Compound Interest

Percentage Accurate: 29.0% → 97.9%

Time: 26.9s

Alternatives: 13

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 29.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $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 28.0%

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

      1. associate-*r/ 28.0%

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

      2. sub-neg 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. metadata-eval 28.0%

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

      6. metadata-eval 28.0%

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

      7. fma-def 28.0%

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

      8. metadata-eval 28.0%

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

   3. Simplified 28.0%

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

      1. fma-udef 28.0%

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

      2. metadata-eval 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. sub-neg 28.0%

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

      6. *-commutative 28.0%

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

      7. add-exp-log 28.0%

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

      8. expm1-def 28.0%

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

      9. log-pow 40.5%

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

      10. log1p-udef 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. associate-/l* 97.5%

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

      2. associate-/r/ 98.1%

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

   8. Applied egg-rr 98.1%

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

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

   1. Initial program 99.9%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-def 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. *-commutative 1.8%

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

      3. fma-neg 1.8%

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

      4. metadata-eval 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 98.5%

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

Alternative 2: 96.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $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 28.0%

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

      1. associate-*r/ 28.0%

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

      2. sub-neg 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. metadata-eval 28.0%

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

      6. metadata-eval 28.0%

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

      7. fma-def 28.0%

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

      8. metadata-eval 28.0%

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

   3. Simplified 28.0%

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

      1. fma-udef 28.0%

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

      2. metadata-eval 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. sub-neg 28.0%

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

      6. *-commutative 28.0%

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

      7. associate-*l/ 28.0%

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

      8. div-inv 28.0%

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

      9. clear-num 27.6%

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

      10. associate-*l* 27.6%

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

      11. add-exp-log 27.6%

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

      12. expm1-def 27.6%

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

      13. log-pow 40.1%

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

      14. log1p-udef 96.7%

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

   6. Applied egg-rr 96.7%

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

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

   1. Initial program 99.9%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-def 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. *-commutative 1.8%

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

      3. fma-neg 1.8%

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

      4. metadata-eval 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 97.5%

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

Alternative 3: 97.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(n * N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $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 28.0%

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

      1. associate-*r/ 28.0%

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

      2. sub-neg 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. metadata-eval 28.0%

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

      6. metadata-eval 28.0%

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

      7. fma-def 28.0%

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

      8. metadata-eval 28.0%

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

   3. Simplified 28.0%

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

      1. fma-udef 28.0%

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

      2. metadata-eval 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. sub-neg 28.0%

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

      6. *-commutative 28.0%

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

      7. add-exp-log 28.0%

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

      8. expm1-def 28.0%

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

      9. log-pow 40.5%

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

      10. log1p-udef 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. associate-/r/ 97.3%

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

      2. *-commutative 97.3%

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

      3. *-un-lft-identity 97.3%

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

      4. times-frac 97.3%

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

      5. metadata-eval 97.3%

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

   8. Applied egg-rr 97.3%

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

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

   1. Initial program 99.9%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-def 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. *-commutative 1.8%

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

      3. fma-neg 1.8%

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

      4. metadata-eval 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 97.9%

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

Alternative 4: 97.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(n * N[(100.0 / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $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 28.0%

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

      1. associate-*r/ 28.0%

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

      2. sub-neg 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. metadata-eval 28.0%

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

      6. metadata-eval 28.0%

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

      7. fma-def 28.0%

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

      8. metadata-eval 28.0%

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

   3. Simplified 28.0%

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

      1. fma-udef 28.0%

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

      2. metadata-eval 28.0%

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

      3. distribute-lft-in 28.0%

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

      4. metadata-eval 28.0%

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

      5. sub-neg 28.0%

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

      6. *-commutative 28.0%

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

      7. add-exp-log 28.0%

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

      8. expm1-def 28.0%

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

      9. log-pow 40.5%

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

      10. log1p-udef 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. associate-/r/ 97.3%

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

      2. *-commutative 97.3%

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

      3. *-un-lft-identity 97.3%

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

      4. times-frac 97.3%

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

      5. metadata-eval 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. clear-num 97.3%

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

      2. un-div-inv 97.4%

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

   10. Applied egg-rr 97.4%

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

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

   1. Initial program 99.9%

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

   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/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-def 0.0%

