Compound Interest

Percentage Accurate: 28.1% → 97.6%

Time: 19.6s

Alternatives: 15

Speedup: 16.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.1% 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: 97.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) (/ i n))
     (if (<= t_1 INFINITY)
       (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
       (/ n (+ 0.01 (* i -0.005)))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) * 100.0) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) * 100.0) / (i / n)
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = n / (0.01 + (i * -0.005))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 24.2%

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

      1. associate-*r/ 24.2%

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

      2. sub-neg 24.2%

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

      3. distribute-lft-in 24.3%

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

      4. fma-def 24.2%

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

      5. metadata-eval 24.2%

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

      6. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

      1. fma-udef 24.3%

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

      2. metadata-eval 24.3%

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

      3. metadata-eval 24.3%

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

      4. distribute-lft-in 24.2%

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

      5. sub-neg 24.2%

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

      6. *-commutative 24.2%

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

      7. pow-to-exp 23.7%

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

      8. expm1-def 36.9%

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

      9. *-commutative 36.9%

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

      10. log1p-udef 99.1%

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

   5. Applied egg-rr 99.1%

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

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

   1. Initial program 92.3%

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

      1. div-sub 92.1%

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

      2. clear-num 92.1%

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

      3. sub-neg 92.1%

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

      4. div-inv 92.6%

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

      5. clear-num 92.6%

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

   3. Applied egg-rr 92.6%

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

      1. sub-neg 92.6%

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

   5. Simplified 92.6%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. *-commutative 1.9%

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

      3. fma-neg 1.9%

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

      4. metadata-eval 1.9%

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

   6. Simplified 1.9%

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

   7. Taylor expanded in i around 0 99.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 99.7%

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

4. Final simplification 98.8%

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

Alternative 2: 95.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* 100.0 (* (expm1 (* n (log1p (/ i n)))) (/ n i)))
     (if (<= t_1 INFINITY)
       (* 100.0 (- (* t_0 (/ n i)) (/ n i)))
       (/ n (+ 0.01 (* i -0.005)))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) * (n / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) * (n / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) * (n / i))
	elif t_1 <= math.inf:
		tmp = 100.0 * ((t_0 * (n / i)) - (n / i))
	else:
		tmp = n / (0.01 + (i * -0.005))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(n / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(Float64(t_0 * Float64(n / i)) - Float64(n / i)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 24.2%

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

      1. clear-num 24.2%

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

      2. associate-/r/ 24.1%

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

      3. clear-num 24.0%

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

      4. pow-to-exp 23.5%

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

      5. expm1-def 36.6%

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

      6. *-commutative 36.6%

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

      7. log1p-udef 96.4%

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

   3. Applied egg-rr 96.4%

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

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

   1. Initial program 92.3%

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

      1. div-sub 92.1%

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

      2. clear-num 92.1%

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

      3. sub-neg 92.1%

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

      4. div-inv 92.6%

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

      5. clear-num 92.6%

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

   3. Applied egg-rr 92.6%

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

      1. sub-neg 92.6%

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

   5. Simplified 92.6%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n))

   1. Initial program 0.0%

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

      1. associate-/r/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. fma-def 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. *-commutative 1.9%

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

      3. fma-neg 1.9%

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

      4. metadata-eval 1.9%

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

   6. Simplified 1.9%

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

   7. Taylor expanded in i around 0 99.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 99.7%

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

4. Final simplification 96.8%

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

Alternative 3: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-11} \lor \neg \left(n \leq 0.09\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{10000 \cdot {n}^{2}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.35e-11) (not (<= n 0.09)))
   (* n (* 100.0 (/ (expm1 i) i)))
   (/
    (* 10000.0 (pow n 2.0))
    (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.35e-11) || !(n <= 0.09)) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = (10000.0 * pow(n, 2.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.35e-11) || !(n <= 0.09)) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = (10000.0 * Math.pow(n, 2.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.35e-11) or not (n <= 0.09):
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = (10000.0 * math.pow(n, 2.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.35e-11) || !(n <= 0.09))
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(10000.0 * (n ^ 2.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.35e-11], N[Not[LessEqual[n, 0.09]], $MachinePrecision]], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(10000.0 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.35000000000000002e-11 or 0.089999999999999997 < n

   1. Initial program 17.3%

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

   2. Taylor expanded in n around inf 39.9%

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

      1. expm1-def 67.9%

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

   4. Simplified 67.9%

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

      1. associate-*r/ 67.9%

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

      2. frac-2neg 67.9%

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

      3. distribute-neg-frac 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. associate-/r/ 95.3%

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

      2. distribute-lft-neg-in 95.3%

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

      3. metadata-eval 95.3%

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

      4. neg-mul-1 95.3%

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

      5. times-frac 95.4%

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

      6. metadata-eval 95.4%

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

   8. Simplified 95.4%

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

   if -1.35000000000000002e-11 < n < 0.089999999999999997

   1. Initial program 35.4%

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

      1. associate-/r/ 35.3%

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

      2. associate-*r* 35.3%

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

      3. *-commutative 35.3%

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

      4. associate-*r/ 35.3%

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

      5. sub-neg 35.3%

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

      6. distribute-lft-in 35.3%

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

      7. fma-def 35.3%

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

      8. metadata-eval 35.3%

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

      9. metadata-eval 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in i around 0 34.4%

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

      1. associate-*r* 34.4%

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

      2. *-commutative 34.4%

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

      3. associate-*r/ 34.4%

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

      4. metadata-eval 34.4%

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

   6. Simplified 34.4%

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

      1. distribute-lft-in 34.4%

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

      2. flip-+ 27.4%

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

   8. Applied egg-rr 27.4%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in i around 0 64.7%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-11} \lor \neg \left(n \leq 0.09\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{10000 \cdot {n}^{2}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 4: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-6} \lor \neg \left(n \leq 0.09\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6.5e-6) (not (<= n 0.09)))
   (* n (* 100.0 (/ (expm1 i) i)))
   (/
    (* n (* n 10000.0))
    (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6.5e-6) || !(n <= 0.09)) {
		tmp = n * (100.0 * (expm1(i) / i));
	} else {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -6.5e-6) || !(n <= 0.09)) {
		tmp = n * (100.0 * (Math.expm1(i) / i));
	} else {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -6.5e-6) or not (n <= 0.09):
		tmp = n * (100.0 * (math.expm1(i) / i))
	else:
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6.5e-6) || !(n <= 0.09))
		tmp = Float64(n * Float64(100.0 * Float64(expm1(i) / i)));
	else
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6.5e-6], N[Not[LessEqual[n, 0.09]], $MachinePrecision]], N[(n * N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.4999999999999996e-6 or 0.089999999999999997 < n

   1. Initial program 17.5%

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

   2. Taylor expanded in n around inf 40.4%

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

      1. expm1-def 67.5%

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

   4. Simplified 67.5%

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

      1. associate-*r/ 67.5%

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

      2. frac-2neg 67.5%

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

      3. distribute-neg-frac 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. associate-/r/ 95.2%

