \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-100}}{n}\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-100}\right), n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      9. metadata-eval 6.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{100}, n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

   9. Applied egg-rr 6.5%

      \[\leadsto \frac{\left(n \cdot -100\right) \cdot \color{blue}{\left(\frac{-0.01}{n} \cdot i\right)} - i \cdot 1}{i \cdot \frac{\frac{i}{n}}{-100}} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-rgt-identity N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

   11. Applied egg-rr 41.5%

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

   if -2.35000000000000009 < i < 2.69999999999999995e196

   1. Initial program 14.7%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 76.0%

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

   6. Taylor expanded in i around 0

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

      1. distribute-lft-in N/A

         \[\leadsto 100 \cdot n + \left(i \cdot \left(50 \cdot n\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left(\left(i \cdot i\right) \cdot \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \left({i}^{2} \cdot \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left({i}^{2}\right), \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\left(i \cdot i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\color{blue}{\frac{25}{6} \cdot \left(i \cdot n\right)} + \frac{50}{3} \cdot n\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 82.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right) \]

   8. Simplified 82.7%

      \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(4.166666666666667 \cdot i + 16.666666666666668\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.35:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{if}\;i \leq -2.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (/ n (/ i -100.0)) (* i (/ -100.0 (/ i (/ n i)))))))
   (if (<= i -2.8)
     t_0
     (if (<= i 2.7e+196)
       (+
        (* n 100.0)
        (*
         i
         (+
          (* n 50.0)
          (* i (* n (+ (* i 4.166666666666667) 16.666666666666668))))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -2.8) {
		tmp = t_0;
	} else if (i <= 2.7e+196) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (n / (i / (-100.0d0))) - (i * ((-100.0d0) / (i / (n / i))))
    if (i <= (-2.8d0)) then
        tmp = t_0
    else if (i <= 2.7d+196) then
        tmp = (n * 100.0d0) + (i * ((n * 50.0d0) + (i * (n * ((i * 4.166666666666667d0) + 16.666666666666668d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -2.8) {
		tmp = t_0;
	} else if (i <= 2.7e+196) {
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))))
	tmp = 0
	if i <= -2.8:
		tmp = t_0
	elif i <= 2.7e+196:
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n / Float64(i / -100.0)) - Float64(i * Float64(-100.0 / Float64(i / Float64(n / i)))))
	tmp = 0.0
	if (i <= -2.8)
		tmp = t_0;
	elseif (i <= 2.7e+196)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(Float64(n * 50.0) + Float64(i * Float64(n * Float64(Float64(i * 4.166666666666667) + 16.666666666666668))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	tmp = 0.0;
	if (i <= -2.8)
		tmp = t_0;
	elseif (i <= 2.7e+196)
		tmp = (n * 100.0) + (i * ((n * 50.0) + (i * (n * ((i * 4.166666666666667) + 16.666666666666668)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision] - N[(i * N[(-100.0 / N[(i / N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.8], t$95$0, If[LessEqual[i, 2.7e+196], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(N[(n * 50.0), $MachinePrecision] + N[(i * N[(n * N[(N[(i * 4.166666666666667), $MachinePrecision] + 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -2.7999999999999998 or 2.69999999999999995e196 < i

   1. Initial program 44.1%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 21.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.2%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{i}{n} \cdot \frac{1}{-100}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \frac{i}{n}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(i \cdot \frac{1}{n}\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(\frac{1}{n} \cdot i\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\left(\frac{1}{-100} \cdot \frac{1}{n}\right) \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{\frac{1}{-100}}{n} \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-100}}{n}\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-100}\right), n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      9. metadata-eval 6.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{100}, n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

   9. Applied egg-rr 6.5%

      \[\leadsto \frac{\left(n \cdot -100\right) \cdot \color{blue}{\left(\frac{-0.01}{n} \cdot i\right)} - i \cdot 1}{i \cdot \frac{\frac{i}{n}}{-100}} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-rgt-identity N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

   11. Applied egg-rr 41.5%

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

   if -2.7999999999999998 < i < 2.69999999999999995e196

   1. Initial program 14.7%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 14.6%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 18.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 18.4%