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

      8. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. *-commutative 1.8%

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

      3. fma-neg 1.8%

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

      4. metadata-eval 1.8%

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

   7. Simplified 1.8%

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

   8. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 98.0%

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

Alternative 5: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -1.4 \cdot 10^{-206}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-200}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -1.4e-206)
     t_0
     (if (<= n 2.35e-200)
       (* (/ 0.0 i) (* n 100.0))
       (if (<= n 1.5e-21) (/ n (+ 0.01 (* i -0.005))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -1.4e-206) {
		tmp = t_0;
	} else if (n <= 2.35e-200) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 1.5e-21) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -1.4e-206) {
		tmp = t_0;
	} else if (n <= 2.35e-200) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 1.5e-21) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -1.4e-206:
		tmp = t_0
	elif n <= 2.35e-200:
		tmp = (0.0 / i) * (n * 100.0)
	elif n <= 1.5e-21:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -1.4e-206)
		tmp = t_0;
	elseif (n <= 2.35e-200)
		tmp = Float64(Float64(0.0 / i) * Float64(n * 100.0));
	elseif (n <= 1.5e-21)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.4e-206], t$95$0, If[LessEqual[n, 2.35e-200], N[(N[(0.0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5e-21], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.4000000000000001e-206 or 1.49999999999999996e-21 < n

   1. Initial program 24.8%

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

      1. associate-/r/ 24.7%

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

      2. sub-neg 24.7%

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

      3. metadata-eval 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in n around inf 33.7%

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

      1. associate-/l* 33.7%

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

      2. expm1-def 86.2%

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

   7. Simplified 86.2%

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

   if -1.4000000000000001e-206 < n < 2.35e-200

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/r/ 65.8%

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

      3. associate-*l* 65.8%

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

      4. sub-neg 65.8%

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

      5. metadata-eval 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in i around 0 85.5%

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

   if 2.35e-200 < n < 1.49999999999999996e-21

   1. Initial program 17.5%

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

      1. associate-*r/ 17.5%

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

      2. sub-neg 17.5%

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

      3. distribute-lft-in 17.5%

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

      4. metadata-eval 17.5%

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

      5. metadata-eval 17.5%

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

      6. metadata-eval 17.5%

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

      7. fma-def 17.5%

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

      8. metadata-eval 17.5%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in n around inf 3.0%

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

      1. associate-/l* 3.0%

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

      2. *-commutative 3.0%

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

      3. fma-neg 3.0%

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

      4. metadata-eval 3.0%

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

   7. Simplified 3.0%

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

   8. Taylor expanded in i around 0 55.7%

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

      1. *-commutative 55.7%

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

   10. Simplified 55.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-206}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 2.35 \cdot 10^{-200}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-38}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -8e-38)
   (* n (/ (* 100.0 (expm1 i)) i))
   (if (<= n 2.3e+30)
     (/ n (+ 0.01 (+ (* i -0.005) (* 0.0008333333333333334 (pow i 2.0)))))
     (* 100.0 (/ n (/ i (expm1 i)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -8e-38) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else if (n <= 2.3e+30) {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * pow(i, 2.0))));
	} else {
		tmp = 100.0 * (n / (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -8e-38) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else if (n <= 2.3e+30) {
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * Math.pow(i, 2.0))));
	} else {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -8e-38:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	elif n <= 2.3e+30:
		tmp = n / (0.01 + ((i * -0.005) + (0.0008333333333333334 * math.pow(i, 2.0))))
	else:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -8e-38)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	elseif (n <= 2.3e+30)
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * -0.005) + Float64(0.0008333333333333334 * (i ^ 2.0)))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -8e-38], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.3e+30], N[(n / N[(0.01 + N[(N[(i * -0.005), $MachinePrecision] + N[(0.0008333333333333334 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.9999999999999997e-38

   1. Initial program 25.6%

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

      1. associate-*r/ 25.6%

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

      2. sub-neg 25.6%

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

      3. distribute-lft-in 25.6%

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

      4. metadata-eval 25.6%

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

      5. metadata-eval 25.6%

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

      6. metadata-eval 25.6%

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

      7. fma-def 25.6%

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

      8. metadata-eval 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in n around inf 43.8%