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

      2. distribute-lft-neg-in 95.2%

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

      3. metadata-eval 95.2%

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

      4. neg-mul-1 95.2%

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

      5. times-frac 95.3%

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

      6. metadata-eval 95.3%

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

   8. Simplified 95.3%

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

   if -6.4999999999999996e-6 < n < 0.089999999999999997

   1. Initial program 34.8%

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

      1. associate-/r/ 34.7%

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

      2. associate-*r* 34.6%

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

      3. *-commutative 34.6%

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

      4. associate-*r/ 34.6%

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

      5. sub-neg 34.6%

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

      6. distribute-lft-in 34.6%

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

      7. fma-def 34.6%

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

      8. metadata-eval 34.6%

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

      9. metadata-eval 34.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in i around 0 35.8%

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

      1. associate-*r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*r/ 35.8%

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

      4. metadata-eval 35.8%

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

   6. Simplified 35.8%

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

      1. distribute-lft-in 35.8%

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

      2. flip-+ 28.9%

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

   8. Applied egg-rr 28.9%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in i around 0 65.4%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.4%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       2. unpow2 65.4%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       3. associate-*l* 65.4%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   11. Simplified 65.4%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-6} \lor \neg \left(n \leq 0.09\right):\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 5: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{if}\;i \leq -5.4 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;n \cdot \left(100 + \frac{i}{n} \cdot -50\right)\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(n \cdot n\right) \cdot \left(10000 + \left(i \cdot i\right) \cdot -2500\right)}{t_0}\\ \mathbf{elif}\;i \leq 6 \cdot 10^{+253}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n))))))))
   (if (<= i -5.4e-38)
     (* 100.0 (* (/ n i) (expm1 i)))
     (if (<= i 2.4e-29)
       (* n (+ 100.0 (* (/ i n) -50.0)))
       (if (<= i 3.5e+65)
         (/ (* (* n n) (+ 10000.0 (* (* i i) -2500.0))) t_0)
         (if (<= i 6e+253) (/ (* n (* n 10000.0)) t_0) (* n (* i 50.0))))))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n)))));
	double tmp;
	if (i <= -5.4e-38) {
		tmp = 100.0 * ((n / i) * expm1(i));
	} else if (i <= 2.4e-29) {
		tmp = n * (100.0 + ((i / n) * -50.0));
	} else if (i <= 3.5e+65) {
		tmp = ((n * n) * (10000.0 + ((i * i) * -2500.0))) / t_0;
	} else if (i <= 6e+253) {
		tmp = (n * (n * 10000.0)) / t_0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n)))));
	double tmp;
	if (i <= -5.4e-38) {
		tmp = 100.0 * ((n / i) * Math.expm1(i));
	} else if (i <= 2.4e-29) {
		tmp = n * (100.0 + ((i / n) * -50.0));
	} else if (i <= 3.5e+65) {
		tmp = ((n * n) * (10000.0 + ((i * i) * -2500.0))) / t_0;
	} else if (i <= 6e+253) {
		tmp = (n * (n * 10000.0)) / t_0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n)))))
	tmp = 0
	if i <= -5.4e-38:
		tmp = 100.0 * ((n / i) * math.expm1(i))
	elif i <= 2.4e-29:
		tmp = n * (100.0 + ((i / n) * -50.0))
	elif i <= 3.5e+65:
		tmp = ((n * n) * (10000.0 + ((i * i) * -2500.0))) / t_0
	elif i <= 6e+253:
		tmp = (n * (n * 10000.0)) / t_0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n))))))
	tmp = 0.0
	if (i <= -5.4e-38)
		tmp = Float64(100.0 * Float64(Float64(n / i) * expm1(i)));
	elseif (i <= 2.4e-29)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i / n) * -50.0)));
	elseif (i <= 3.5e+65)
		tmp = Float64(Float64(Float64(n * n) * Float64(10000.0 + Float64(Float64(i * i) * -2500.0))) / t_0);
	elseif (i <= 6e+253)
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / t_0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5.4e-38], N[(100.0 * N[(N[(n / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.4e-29], N[(n * N[(100.0 + N[(N[(i / n), $MachinePrecision] * -50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.5e+65], N[(N[(N[(n * n), $MachinePrecision] * N[(10000.0 + N[(N[(i * i), $MachinePrecision] * -2500.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[i, 6e+253], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -5.40000000000000011e-38

   1. Initial program 42.1%

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

      1. associate-*r/ 42.2%

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

      2. sub-neg 42.2%

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

      3. distribute-lft-in 42.3%

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

      4. fma-def 42.2%

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

      5. metadata-eval 42.2%

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

      6. metadata-eval 42.2%

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

   3. Simplified 42.2%

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

      1. fma-udef 42.3%

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

      2. metadata-eval 42.3%

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

      3. metadata-eval 42.3%

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

      4. distribute-lft-in 42.2%

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

      5. sub-neg 42.2%

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

      6. *-commutative 42.2%

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

      7. pow-to-exp 40.7%

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

      8. expm1-def 51.1%

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

      9. *-commutative 51.1%

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

      10. log1p-udef 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in n around inf 75.9%

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

      1. expm1-def 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*r/ 78.7%

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

      4. *-commutative 78.7%

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

   8. Simplified 78.7%

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

   if -5.40000000000000011e-38 < i < 2.39999999999999992e-29

   1. Initial program 7.5%

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

      1. associate-/r/ 8.1%

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

      2. associate-*r* 8.1%

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

      3. *-commutative 8.1%

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

      4. associate-*r/ 8.1%

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

      5. sub-neg 8.1%

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

      6. distribute-lft-in 8.1%

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

      7. fma-def 8.1%

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

      8. metadata-eval 8.1%

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

      9. metadata-eval 8.1%

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

   3. Simplified 8.1%

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

   4. Taylor expanded in i around 0 86.5%

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

      1. associate-*r* 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r/ 86.5%

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

      4. metadata-eval 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in n around 0 86.5%

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

   if 2.39999999999999992e-29 < i < 3.5000000000000001e65

   1. Initial program 20.7%

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

      1. associate-/r/ 20.7%

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

      2. associate-*r* 20.7%

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

      3. *-commutative 20.7%

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

      4. associate-*r/ 20.7%

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

      5. sub-neg 20.7%

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

      6. distribute-lft-in 20.4%

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

      7. fma-def 20.7%

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

      8. metadata-eval 20.7%

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

      9. metadata-eval 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in i around 0 19.9%

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

      1. associate-*r* 19.9%

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

      2. *-commutative 19.9%

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

      3. associate-*r/ 19.9%

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

      4. metadata-eval 19.9%

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

   6. Simplified 19.9%

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

      1. distribute-lft-in 19.8%

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

      2. flip-+ 16.4%

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

   8. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in n around inf 75.5%

      \[\leadsto \frac{\color{blue}{{n}^{2} \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 75.5%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       2. cancel-sign-sub-inv 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \color{blue}{\left(10000 + \left(-2500\right) \cdot {i}^{2}\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       3. metadata-eval 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \left(10000 + \color{blue}{-2500} \cdot {i}^{2}\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       4. *-commutative 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \left(10000 + \color{blue}{{i}^{2} \cdot -2500}\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       5. unpow2 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \left(10000 + \color{blue}{\left(i \cdot i\right)} \cdot -2500\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   11. Simplified 75.5%

       \[\leadsto \frac{\color{blue}{\left(n \cdot n\right) \cdot \left(10000 + \left(i \cdot i\right) \cdot -2500\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   if 3.5000000000000001e65 < i < 5.9999999999999996e253