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

   8. Taylor expanded in i around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot 100\right), \left(\color{blue}{i} \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \left(\color{blue}{i} \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \color{blue}{\left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(50 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(n \cdot 50\right), \left(\color{blue}{i} \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \left(\color{blue}{i} \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 82.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 50\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{25}{6}, i\right), \frac{50}{3}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 82.7%

       \[\leadsto \color{blue}{n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(4.166666666666667 \cdot i + 16.666666666666668\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.8:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot 50 + i \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{if}\;i \leq -9.5 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (/ n (/ i -100.0)) (* i (/ -100.0 (/ i (/ n i)))))))
   (if (<= i -9.5e+24)
     t_0
     (if (<= i 2.7e+196)
       (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
       t_0))))

C

double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -9.5e+24) {
		tmp = t_0;
	} else if (i <= 2.7e+196) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (n / (i / (-100.0d0))) - (i * ((-100.0d0) / (i / (n / i))))
    if (i <= (-9.5d+24)) then
        tmp = t_0
    else if (i <= 2.7d+196) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	double tmp;
	if (i <= -9.5e+24) {
		tmp = t_0;
	} else if (i <= 2.7e+196) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))))
	tmp = 0
	if i <= -9.5e+24:
		tmp = t_0
	elif i <= 2.7e+196:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n / Float64(i / -100.0)) - Float64(i * Float64(-100.0 / Float64(i / Float64(n / i)))))
	tmp = 0.0
	if (i <= -9.5e+24)
		tmp = t_0;
	elseif (i <= 2.7e+196)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n / (i / -100.0)) - (i * (-100.0 / (i / (n / i))));
	tmp = 0.0;
	if (i <= -9.5e+24)
		tmp = t_0;
	elseif (i <= 2.7e+196)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n / N[(i / -100.0), $MachinePrecision]), $MachinePrecision] - N[(i * N[(-100.0 / N[(i / N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -9.5e+24], t$95$0, If[LessEqual[i, 2.7e+196], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if i < -9.5000000000000001e24 or 2.69999999999999995e196 < i

   1. Initial program 46.9%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 22.7%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.4%

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

      1. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{i}{n} \cdot \frac{1}{-100}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \frac{i}{n}\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(i \cdot \frac{1}{n}\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{1}{-100} \cdot \left(\frac{1}{n} \cdot i\right)\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\left(\frac{1}{-100} \cdot \frac{1}{n}\right) \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \left(\frac{\frac{1}{-100}}{n} \cdot i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{-100}}{n}\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-100}\right), n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

      9. metadata-eval 6.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{100}, n\right), i\right)\right), \mathsf{*.f64}\left(i, 1\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]

   9. Applied egg-rr 6.6%

      \[\leadsto \frac{\left(n \cdot -100\right) \cdot \color{blue}{\left(\frac{-0.01}{n} \cdot i\right)} - i \cdot 1}{i \cdot \frac{\frac{i}{n}}{-100}} \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

       3. *-rgt-identity N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

   11. Applied egg-rr 44.1%

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

   if -9.5000000000000001e24 < i < 2.69999999999999995e196

   1. Initial program 14.4%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 14.3%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 20.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 20.1%

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

   8. Taylor expanded in i around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot 100\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 78.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   10. Simplified 78.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -9.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \mathbf{elif}\;i \leq 2.7 \cdot 10^{+196}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{-100}} - i \cdot \frac{-100}{\frac{i}{\frac{n}{i}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.8:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))))
   (if (<= n -1.9e-7) t_0 (if (<= n 1.8) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -1.9e-7) {
		tmp = t_0;
	} else if (n <= 1.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    if (n <= (-1.9d-7)) then
        tmp = t_0
    else if (n <= 1.8d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	double tmp;
	if (n <= -1.9e-7) {
		tmp = t_0;
	} else if (n <= 1.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	tmp = 0
	if n <= -1.9e-7:
		tmp = t_0
	elif n <= 1.8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))))
	tmp = 0.0
	if (n <= -1.9e-7)
		tmp = t_0;
	elseif (n <= 1.8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	tmp = 0.0;
	if (n <= -1.9e-7)
		tmp = t_0;
	elseif (n <= 1.8)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.9e-7], t$95$0, If[LessEqual[n, 1.8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.90000000000000007e-7 or 1.80000000000000004 < n

   1. Initial program 22.0%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 21.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 39.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 39.3%

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

   8. Taylor expanded in i around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot 100\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 69.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, 100\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   10. Simplified 69.4%