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

      1. associate-/l* 43.8%

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

      2. *-commutative 43.8%

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

      3. fma-neg 43.8%

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

      4. metadata-eval 43.8%

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

   7. Simplified 43.8%

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

      1. clear-num 43.8%

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

      2. associate-/r/ 43.9%

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

      3. clear-num 43.9%

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

      4. fma-udef 43.9%

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

      5. metadata-eval 43.9%

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

      6. metadata-eval 43.9%

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

      7. distribute-rgt-in 43.9%

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

      8. sub-neg 43.9%

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

      9. expm1-def 88.2%

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

   9. Applied egg-rr 88.2%

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

   if -7.9999999999999997e-38 < n < 2.3e30

   1. Initial program 32.7%

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

      1. associate-*r/ 32.7%

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

      2. sub-neg 32.7%

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

      3. distribute-lft-in 32.7%

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

      4. metadata-eval 32.7%

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

      5. metadata-eval 32.7%

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

      6. metadata-eval 32.7%

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

      7. fma-def 32.7%

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

      8. metadata-eval 32.7%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in n around inf 20.1%

      \[\leadsto \color{blue}{\frac{n \cdot \left(100 \cdot e^{i} - 100\right)}{i}} \]
   6. 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%

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

   7. Simplified 20.1%

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

   8. Taylor expanded in i around 0 67.3%

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

   if 2.3e30 < n

   1. Initial program 21.6%

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

      1. associate-/r/ 22.4%

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

      2. sub-neg 22.4%

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

      3. metadata-eval 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in n around inf 33.1%

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

      1. associate-/l* 33.1%

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

      2. expm1-def 96.5%

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

   7. Simplified 96.5%

      \[\leadsto \color{blue}{100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-38}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot -0.005 + 0.0008333333333333334 \cdot {i}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ t_1 := \frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{if}\;n \leq -3 \cdot 10^{+209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))) (t_1 (/ n (+ 0.01 (* i -0.005)))))
   (if (<= n -3e+209)
     t_0
     (if (<= n -2.35e-206)
       t_1
       (if (<= n 3.8e-195)
         (* (/ 0.0 i) (* n 100.0))
         (if (<= n 1.7e-51) t_1 t_0))))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = n / (0.01 + (i * -0.005));
	double tmp;
	if (n <= -3e+209) {
		tmp = t_0;
	} else if (n <= -2.35e-206) {
		tmp = t_1;
	} else if (n <= 3.8e-195) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 1.7e-51) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    t_1 = n / (0.01d0 + (i * (-0.005d0)))
    if (n <= (-3d+209)) then
        tmp = t_0
    else if (n <= (-2.35d-206)) then
        tmp = t_1
    else if (n <= 3.8d-195) then
        tmp = (0.0d0 / i) * (n * 100.0d0)
    else if (n <= 1.7d-51) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double t_1 = n / (0.01 + (i * -0.005));
	double tmp;
	if (n <= -3e+209) {
		tmp = t_0;
	} else if (n <= -2.35e-206) {
		tmp = t_1;
	} else if (n <= 3.8e-195) {
		tmp = (0.0 / i) * (n * 100.0);
	} else if (n <= 1.7e-51) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	t_1 = n / (0.01 + (i * -0.005))
	tmp = 0
	if n <= -3e+209:
		tmp = t_0
	elif n <= -2.35e-206:
		tmp = t_1
	elif n <= 3.8e-195:
		tmp = (0.0 / i) * (n * 100.0)
	elif n <= 1.7e-51:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	t_1 = Float64(n / Float64(0.01 + Float64(i * -0.005)))
	tmp = 0.0
	if (n <= -3e+209)
		tmp = t_0;
	elseif (n <= -2.35e-206)
		tmp = t_1;
	elseif (n <= 3.8e-195)
		tmp = Float64(Float64(0.0 / i) * Float64(n * 100.0));
	elseif (n <= 1.7e-51)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	t_1 = n / (0.01 + (i * -0.005));
	tmp = 0.0;
	if (n <= -3e+209)
		tmp = t_0;
	elseif (n <= -2.35e-206)
		tmp = t_1;
	elseif (n <= 3.8e-195)
		tmp = (0.0 / i) * (n * 100.0);
	elseif (n <= 1.7e-51)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3e+209], t$95$0, If[LessEqual[n, -2.35e-206], t$95$1, If[LessEqual[n, 3.8e-195], N[(N[(0.0 / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.7e-51], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.99999999999999985e209 or 1.70000000000000001e-51 < n

   1. Initial program 18.5%

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

      1. associate-*r/ 18.4%

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

      2. sub-neg 18.4%

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

      3. distribute-lft-in 18.4%

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

      4. metadata-eval 18.4%

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

      5. metadata-eval 18.4%

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

      6. metadata-eval 18.4%

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

      7. fma-def 18.4%

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

      8. metadata-eval 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in n around inf 35.8%