   1. Initial program 47.6%

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

      1. associate-/r/ 47.8%

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

      2. associate-*r* 47.9%

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

      3. *-commutative 47.9%

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

      4. associate-*r/ 47.9%

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

      5. sub-neg 47.9%

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

      6. distribute-lft-in 47.9%

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

      7. fma-def 47.9%

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

      8. metadata-eval 47.9%

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

      9. metadata-eval 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in i around 0 21.6%

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

      1. associate-*r* 21.6%

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

      2. *-commutative 21.6%

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

      3. associate-*r/ 21.6%

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

      4. metadata-eval 21.6%

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

   6. Simplified 21.6%

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

      1. distribute-lft-in 21.6%

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

      2. flip-+ 7.5%

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

   8. Applied egg-rr 7.5%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in i around 0 51.1%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 51.1%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       2. unpow2 51.1%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       3. associate-*l* 51.1%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   11. Simplified 51.1%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   if 5.9999999999999996e253 < i

   1. Initial program 99.5%

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

   2. Taylor expanded in n around inf 66.9%

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

      1. expm1-def 66.9%

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

   4. Simplified 66.9%

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

   5. Taylor expanded in i around 0 66.9%

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

      1. associate-*r* 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in i around inf 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-*l* 66.9%

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

   10. Simplified 66.9%

       \[\leadsto \color{blue}{n \cdot \left(i \cdot 50\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.4 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \left(\frac{n}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;i \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;n \cdot \left(100 + \frac{i}{n} \cdot -50\right)\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(n \cdot n\right) \cdot \left(10000 + \left(i \cdot i\right) \cdot -2500\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{elif}\;i \leq 6 \cdot 10^{+253}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]

Alternative 6: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\\ \mathbf{if}\;i \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2 \cdot 10^{-28}:\\ \;\;\;\;n \cdot \left(100 + \frac{i}{n} \cdot -50\right)\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(n \cdot n\right) \cdot \left(10000 + \left(i \cdot i\right) \cdot -2500\right)}{t_0}\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+253}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n))))))))
   (if (<= i -8.5e-39)
     (* 100.0 (/ (expm1 i) (/ i n)))
     (if (<= i 2e-28)
       (* n (+ 100.0 (* (/ i n) -50.0)))
       (if (<= i 3.5e+65)
         (/ (* (* n n) (+ 10000.0 (* (* i i) -2500.0))) t_0)
         (if (<= i 7.5e+253) (/ (* n (* n 10000.0)) t_0) (* n (* i 50.0))))))))

C

double code(double i, double n) {
	double t_0 = (n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n)))));
	double tmp;
	if (i <= -8.5e-39) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else if (i <= 2e-28) {
		tmp = n * (100.0 + ((i / n) * -50.0));
	} else if (i <= 3.5e+65) {
		tmp = ((n * n) * (10000.0 + ((i * i) * -2500.0))) / t_0;
	} else if (i <= 7.5e+253) {
		tmp = (n * (n * 10000.0)) / t_0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n)))));
	double tmp;
	if (i <= -8.5e-39) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else if (i <= 2e-28) {
		tmp = n * (100.0 + ((i / n) * -50.0));
	} else if (i <= 3.5e+65) {
		tmp = ((n * n) * (10000.0 + ((i * i) * -2500.0))) / t_0;
	} else if (i <= 7.5e+253) {
		tmp = (n * (n * 10000.0)) / t_0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n)))))
	tmp = 0
	if i <= -8.5e-39:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	elif i <= 2e-28:
		tmp = n * (100.0 + ((i / n) * -50.0))
	elif i <= 3.5e+65:
		tmp = ((n * n) * (10000.0 + ((i * i) * -2500.0))) / t_0
	elif i <= 7.5e+253:
		tmp = (n * (n * 10000.0)) / t_0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n))))))
	tmp = 0.0
	if (i <= -8.5e-39)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (i <= 2e-28)
		tmp = Float64(n * Float64(100.0 + Float64(Float64(i / n) * -50.0)));
	elseif (i <= 3.5e+65)
		tmp = Float64(Float64(Float64(n * n) * Float64(10000.0 + Float64(Float64(i * i) * -2500.0))) / t_0);
	elseif (i <= 7.5e+253)
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / t_0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -8.5e-39], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2e-28], N[(n * N[(100.0 + N[(N[(i / n), $MachinePrecision] * -50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.5e+65], N[(N[(N[(n * n), $MachinePrecision] * N[(10000.0 + N[(N[(i * i), $MachinePrecision] * -2500.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[i, 7.5e+253], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if i < -8.5000000000000005e-39

   1. Initial program 42.1%

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

   2. Taylor expanded in n around inf 76.2%

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

      1. expm1-def 79.0%

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

   4. Simplified 79.0%

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

   if -8.5000000000000005e-39 < i < 1.99999999999999994e-28

   1. Initial program 7.5%

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

      1. associate-/r/ 8.1%

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

      2. associate-*r* 8.1%

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

      3. *-commutative 8.1%

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

      4. associate-*r/ 8.1%

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

      5. sub-neg 8.1%

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

      6. distribute-lft-in 8.1%

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

      7. fma-def 8.1%

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

      8. metadata-eval 8.1%

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

      9. metadata-eval 8.1%

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

   3. Simplified 8.1%

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

   4. Taylor expanded in i around 0 86.5%

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

      1. associate-*r* 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r/ 86.5%

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

      4. metadata-eval 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in n around 0 86.5%

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

   if 1.99999999999999994e-28 < i < 3.5000000000000001e65

   1. Initial program 20.7%

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

      1. associate-/r/ 20.7%

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

      2. associate-*r* 20.7%

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

      3. *-commutative 20.7%

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

      4. associate-*r/ 20.7%

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

      5. sub-neg 20.7%

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

      6. distribute-lft-in 20.4%

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

      7. fma-def 20.7%

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

      8. metadata-eval 20.7%

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

      9. metadata-eval 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in i around 0 19.9%

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

      1. associate-*r* 19.9%

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

      2. *-commutative 19.9%

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

      3. associate-*r/ 19.9%

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

      4. metadata-eval 19.9%

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

   6. Simplified 19.9%

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

      1. distribute-lft-in 19.8%

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

      2. flip-+ 16.4%

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

   8. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in n around inf 75.5%

      \[\leadsto \frac{\color{blue}{{n}^{2} \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 75.5%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       2. cancel-sign-sub-inv 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \color{blue}{\left(10000 + \left(-2500\right) \cdot {i}^{2}\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       3. metadata-eval 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \left(10000 + \color{blue}{-2500} \cdot {i}^{2}\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       4. *-commutative 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \left(10000 + \color{blue}{{i}^{2} \cdot -2500}\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       5. unpow2 75.5%

          \[\leadsto \frac{\left(n \cdot n\right) \cdot \left(10000 + \color{blue}{\left(i \cdot i\right)} \cdot -2500\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   11. Simplified 75.5%

       \[\leadsto \frac{\color{blue}{\left(n \cdot n\right) \cdot \left(10000 + \left(i \cdot i\right) \cdot -2500\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   if 3.5000000000000001e65 < i < 7.50000000000000017e253