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

   if -1.90000000000000007e-7 < n < 1.80000000000000004

   1. Initial program 22.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 68.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 1.8:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3700:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* i (* n (+ 100.0 (* i 50.0)))) i)))
   (if (<= n -3.8e-7) t_0 (if (<= n 3700.0) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -3.8e-7) {
		tmp = t_0;
	} else if (n <= 3700.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    if (n <= (-3.8d-7)) then
        tmp = t_0
    else if (n <= 3700.0d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -3.8e-7) {
		tmp = t_0;
	} else if (n <= 3700.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i
	tmp = 0
	if n <= -3.8e-7:
		tmp = t_0
	elif n <= 3700.0:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i)
	tmp = 0.0
	if (n <= -3.8e-7)
		tmp = t_0;
	elseif (n <= 3700.0)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	tmp = 0.0;
	if (n <= -3.8e-7)
		tmp = t_0;
	elseif (n <= 3700.0)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -3.8e-7], t$95$0, If[LessEqual[n, 3700.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.80000000000000015e-7 or 3700 < n

   1. Initial program 22.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 93.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 93.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right)\right), i\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right)\right), i\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right)\right), i\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right)\right), i\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right)\right), i\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

      13. *-lowering-*.f64 68.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, i\right)\right)\right)\right), i\right) \]

   8. Simplified 68.2%

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

   if -3.80000000000000015e-7 < n < 3700

   1. Initial program 21.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 68.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;n \leq 3700:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.00021:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 102000000:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot n\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.00021)
   0.0
   (if (<= i 102000000.0) (* n (+ 100.0 (* i 50.0))) (/ (* 100.0 (* n n)) n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.00021) {
		tmp = 0.0;
	} else if (i <= 102000000.0) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (100.0 * (n * n)) / 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 <= (-0.00021d0)) then
        tmp = 0.0d0
    else if (i <= 102000000.0d0) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = (100.0d0 * (n * n)) / n
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.00021) {
		tmp = 0.0;
	} else if (i <= 102000000.0) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (100.0 * (n * n)) / n;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.00021:
		tmp = 0.0
	elif i <= 102000000.0:
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = (100.0 * (n * n)) / n
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.00021)
		tmp = 0.0;
	elseif (i <= 102000000.0)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(Float64(100.0 * Float64(n * n)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.00021)
		tmp = 0.0;
	elseif (i <= 102000000.0)
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = (100.0 * (n * n)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.00021], 0.0, If[LessEqual[i, 102000000.0], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(n * n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.1000000000000001e-4

   1. Initial program 48.5%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 25.5%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 8.0%

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

   9. Applied egg-rr 18.1%

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

   if -2.1000000000000001e-4 < i < 1.02e8

   1. Initial program 6.9%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 6.8%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 9.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 9.0%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + 50 \cdot i\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right) \]

      6. *-lowering-*.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   10. Simplified 87.7%

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

   if 1.02e8 < i

   1. Initial program 44.6%

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

      1. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i \cdot \frac{1}{n}} - \frac{1}{\frac{i}{n}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}{\frac{1}{n}} - \frac{\color{blue}{1}}{\frac{i}{n}}\right)\right) \]

      4. frac-sub N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot \frac{i}{n} - \frac{1}{n} \cdot 1}{\color{blue}{\frac{1}{n} \cdot \frac{i}{n}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot \frac{i}{n} - \frac{1}{n} \cdot 1\right), \color{blue}{\left(\frac{1}{n} \cdot \frac{i}{n}\right)}\right)\right) \]

   4. Applied egg-rr 44.3%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(\frac{i}{n}\right)}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(i, n\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 17.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, n\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right)\right) \]

   7. Simplified 17.5%

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

      1. *-commutative N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\frac{1}{n} \cdot \frac{i}{n}} \cdot 100\right), \color{blue}{n}\right) \]

   9. Applied egg-rr 41.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00021:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 102000000:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(n \cdot n\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100\right)}{i}\\ \mathbf{elif}\;n \leq 1.45:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2e-12)
   (/ (* i (* n 100.0)) i)
   (if (<= n 1.45) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2e-12) {
		tmp = (i * (n * 100.0)) / i;
	} else if (n <= 1.45) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2d-12)) then
        tmp = (i * (n * 100.0d0)) / i
    else if (n <= 1.45d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2e-12) {
		tmp = (i * (n * 100.0)) / i;
	} else if (n <= 1.45) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2e-12:
		tmp = (i * (n * 100.0)) / i
	elif n <= 1.45:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2e-12)
		tmp = Float64(Float64(i * Float64(n * 100.0)) / i);
	elseif (n <= 1.45)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2e-12)
		tmp = (i * (n * 100.0)) / i;
	elseif (n <= 1.45)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2e-12], N[(N[(i * N[(n * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 1.45], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.99999999999999996e-12