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

      1. associate-/l* 35.8%

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

      2. *-commutative 35.8%

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

      3. fma-neg 35.8%

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

      4. metadata-eval 35.8%

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

   7. Simplified 35.8%

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

      1. clear-num 35.8%

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

      2. associate-/r/ 35.8%

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

      3. clear-num 35.8%

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

      4. fma-udef 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. distribute-rgt-in 35.8%

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

      8. sub-neg 35.8%

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

      9. expm1-def 92.1%

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

   9. Applied egg-rr 92.1%

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

   10. Taylor expanded in i around 0 74.6%

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

       1. *-commutative 74.6%

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

   12. Simplified 74.6%

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

   if -2.99999999999999985e209 < n < -2.3499999999999999e-206 or 3.80000000000000013e-195 < n < 1.70000000000000001e-51

   1. Initial program 27.9%

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

      1. associate-*r/ 27.9%

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

      2. sub-neg 27.9%

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

      3. distribute-lft-in 27.9%

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

      4. metadata-eval 27.9%

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

      5. metadata-eval 27.9%

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

      6. metadata-eval 27.9%

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

      7. fma-def 27.9%

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

      8. metadata-eval 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in n around inf 22.9%

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

      1. associate-/l* 22.9%

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

      2. *-commutative 22.9%

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

      3. fma-neg 22.9%

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

      4. metadata-eval 22.9%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in i around 0 59.9%

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

      1. *-commutative 59.9%

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

   10. Simplified 59.9%

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

   if -2.3499999999999999e-206 < n < 3.80000000000000013e-195

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/r/ 65.8%

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

      3. associate-*l* 65.8%

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

      4. sub-neg 65.8%

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

      5. metadata-eval 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in i around 0 85.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+209}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -2.35 \cdot 10^{-206}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{0}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.1 \cdot 10^{+208}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -6.1e+208)
   (* n (+ 100.0 (* i 50.0)))
   (if (<= n 1.35e-43)
     (/ n (+ 0.01 (* i -0.005)))
     (* 100.0 (+ n (* (* i n) (- 0.5 (/ 0.5 n))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -6.1e+208) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.35e-43) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 100.0 * (n + ((i * n) * (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 <= (-6.1d+208)) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else if (n <= 1.35d-43) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = 100.0d0 * (n + ((i * n) * (0.5d0 - (0.5d0 / n))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -6.1e+208) {
		tmp = n * (100.0 + (i * 50.0));
	} else if (n <= 1.35e-43) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -6.1e+208:
		tmp = n * (100.0 + (i * 50.0))
	elif n <= 1.35e-43:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -6.1e+208)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	elseif (n <= 1.35e-43)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(Float64(i * n) * Float64(0.5 - Float64(0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -6.1e+208)
		tmp = n * (100.0 + (i * 50.0));
	elseif (n <= 1.35e-43)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = 100.0 * (n + ((i * n) * (0.5 - (0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -6.1e+208], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.35e-43], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(N[(i * n), $MachinePrecision] * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.09999999999999975e208

   1. Initial program 14.6%

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

      1. associate-*r/ 14.5%

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

      2. sub-neg 14.5%

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

      3. distribute-lft-in 14.5%

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

      4. metadata-eval 14.5%

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

      5. metadata-eval 14.5%

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

      6. metadata-eval 14.5%

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

      7. fma-def 14.5%

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

      8. metadata-eval 14.5%

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

   3. Simplified 14.5%

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

   5. Taylor expanded in n around inf 59.8%

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

      1. associate-/l* 59.7%

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

      2. *-commutative 59.7%

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

      3. fma-neg 59.7%

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

      4. metadata-eval 59.7%

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

   7. Simplified 59.7%

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

      1. clear-num 59.7%

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

      2. associate-/r/ 59.8%

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

      3. clear-num 59.9%

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

      4. fma-udef 59.8%

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

      5. metadata-eval 59.8%

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

      6. metadata-eval 59.8%

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

      7. distribute-rgt-in 59.9%

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

      8. sub-neg 59.9%

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

      9. expm1-def 96.5%

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

   9. Applied egg-rr 96.5%

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

   10. Taylor expanded in i around 0 69.4%

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

       1. *-commutative 69.4%

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

   12. Simplified 69.4%

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

   if -6.09999999999999975e208 < n < 1.34999999999999996e-43

   1. Initial program 33.5%

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

      1. associate-*r/ 33.5%

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

      2. sub-neg 33.5%

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

      3. distribute-lft-in 33.5%

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

      4. metadata-eval 33.5%

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

      5. metadata-eval 33.5%

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

      6. metadata-eval 33.5%

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

      7. fma-def 33.5%

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

      8. metadata-eval 33.5%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in n around inf 27.6%