   1. Initial program 47.6%

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

      1. associate-/r/ 47.8%

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

      2. associate-*r* 47.9%

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

      3. *-commutative 47.9%

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

      4. associate-*r/ 47.9%

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

      5. sub-neg 47.9%

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

      6. distribute-lft-in 47.9%

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

      7. fma-def 47.9%

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

      8. metadata-eval 47.9%

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

      9. metadata-eval 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in i around 0 21.6%

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

      1. associate-*r* 21.6%

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

      2. *-commutative 21.6%

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

      3. associate-*r/ 21.6%

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

      4. metadata-eval 21.6%

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

   6. Simplified 21.6%

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

      1. distribute-lft-in 21.6%

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

      2. flip-+ 7.5%

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

   8. Applied egg-rr 7.5%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in i around 0 51.1%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 51.1%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       2. unpow2 51.1%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       3. associate-*l* 51.1%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   11. Simplified 51.1%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   if 7.50000000000000017e253 < i

   1. Initial program 99.5%

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

   2. Taylor expanded in n around inf 66.9%

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

      1. expm1-def 66.9%

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

   4. Simplified 66.9%

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

   5. Taylor expanded in i around 0 66.9%

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

      1. associate-*r* 66.9%

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

   7. Simplified 66.9%

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

   8. Taylor expanded in i around inf 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-*l* 66.9%

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

   10. Simplified 66.9%

       \[\leadsto \color{blue}{n \cdot \left(i \cdot 50\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2 \cdot 10^{-28}:\\ \;\;\;\;n \cdot \left(100 + \frac{i}{n} \cdot -50\right)\\ \mathbf{elif}\;i \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(n \cdot n\right) \cdot \left(10000 + \left(i \cdot i\right) \cdot -2500\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{+253}:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]

Alternative 7: 68.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.0036:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.086:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -0.0036)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 0.086)
     (/
      (* n (* n 10000.0))
      (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))
     (*
      100.0
      (+
       n
       (*
        n
        (+
         (*
          (* i i)
          (+ (/ 0.3333333333333333 (* n n)) (- 0.16666666666666666 (/ 0.5 n))))
         (* i (- 0.5 (/ 0.5 n))))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -0.0036) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.086) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-0.0036d0)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 0.086d0) then
        tmp = (n * (n * 10000.0d0)) / ((n * 100.0d0) - (n * (i * (50.0d0 + (100.0d0 * ((-0.5d0) / n))))))
    else
        tmp = 100.0d0 * (n + (n * (((i * i) * ((0.3333333333333333d0 / (n * n)) + (0.16666666666666666d0 - (0.5d0 / n)))) + (i * (0.5d0 - (0.5d0 / n))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -0.0036) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.086) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -0.0036:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 0.086:
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	else:
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -0.0036)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 0.086)
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	else
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(Float64(i * i) * Float64(Float64(0.3333333333333333 / Float64(n * n)) + Float64(0.16666666666666666 - Float64(0.5 / n)))) + Float64(i * Float64(0.5 - Float64(0.5 / n)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -0.0036)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 0.086)
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	else
		tmp = 100.0 * (n + (n * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 - (0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -0.0036], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.086], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(n * N[(N[(N[(i * i), $MachinePrecision] * N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.0035999999999999999

   1. Initial program 20.8%

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

      1. associate-/r/ 21.2%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

      4. associate-*r/ 21.2%

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

      5. sub-neg 21.2%

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

      6. distribute-lft-in 21.2%

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

      7. fma-def 21.2%

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

      8. metadata-eval 21.2%

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

      9. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in n around inf 39.4%

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

      1. associate-/l* 39.5%

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

      2. *-commutative 39.5%

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

      3. fma-neg 39.4%

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

      4. metadata-eval 39.4%

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

   6. Simplified 39.4%

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

   7. Taylor expanded in i around 0 63.1%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 63.1%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 63.1%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -0.0035999999999999999 < n < 0.085999999999999993

   1. Initial program 34.8%

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

      1. associate-/r/ 34.7%

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

      2. associate-*r* 34.6%

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

      3. *-commutative 34.6%

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

      4. associate-*r/ 34.6%

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

      5. sub-neg 34.6%

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

      6. distribute-lft-in 34.6%

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

      7. fma-def 34.6%

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

      8. metadata-eval 34.6%

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

      9. metadata-eval 34.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in i around 0 35.8%

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

      1. associate-*r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*r/ 35.8%

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

      4. metadata-eval 35.8%

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

   6. Simplified 35.8%

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

      1. distribute-lft-in 35.8%

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

      2. flip-+ 28.9%

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

   8. Applied egg-rr 28.9%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in i around 0 65.4%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.4%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       2. unpow2 65.4%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       3. associate-*l* 65.4%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   11. Simplified 65.4%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   if 0.085999999999999993 < n

   1. Initial program 13.6%

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

   2. Taylor expanded in i around 0 73.0%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right)\right) + n \cdot \left(i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. distribute-lft-out 73.1%

         \[\leadsto 100 \cdot \left(n + \color{blue}{n \cdot \left({i}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)}\right) \]

      2. unpow2 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\color{blue}{\left(i \cdot i\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + 0.16666666666666666\right) - 0.5 \cdot \frac{1}{n}\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      3. associate--l+ 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right)} + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      4. associate-*r/ 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{n}^{2}}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      5. metadata-eval 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{\color{blue}{0.3333333333333333}}{{n}^{2}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      6. unpow2 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{\color{blue}{n \cdot n}} + \left(0.16666666666666666 - 0.5 \cdot \frac{1}{n}\right)\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      7. associate-*r/ 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      8. metadata-eval 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{\color{blue}{0.5}}{n}\right)\right) + i \cdot \left(0.5 - 0.5 \cdot \frac{1}{n}\right)\right)\right) \]

      9. associate-*r/ 73.1%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right) + i \cdot \left(0.5 - \color{blue}{\frac{0.5 \cdot 1}{n}}\right)\right)\right) \]