   1. Initial program 26.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 88.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 88.8%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(100 \cdot \left(i \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(i \cdot n\right) \cdot 100\right), i\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(n \cdot 100\right)\right), i\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n\right)\right), i\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot 100\right)\right), i\right) \]

      6. *-lowering-*.f64 60.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, 100\right)\right), i\right) \]

   8. Simplified 60.2%

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

   if -1.99999999999999996e-12 < n < 1.44999999999999996

   1. Initial program 22.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 68.4%

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

   if 1.44999999999999996 < n

   1. Initial program 16.4%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 16.3%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 41.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 41.1%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + 50 \cdot i\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right) \]

      6. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   10. Simplified 68.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100\right)}{i}\\ \mathbf{elif}\;n \leq 1.45:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.45:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -3.8e-7) t_0 (if (<= n 1.45) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -3.8e-7) {
		tmp = t_0;
	} else if (n <= 1.45) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-3.8d-7)) then
        tmp = t_0
    else if (n <= 1.45d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -3.8e-7) {
		tmp = t_0;
	} else if (n <= 1.45) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -3.8e-7:
		tmp = t_0
	elif n <= 1.45:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -3.8e-7)
		tmp = t_0;
	elseif (n <= 1.45)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -3.8e-7)
		tmp = t_0;
	elseif (n <= 1.45)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.8e-7], t$95$0, If[LessEqual[n, 1.45], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.80000000000000015e-7 or 1.44999999999999996 < n

   1. Initial program 22.0%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 21.9%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 39.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 39.3%

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

   8. Taylor expanded in i around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 + 50 \cdot i\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(50 \cdot i\right)}\right)\right) \]

      6. *-lowering-*.f64 63.4%

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   10. Simplified 63.4%

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

   if -3.80000000000000015e-7 < n < 1.44999999999999996

   1. Initial program 22.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 68.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.45:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.00021:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{+126}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.00021) 0.0 (if (<= i 1.55e+126) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.00021) {
		tmp = 0.0;
	} else if (i <= 1.55e+126) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-0.00021d0)) then
        tmp = 0.0d0
    else if (i <= 1.55d+126) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.00021) {
		tmp = 0.0;
	} else if (i <= 1.55e+126) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.00021:
		tmp = 0.0
	elif i <= 1.55e+126:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.00021)
		tmp = 0.0;
	elseif (i <= 1.55e+126)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -0.00021)
		tmp = 0.0;
	elseif (i <= 1.55e+126)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.00021], 0.0, If[LessEqual[i, 1.55e+126], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -2.1000000000000001e-4 or 1.55e126 < i

   1. Initial program 48.1%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. associate-/r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. div-sub N/A

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

      12. clear-num N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

   4. Applied egg-rr 28.0%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 6.6%

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

   9. Applied egg-rr 26.8%

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

   if -2.1000000000000001e-4 < i < 1.55e126

   1. Initial program 9.7%

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

   3. Taylor expanded in i around 0

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

      1. *-lowering-*.f64 79.8%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 79.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.00021:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.55 \cdot 10^{+126}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 18.2% accurate, 114.0× speedup

Math

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

FPCore

(FPCore (i n) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

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

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 22.0%

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

   1. *-commutative N/A

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

   2. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \frac{0 - \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

   4. associate-+l- N/A

      \[\leadsto \frac{\left(0 - {\left(1 + \frac{i}{n}\right)}^{n}\right) + 1}{\mathsf{neg}\left(\frac{i}{n}\right)} \cdot 100 \]

   5. neg-sub0 N/A

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

   6. +-commutative N/A

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

   7. sub-neg N/A

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

   8. associate-/r/ N/A

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

   9. metadata-eval N/A

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

   10. frac-2neg N/A

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

   11. div-sub N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

   14. associate-*l/ N/A

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

4. Applied egg-rr 15.2%

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

5. Taylor expanded in i around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, -100\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right), \mathsf{*.f64}\left(i, \color{blue}{1}\right)\right), \mathsf{*.f64}\left(i, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, n\right), -100\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 6.2%

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

9. Applied egg-rr 13.3%

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

Developer Target 1: 34.1% 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 2024164 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))