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

      1. associate-/l* 27.6%

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

      2. *-commutative 27.6%

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

      3. fma-neg 27.6%

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

      4. metadata-eval 27.6%

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

   7. Simplified 27.6%

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

   8. Taylor expanded in i around 0 58.7%

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

      1. *-commutative 58.7%

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

   10. Simplified 58.7%

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

   if 1.34999999999999996e-43 < n

   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/ 21.3%

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

      2. sub-neg 21.3%

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

      3. metadata-eval 21.3%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in i around 0 78.6%

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

      1. distribute-lft-out 78.6%

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

      2. associate-*r* 78.6%

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

      3. associate-*r/ 78.6%

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

      4. metadata-eval 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.1 \cdot 10^{+208}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(i \cdot n\right) \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+209} \lor \neg \left(n \leq 7 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2e+209) (not (<= n 7e-52)))
   (* n (+ 100.0 (* i 50.0)))
   (/ n (+ 0.01 (* i -0.005)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2e+209) || !(n <= 7e-52)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2d+209)) .or. (.not. (n <= 7d-52))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2e+209) || !(n <= 7e-52)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2e+209) or not (n <= 7e-52):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = n / (0.01 + (i * -0.005))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2e+209) || !(n <= 7e-52))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2e+209) || ~((n <= 7e-52)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = n / (0.01 + (i * -0.005));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2e+209], N[Not[LessEqual[n, 7e-52]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.0000000000000001e209 or 7.0000000000000001e-52 < n

   1. Initial program 18.5%

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

      1. associate-*r/ 18.4%

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

      2. sub-neg 18.4%

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

      3. distribute-lft-in 18.4%

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

      4. metadata-eval 18.4%

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

      5. metadata-eval 18.4%

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

      6. metadata-eval 18.4%

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

      7. fma-def 18.4%

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

      8. metadata-eval 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in n around inf 35.8%

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

      1. associate-/l* 35.8%

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

      2. *-commutative 35.8%

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

      3. fma-neg 35.8%

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

      4. metadata-eval 35.8%

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

   7. Simplified 35.8%

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

      1. clear-num 35.8%

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

      2. associate-/r/ 35.8%

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

      3. clear-num 35.8%

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

      4. fma-udef 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. distribute-rgt-in 35.8%

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

      8. sub-neg 35.8%

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

      9. expm1-def 92.1%

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

   9. Applied egg-rr 92.1%

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

   10. Taylor expanded in i around 0 74.6%

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

       1. *-commutative 74.6%

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

   12. Simplified 74.6%

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

   if -2.0000000000000001e209 < n < 7.0000000000000001e-52

   1. Initial program 34.1%

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

      1. associate-*r/ 34.1%

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

      2. sub-neg 34.1%

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

      3. distribute-lft-in 34.1%

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

      4. metadata-eval 34.1%

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

      5. metadata-eval 34.1%

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

      6. metadata-eval 34.1%

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

      7. fma-def 34.1%

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

      8. metadata-eval 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in n around inf 28.1%

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

      1. associate-/l* 28.1%

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

      2. *-commutative 28.1%

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

      3. fma-neg 28.1%

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

      4. metadata-eval 28.1%

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

   7. Simplified 28.1%

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

   8. Taylor expanded in i around 0 59.2%

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

      1. *-commutative 59.2%

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

   10. Simplified 59.2%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+209} \lor \neg \left(n \leq 7 \cdot 10^{-52}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.6e+64) (* n 100.0) (* n (* i 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.6e+64) {
		tmp = n * 100.0;
	} else {
		tmp = n * (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 (i <= 1.6d+64) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.6e+64) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.6e+64:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.6e+64)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.6e+64)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.6e+64], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.60000000000000009e64

   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%

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

      2. sub-neg 24.2%

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

      3. metadata-eval 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in i around 0 59.6%

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

      1. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   if 1.60000000000000009e64 < i

   1. Initial program 43.4%

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

      1. associate-/r/ 43.8%

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

      2. sub-neg 43.8%

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

      3. metadata-eval 43.8%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in i around 0 38.8%