      10. metadata-eval 73.1%

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

   4. Simplified 73.1%

      \[\leadsto 100 \cdot \color{blue}{\left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.0036:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.086:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 - \frac{0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 8: 68.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.0062:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.09:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 + i \cdot -50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -0.0062)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 0.09)
     (/
      (* n (* n 10000.0))
      (- (* n 100.0) (* n (* i (+ 50.0 (* 100.0 (/ -0.5 n)))))))
     (/ n (/ (+ 100.0 (* i -50.0)) (+ 10000.0 (* (* i i) -2500.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -0.0062) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.09) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-0.0062d0)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 0.09d0) then
        tmp = (n * (n * 10000.0d0)) / ((n * 100.0d0) - (n * (i * (50.0d0 + (100.0d0 * ((-0.5d0) / n))))))
    else
        tmp = n / ((100.0d0 + (i * (-50.0d0))) / (10000.0d0 + ((i * i) * (-2500.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -0.0062) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.09) {
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	} else {
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -0.0062:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 0.09:
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))))
	else:
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -0.0062)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 0.09)
		tmp = Float64(Float64(n * Float64(n * 10000.0)) / Float64(Float64(n * 100.0) - Float64(n * Float64(i * Float64(50.0 + Float64(100.0 * Float64(-0.5 / n)))))));
	else
		tmp = Float64(n / Float64(Float64(100.0 + Float64(i * -50.0)) / Float64(10000.0 + Float64(Float64(i * i) * -2500.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -0.0062)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 0.09)
		tmp = (n * (n * 10000.0)) / ((n * 100.0) - (n * (i * (50.0 + (100.0 * (-0.5 / n))))));
	else
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -0.0062], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.09], N[(N[(n * N[(n * 10000.0), $MachinePrecision]), $MachinePrecision] / N[(N[(n * 100.0), $MachinePrecision] - N[(n * N[(i * N[(50.0 + N[(100.0 * N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(100.0 + N[(i * -50.0), $MachinePrecision]), $MachinePrecision] / N[(10000.0 + N[(N[(i * i), $MachinePrecision] * -2500.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.00619999999999999978

   1. Initial program 20.8%

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

      1. associate-/r/ 21.2%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

      4. associate-*r/ 21.2%

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

      5. sub-neg 21.2%

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

      6. distribute-lft-in 21.2%

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

      7. fma-def 21.2%

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

      8. metadata-eval 21.2%

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

      9. metadata-eval 21.2%

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

   3. Simplified 21.2%

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

   4. Taylor expanded in n around inf 39.4%

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

      1. associate-/l* 39.5%

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

      2. *-commutative 39.5%

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

      3. fma-neg 39.4%

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

      4. metadata-eval 39.4%

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

   6. Simplified 39.4%

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

   7. Taylor expanded in i around 0 63.1%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 63.1%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 63.1%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -0.00619999999999999978 < n < 0.089999999999999997

   1. Initial program 34.8%

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

      1. associate-/r/ 34.7%

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

      2. associate-*r* 34.6%

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

      3. *-commutative 34.6%

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

      4. associate-*r/ 34.6%

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

      5. sub-neg 34.6%

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

      6. distribute-lft-in 34.6%

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

      7. fma-def 34.6%

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

      8. metadata-eval 34.6%

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

      9. metadata-eval 34.6%

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

   3. Simplified 34.6%

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

   4. Taylor expanded in i around 0 35.8%

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

      1. associate-*r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*r/ 35.8%

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

      4. metadata-eval 35.8%

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

   6. Simplified 35.8%

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

      1. distribute-lft-in 35.8%

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

      2. flip-+ 28.9%

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

   8. Applied egg-rr 28.9%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in i around 0 65.4%

      \[\leadsto \frac{\color{blue}{10000 \cdot {n}^{2}}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 65.4%

          \[\leadsto \frac{\color{blue}{{n}^{2} \cdot 10000}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       2. unpow2 65.4%

          \[\leadsto \frac{\color{blue}{\left(n \cdot n\right)} \cdot 10000}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

       3. associate-*l* 65.4%

          \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   11. Simplified 65.4%

       \[\leadsto \frac{\color{blue}{n \cdot \left(n \cdot 10000\right)}}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)} \]

   if 0.089999999999999997 < n

   1. Initial program 13.6%

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

      1. associate-/r/ 14.2%

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

      2. associate-*r* 14.2%

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

      3. *-commutative 14.2%

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

      4. associate-*r/ 14.2%

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

      5. sub-neg 14.2%

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

      6. distribute-lft-in 14.2%

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

      7. fma-def 14.2%

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

      8. metadata-eval 14.2%

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

      9. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

   4. Taylor expanded in i around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 71.8%

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

      4. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. distribute-lft-in 71.8%

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

      2. flip-+ 31.4%

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

   8. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in n around inf 71.8%

      \[\leadsto \color{blue}{\frac{n \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}{100 - 50 \cdot i}} \]
   10. Step-by-step derivation

       1. associate-/l* 73.0%

          \[\leadsto \color{blue}{\frac{n}{\frac{100 - 50 \cdot i}{10000 - 2500 \cdot {i}^{2}}}} \]

       2. cancel-sign-sub-inv 73.0%

          \[\leadsto \frac{n}{\frac{\color{blue}{100 + \left(-50\right) \cdot i}}{10000 - 2500 \cdot {i}^{2}}} \]

       3. metadata-eval 73.0%

          \[\leadsto \frac{n}{\frac{100 + \color{blue}{-50} \cdot i}{10000 - 2500 \cdot {i}^{2}}} \]

       4. *-commutative 73.0%

          \[\leadsto \frac{n}{\frac{100 + \color{blue}{i \cdot -50}}{10000 - 2500 \cdot {i}^{2}}} \]

       5. cancel-sign-sub-inv 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{\color{blue}{10000 + \left(-2500\right) \cdot {i}^{2}}}} \]

       6. metadata-eval 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{10000 + \color{blue}{-2500} \cdot {i}^{2}}} \]

       7. *-commutative 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{10000 + \color{blue}{{i}^{2} \cdot -2500}}} \]

       8. unpow2 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{10000 + \color{blue}{\left(i \cdot i\right)} \cdot -2500}} \]

   11. Simplified 73.0%

       \[\leadsto \color{blue}{\frac{n}{\frac{100 + i \cdot -50}{10000 + \left(i \cdot i\right) \cdot -2500}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.0062:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.09:\\ \;\;\;\;\frac{n \cdot \left(n \cdot 10000\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 + i \cdot -50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \]

Alternative 9: 66.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-21}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.09:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot i\right) \cdot 0.0008333333333333334}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 + i \cdot -50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7e-21)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 0.09)
     (/ n (+ 0.01 (* (* i i) 0.0008333333333333334)))
     (/ n (/ (+ 100.0 (* i -50.0)) (+ 10000.0 (* (* i i) -2500.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7e-21) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.09) {
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334));
	} else {
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-7d-21)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 0.09d0) then
        tmp = n / (0.01d0 + ((i * i) * 0.0008333333333333334d0))
    else
        tmp = n / ((100.0d0 + (i * (-50.0d0))) / (10000.0d0 + ((i * i) * (-2500.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7e-21) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.09) {
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334));
	} else {
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7e-21:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 0.09:
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334))
	else:
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7e-21)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 0.09)
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * i) * 0.0008333333333333334)));
	else
		tmp = Float64(n / Float64(Float64(100.0 + Float64(i * -50.0)) / Float64(10000.0 + Float64(Float64(i * i) * -2500.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -7e-21)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 0.09)
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334));
	else
		tmp = n / ((100.0 + (i * -50.0)) / (10000.0 + ((i * i) * -2500.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7e-21], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.09], N[(n / N[(0.01 + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(100.0 + N[(i * -50.0), $MachinePrecision]), $MachinePrecision] / N[(10000.0 + N[(N[(i * i), $MachinePrecision] * -2500.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.0000000000000007e-21

   1. Initial program 21.0%

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

      1. associate-/r/ 21.4%

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

      2. associate-*r* 21.5%

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

      3. *-commutative 21.5%

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

      4. associate-*r/ 21.5%

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

      5. sub-neg 21.5%

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

      6. distribute-lft-in 21.5%

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

      7. fma-def 21.5%

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

      8. metadata-eval 21.5%

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

      9. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

   4. Taylor expanded in n around inf 38.0%

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

      1. associate-/l* 38.0%

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

      2. *-commutative 38.0%

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

      3. fma-neg 38.0%

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

      4. metadata-eval 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in i around 0 62.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 62.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -7.0000000000000007e-21 < n < 0.089999999999999997

   1. Initial program 35.1%

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

      1. associate-/r/ 35.0%

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

      2. associate-*r* 35.0%

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

      3. *-commutative 35.0%

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

      4. associate-*r/ 35.0%

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

      5. sub-neg 35.0%

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

      6. distribute-lft-in 34.9%

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

      7. fma-def 35.0%

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

      8. metadata-eval 35.0%

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

      9. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in n around inf 19.8%

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

      1. associate-/l* 19.8%

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

      2. *-commutative 19.8%

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

      3. fma-neg 19.8%

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

      4. metadata-eval 19.8%

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

   6. Simplified 19.8%

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

   7. Taylor expanded in i around 0 62.2%

      \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 62.2%

         \[\leadsto \frac{n}{\color{blue}{\left(0.01 + -0.005 \cdot i\right) + 0.0008333333333333334 \cdot {i}^{2}}} \]

      2. +-commutative 62.2%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.005 \cdot i + 0.01\right)} + 0.0008333333333333334 \cdot {i}^{2}} \]

      3. *-commutative 62.2%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.005} + 0.01\right) + 0.0008333333333333334 \cdot {i}^{2}} \]

      4. fma-def 62.2%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.005, 0.01\right)} + 0.0008333333333333334 \cdot {i}^{2}} \]

      5. *-commutative 62.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \color{blue}{{i}^{2} \cdot 0.0008333333333333334}} \]

      6. unpow2 62.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334} \]

   9. Simplified 62.2%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \left(i \cdot i\right) \cdot 0.0008333333333333334}} \]

   10. Taylor expanded in i around 0 62.2%

       \[\leadsto \frac{n}{\color{blue}{0.01} + \left(i \cdot i\right) \cdot 0.0008333333333333334} \]

   if 0.089999999999999997 < n

   1. Initial program 13.6%

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

      1. associate-/r/ 14.2%

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

      2. associate-*r* 14.2%

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

      3. *-commutative 14.2%

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

      4. associate-*r/ 14.2%

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

      5. sub-neg 14.2%

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

      6. distribute-lft-in 14.2%

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

      7. fma-def 14.2%

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

      8. metadata-eval 14.2%

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

      9. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

   4. Taylor expanded in i around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 71.8%

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

      4. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

      1. distribute-lft-in 71.8%

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

      2. flip-+ 31.4%

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

   8. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{\frac{\left(n \cdot 100\right) \cdot \left(n \cdot 100\right) - \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right) \cdot \left(n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)\right)}{n \cdot 100 - n \cdot \left(i \cdot \left(50 + 100 \cdot \frac{-0.5}{n}\right)\right)}} \]

   9. Taylor expanded in n around inf 71.8%

      \[\leadsto \color{blue}{\frac{n \cdot \left(10000 - 2500 \cdot {i}^{2}\right)}{100 - 50 \cdot i}} \]
   10. Step-by-step derivation

       1. associate-/l* 73.0%

          \[\leadsto \color{blue}{\frac{n}{\frac{100 - 50 \cdot i}{10000 - 2500 \cdot {i}^{2}}}} \]

       2. cancel-sign-sub-inv 73.0%

          \[\leadsto \frac{n}{\frac{\color{blue}{100 + \left(-50\right) \cdot i}}{10000 - 2500 \cdot {i}^{2}}} \]

       3. metadata-eval 73.0%

          \[\leadsto \frac{n}{\frac{100 + \color{blue}{-50} \cdot i}{10000 - 2500 \cdot {i}^{2}}} \]

       4. *-commutative 73.0%

          \[\leadsto \frac{n}{\frac{100 + \color{blue}{i \cdot -50}}{10000 - 2500 \cdot {i}^{2}}} \]

       5. cancel-sign-sub-inv 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{\color{blue}{10000 + \left(-2500\right) \cdot {i}^{2}}}} \]

       6. metadata-eval 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{10000 + \color{blue}{-2500} \cdot {i}^{2}}} \]

       7. *-commutative 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{10000 + \color{blue}{{i}^{2} \cdot -2500}}} \]

       8. unpow2 73.0%

          \[\leadsto \frac{n}{\frac{100 + i \cdot -50}{10000 + \color{blue}{\left(i \cdot i\right)} \cdot -2500}} \]

   11. Simplified 73.0%

       \[\leadsto \color{blue}{\frac{n}{\frac{100 + i \cdot -50}{10000 + \left(i \cdot i\right) \cdot -2500}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{-21}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.09:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot i\right) \cdot 0.0008333333333333334}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{100 + i \cdot -50}{10000 + \left(i \cdot i\right) \cdot -2500}}\\ \end{array} \]

Alternative 10: 64.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.01 + i \cdot -0.005\\ \mathbf{if}\;n \leq -6.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{n}{t_0}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{n}{\left(i \cdot i\right) \cdot 0.0008333333333333334 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 0.01 (* i -0.005))))
   (if (<= n -6.8e-17)
     (/ n t_0)
     (if (<= n 1.8e+57)
       (/ n (+ (* (* i i) 0.0008333333333333334) t_0))
       (/ (* 100.0 (* i n)) i)))))

C

double code(double i, double n) {
	double t_0 = 0.01 + (i * -0.005);
	double tmp;
	if (n <= -6.8e-17) {
		tmp = n / t_0;
	} else if (n <= 1.8e+57) {
		tmp = n / (((i * i) * 0.0008333333333333334) + t_0);
	} else {
		tmp = (100.0 * (i * n)) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.01d0 + (i * (-0.005d0))
    if (n <= (-6.8d-17)) then
        tmp = n / t_0
    else if (n <= 1.8d+57) then
        tmp = n / (((i * i) * 0.0008333333333333334d0) + t_0)
    else
        tmp = (100.0d0 * (i * n)) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 0.01 + (i * -0.005);
	double tmp;
	if (n <= -6.8e-17) {
		tmp = n / t_0;
	} else if (n <= 1.8e+57) {
		tmp = n / (((i * i) * 0.0008333333333333334) + t_0);
	} else {
		tmp = (100.0 * (i * n)) / i;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 0.01 + (i * -0.005)
	tmp = 0
	if n <= -6.8e-17:
		tmp = n / t_0
	elif n <= 1.8e+57:
		tmp = n / (((i * i) * 0.0008333333333333334) + t_0)
	else:
		tmp = (100.0 * (i * n)) / i
	return tmp