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

      1. distribute-lft-out 38.8%

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

      2. associate-*r* 38.8%

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

      3. associate-*r/ 38.8%

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

      4. metadata-eval 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in n around inf 39.2%

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

      1. associate-*r* 39.2%

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

      2. *-commutative 39.2%

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

      3. *-commutative 39.2%

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

   10. Simplified 39.2%

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

   11. Taylor expanded in i around inf 39.2%

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

       1. *-commutative 39.2%

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

       2. *-commutative 39.2%

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

       3. associate-*l* 39.2%

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

   13. Simplified 39.2%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.6 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.7% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ n \cdot \left(100 + i \cdot 50\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n (+ 100.0 (* i 50.0))))

C

double code(double i, double n) {
	return n * (100.0 + (i * 50.0));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * (100.0d0 + (i * 50.0d0))
end function

Java

public static double code(double i, double n) {
	return n * (100.0 + (i * 50.0));
}

Python

def code(i, n):
	return n * (100.0 + (i * 50.0))

Julia

function code(i, n)
	return Float64(n * Float64(100.0 + Float64(i * 50.0)))
end

MATLAB

function tmp = code(i, n)
	tmp = n * (100.0 + (i * 50.0));
end

Wolfram

code[i_, n_] := N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

   1. associate-*r/ 27.7%

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

   2. sub-neg 27.7%

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

   3. distribute-lft-in 27.7%

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

   4. metadata-eval 27.7%

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

   5. metadata-eval 27.7%

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

   6. metadata-eval 27.7%

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

   7. fma-def 27.7%

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

   8. metadata-eval 27.7%

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

3. Simplified 27.7%

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

5. Taylor expanded in n around inf 31.3%

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

   1. associate-/l* 31.3%

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

   2. *-commutative 31.3%

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

   3. fma-neg 31.3%

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

   4. metadata-eval 31.3%

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

7. Simplified 31.3%

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

   1. clear-num 31.5%

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

   2. associate-/r/ 31.3%

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

   3. clear-num 31.3%

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

   4. fma-udef 31.3%

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

   5. metadata-eval 31.3%

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

   6. metadata-eval 31.3%

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

   7. distribute-rgt-in 31.3%

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

   8. sub-neg 31.3%

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

   9. expm1-def 74.4%

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

9. Applied egg-rr 74.4%

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

10. Taylor expanded in i around 0 55.5%

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

    1. *-commutative 55.5%

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

12. Simplified 55.5%

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

13. Final simplification 55.5%

    \[\leadsto n \cdot \left(100 + i \cdot 50\right) \]
14. Add Preprocessing

Alternative 12: 2.8% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

   1. associate-/r/ 27.7%

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

   2. sub-neg 27.7%

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

   3. metadata-eval 27.7%

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

3. Simplified 27.7%

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

5. Taylor expanded in i around 0 55.2%

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

   1. distribute-lft-out 55.2%

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

   2. associate-*r* 55.4%

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

   3. associate-*r/ 55.4%

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

   4. metadata-eval 55.4%

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

7. Simplified 55.4%

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

8. Taylor expanded in n around 0 2.7%

   \[\leadsto \color{blue}{-50 \cdot i} \]
9. Step-by-step derivation

   1. *-commutative 2.7%

      \[\leadsto \color{blue}{i \cdot -50} \]

10. Simplified 2.7%

    \[\leadsto \color{blue}{i \cdot -50} \]

11. Final simplification 2.7%

    \[\leadsto i \cdot -50 \]
12. Add Preprocessing

Alternative 13: 49.3% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ n \cdot 100 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n 100.0))

C

double code(double i, double n) {
	return n * 100.0;
}

Fortran

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

Java

public static double code(double i, double n) {
	return n * 100.0;
}

Python

def code(i, n):
	return n * 100.0

Julia

function code(i, n)
	return Float64(n * 100.0)
end

MATLAB

function tmp = code(i, n)
	tmp = n * 100.0;
end

Wolfram

code[i_, n_] := N[(n * 100.0), $MachinePrecision]

Derivation

1. Initial program 27.7%

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

   1. associate-/r/ 27.7%

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

   2. sub-neg 27.7%

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

   3. metadata-eval 27.7%

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

3. Simplified 27.7%

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

5. Taylor expanded in i around 0 49.8%

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

   1. *-commutative 49.8%

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

7. Simplified 49.8%

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

8. Final simplification 49.8%

   \[\leadsto n \cdot 100 \]
9. Add Preprocessing

Developer target: 33.9% 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 2024027 
(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))))