Julia

function code(i, n)
	t_0 = Float64(0.01 + Float64(i * -0.005))
	tmp = 0.0
	if (n <= -6.8e-17)
		tmp = Float64(n / t_0);
	elseif (n <= 1.8e+57)
		tmp = Float64(n / Float64(Float64(Float64(i * i) * 0.0008333333333333334) + t_0));
	else
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 0.01 + (i * -0.005);
	tmp = 0.0;
	if (n <= -6.8e-17)
		tmp = n / t_0;
	elseif (n <= 1.8e+57)
		tmp = n / (((i * i) * 0.0008333333333333334) + t_0);
	else
		tmp = (100.0 * (i * n)) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.8e-17], N[(n / t$95$0), $MachinePrecision], If[LessEqual[n, 1.8e+57], N[(n / N[(N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.7999999999999996e-17

   1. Initial program 21.0%

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

      1. associate-/r/ 21.4%

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

      2. associate-*r* 21.5%

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

      3. *-commutative 21.5%

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

      4. associate-*r/ 21.5%

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

      5. sub-neg 21.5%

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

      6. distribute-lft-in 21.5%

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

      7. fma-def 21.5%

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

      8. metadata-eval 21.5%

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

      9. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

   4. Taylor expanded in n around inf 38.0%

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

      1. associate-/l* 38.0%

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

      2. *-commutative 38.0%

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

      3. fma-neg 38.0%

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

      4. metadata-eval 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in i around 0 62.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 62.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -6.7999999999999996e-17 < n < 1.8000000000000001e57

   1. Initial program 33.9%

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

      1. associate-/r/ 33.8%

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

      2. associate-*r* 33.8%

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

      3. *-commutative 33.8%

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

      4. associate-*r/ 33.8%

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

      5. sub-neg 33.8%

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

      6. distribute-lft-in 33.7%

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

      7. fma-def 33.8%

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

      8. metadata-eval 33.8%

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

      9. metadata-eval 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in n around inf 20.1%

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

      1. associate-/l* 20.1%

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

      2. *-commutative 20.1%

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

      3. fma-neg 20.1%

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

      4. metadata-eval 20.1%

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

   6. Simplified 20.1%

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

   7. Taylor expanded in i around 0 64.1%

      \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 64.1%

         \[\leadsto \frac{n}{\color{blue}{\left(0.01 + -0.005 \cdot i\right) + 0.0008333333333333334 \cdot {i}^{2}}} \]

      2. +-commutative 64.1%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.005 \cdot i + 0.01\right)} + 0.0008333333333333334 \cdot {i}^{2}} \]

      3. *-commutative 64.1%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.005} + 0.01\right) + 0.0008333333333333334 \cdot {i}^{2}} \]

      4. fma-def 64.1%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.005, 0.01\right)} + 0.0008333333333333334 \cdot {i}^{2}} \]

      5. *-commutative 64.1%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \color{blue}{{i}^{2} \cdot 0.0008333333333333334}} \]

      6. unpow2 64.1%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334} \]

   9. Simplified 64.1%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \left(i \cdot i\right) \cdot 0.0008333333333333334}} \]
   10. Step-by-step derivation

       1. fma-udef 64.1%

          \[\leadsto \frac{n}{\color{blue}{\left(i \cdot -0.005 + 0.01\right)} + \left(i \cdot i\right) \cdot 0.0008333333333333334} \]

   11. Applied egg-rr 64.1%

       \[\leadsto \frac{n}{\color{blue}{\left(i \cdot -0.005 + 0.01\right)} + \left(i \cdot i\right) \cdot 0.0008333333333333334} \]

   if 1.8000000000000001e57 < n

   1. Initial program 11.1%

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

      1. associate-/r/ 11.8%

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

      2. associate-*r* 11.8%

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

      3. *-commutative 11.8%

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

      4. associate-*r/ 11.8%

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

      5. sub-neg 11.8%

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

      6. distribute-lft-in 11.8%

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

      7. fma-def 11.8%

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

      8. metadata-eval 11.8%

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

      9. metadata-eval 11.8%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in n around inf 47.3%

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

   5. Taylor expanded in i around 0 70.9%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{n}{\left(i \cdot i\right) \cdot 0.0008333333333333334 + \left(0.01 + i \cdot -0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \end{array} \]

Alternative 11: 65.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.065:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot i\right) \cdot 0.0008333333333333334}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.8e-19)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 0.065)
     (/ n (+ 0.01 (* (* i i) 0.0008333333333333334)))
     (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.8e-19) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.065) {
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334));
	} 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 <= (-7.8d-19)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 0.065d0) then
        tmp = n / (0.01d0 + ((i * i) * 0.0008333333333333334d0))
    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 <= -7.8e-19) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 0.065) {
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.8e-19:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 0.065:
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.8e-19)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 0.065)
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * i) * 0.0008333333333333334)));
	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 <= -7.8e-19)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 0.065)
		tmp = n / (0.01 + ((i * i) * 0.0008333333333333334));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.8e-19], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.065], N[(n / N[(0.01 + N[(N[(i * i), $MachinePrecision] * 0.0008333333333333334), $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 < -7.7999999999999999e-19

   1. Initial program 21.0%

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

      1. associate-/r/ 21.4%

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

      2. associate-*r* 21.5%

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

      3. *-commutative 21.5%

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

      4. associate-*r/ 21.5%

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

      5. sub-neg 21.5%

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

      6. distribute-lft-in 21.5%

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

      7. fma-def 21.5%

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

      8. metadata-eval 21.5%

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

      9. metadata-eval 21.5%

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

   3. Simplified 21.5%

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

   4. Taylor expanded in n around inf 38.0%

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

      1. associate-/l* 38.0%

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

      2. *-commutative 38.0%

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

      3. fma-neg 38.0%

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

      4. metadata-eval 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in i around 0 62.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 62.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -7.7999999999999999e-19 < n < 0.065000000000000002

   1. Initial program 35.1%

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

      1. associate-/r/ 35.0%

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

      2. associate-*r* 35.0%

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

      3. *-commutative 35.0%

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

      4. associate-*r/ 35.0%

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

      5. sub-neg 35.0%

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

      6. distribute-lft-in 34.9%

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

      7. fma-def 35.0%

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

      8. metadata-eval 35.0%

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

      9. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in n around inf 19.8%

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

      1. associate-/l* 19.8%

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

      2. *-commutative 19.8%

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

      3. fma-neg 19.8%

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

      4. metadata-eval 19.8%

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

   6. Simplified 19.8%

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

   7. Taylor expanded in i around 0 62.2%

      \[\leadsto \frac{n}{\color{blue}{0.01 + \left(-0.005 \cdot i + 0.0008333333333333334 \cdot {i}^{2}\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 62.2%

         \[\leadsto \frac{n}{\color{blue}{\left(0.01 + -0.005 \cdot i\right) + 0.0008333333333333334 \cdot {i}^{2}}} \]

      2. +-commutative 62.2%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.005 \cdot i + 0.01\right)} + 0.0008333333333333334 \cdot {i}^{2}} \]

      3. *-commutative 62.2%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.005} + 0.01\right) + 0.0008333333333333334 \cdot {i}^{2}} \]

      4. fma-def 62.2%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.005, 0.01\right)} + 0.0008333333333333334 \cdot {i}^{2}} \]

      5. *-commutative 62.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \color{blue}{{i}^{2} \cdot 0.0008333333333333334}} \]

      6. unpow2 62.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.0008333333333333334} \]

   9. Simplified 62.2%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.005, 0.01\right) + \left(i \cdot i\right) \cdot 0.0008333333333333334}} \]

   10. Taylor expanded in i around 0 62.2%

       \[\leadsto \frac{n}{\color{blue}{0.01} + \left(i \cdot i\right) \cdot 0.0008333333333333334} \]

   if 0.065000000000000002 < n

   1. Initial program 13.6%

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

      1. associate-/r/ 14.2%

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

      2. associate-*r* 14.2%

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

      3. *-commutative 14.2%

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

      4. associate-*r/ 14.2%

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

      5. sub-neg 14.2%

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

      6. distribute-lft-in 14.2%

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

      7. fma-def 14.2%

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

      8. metadata-eval 14.2%

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

      9. metadata-eval 14.2%

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

   3. Simplified 14.2%

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

   4. Taylor expanded in i around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 71.8%

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

      4. metadata-eval 71.8%

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

   6. Simplified 71.8%

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

   7. Taylor expanded in n around inf 71.8%

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

      1. *-commutative 71.8%

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

   9. Simplified 71.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 0.065:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot i\right) \cdot 0.0008333333333333334}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 12: 63.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-129} \lor \neg \left(n \leq 2.1 \cdot 10^{-116}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.4e-129) (not (<= n 2.1e-116)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e-129) || !(n <= 2.1e-116)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.4d-129)) .or. (.not. (n <= 2.1d-116))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.4e-129) || !(n <= 2.1e-116)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.4e-129) or not (n <= 2.1e-116):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.4e-129) || !(n <= 2.1e-116))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.4e-129) || ~((n <= 2.1e-116)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.4e-129], N[Not[LessEqual[n, 2.1e-116]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e-129 or 2.0999999999999999e-116 < n

   1. Initial program 17.7%

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

      1. associate-/r/ 18.1%

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

      2. associate-*r* 18.1%

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

      3. *-commutative 18.1%

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

      4. associate-*r/ 18.1%

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

      5. sub-neg 18.1%

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

      6. distribute-lft-in 18.1%

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

      7. fma-def 18.1%

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

      8. metadata-eval 18.1%

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

      9. metadata-eval 18.1%

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

   3. Simplified 18.1%

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

   4. Taylor expanded in i around 0 60.5%

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

      1. associate-*r* 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-*r/ 60.5%

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

      4. metadata-eval 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in n around inf 60.9%

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

      1. *-commutative 60.9%

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

   9. Simplified 60.9%

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

   if -1.4e-129 < n < 2.0999999999999999e-116

   1. Initial program 48.6%

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

   2. Taylor expanded in i around 0 62.1%

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

   3. Taylor expanded in i around 0 62.1%

      \[\leadsto 100 \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-129} \lor \neg \left(n \leq 2.1 \cdot 10^{-116}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 63.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-240}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-116}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9e-240)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 2.1e-116) 0.0 (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9e-240) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 2.1e-116) {
		tmp = 0.0;
	} 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 <= (-9d-240)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 2.1d-116) then
        tmp = 0.0d0
    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 <= -9e-240) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 2.1e-116) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9e-240:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 2.1e-116:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9e-240)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 2.1e-116)
		tmp = 0.0;
	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 <= -9e-240)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 2.1e-116)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9e-240], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.1e-116], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.0000000000000003e-240

   1. Initial program 23.9%

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

      1. associate-/r/ 24.1%

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

      2. associate-*r* 24.1%

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

      3. *-commutative 24.1%

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

      4. associate-*r/ 24.1%

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

      5. sub-neg 24.1%

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

      6. distribute-lft-in 24.2%

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

      7. fma-def 24.1%

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

      8. metadata-eval 24.1%

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

      9. metadata-eval 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in n around inf 32.5%

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

      1. associate-/l* 32.5%

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

      2. *-commutative 32.5%

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

      3. fma-neg 32.5%

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

      4. metadata-eval 32.5%

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

   6. Simplified 32.5%

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

   7. Taylor expanded in i around 0 59.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
   8. Step-by-step derivation

      1. *-commutative 59.8%

         \[\leadsto \frac{n}{0.01 + \color{blue}{i \cdot -0.005}} \]

   9. Simplified 59.8%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]

   if -9.0000000000000003e-240 < n < 2.0999999999999999e-116

   1. Initial program 47.8%

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

   2. Taylor expanded in i around 0 67.8%

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

   3. Taylor expanded in i around 0 67.8%

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

   if 2.0999999999999999e-116 < n

   1. Initial program 15.1%

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

      1. associate-/r/ 15.6%

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

      2. associate-*r* 15.6%

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

      3. *-commutative 15.6%

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

      4. associate-*r/ 15.6%

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

      5. sub-neg 15.6%

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

      6. distribute-lft-in 15.6%

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

      7. fma-def 15.6%

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

      8. metadata-eval 15.6%

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

      9. metadata-eval 15.6%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in i around 0 66.6%

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

      1. associate-*r* 66.6%

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

      2. *-commutative 66.6%

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

      3. associate-*r/ 66.6%

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

      4. metadata-eval 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in n around inf 67.0%

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

      1. *-commutative 67.0%

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

   9. Simplified 67.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{-240}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{-116}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14: 59.7% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -8.2 \cdot 10^{+34}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -8.2e+34) 0.0 (if (<= i 5.8e+24) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -8.2e+34) {
		tmp = 0.0;
	} else if (i <= 5.8e+24) {
		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 <= (-8.2d+34)) then
        tmp = 0.0d0
    else if (i <= 5.8d+24) 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 <= -8.2e+34) {
		tmp = 0.0;
	} else if (i <= 5.8e+24) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -8.2e+34:
		tmp = 0.0
	elif i <= 5.8e+24:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -8.2e+34)
		tmp = 0.0;
	elseif (i <= 5.8e+24)
		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 <= -8.2e+34)
		tmp = 0.0;
	elseif (i <= 5.8e+24)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -8.2e+34], 0.0, If[LessEqual[i, 5.8e+24], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -8.1999999999999997e34 or 5.79999999999999958e24 < i

   1. Initial program 51.5%

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

   2. Taylor expanded in i around 0 32.8%

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

   3. Taylor expanded in i around 0 32.8%

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

   if -8.1999999999999997e34 < i < 5.79999999999999958e24

   1. Initial program 8.5%

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

   2. Taylor expanded in i around 0 76.0%

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

      1. *-commutative 76.0%

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

   4. Simplified 76.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.2 \cdot 10^{+34}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+24}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 17.4% 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 24.0%

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

2. Taylor expanded in i around 0 16.0%

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

3. Taylor expanded in i around 0 16.4%

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

4. Final simplification 16.4%

   \[\leadsto 0 \]

Developer target: 33.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023230 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :herbie-target
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))