Compound Interest

Percentage Accurate: 28.0% → 94.5%

Time: 18.1s

Alternatives: 16

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: 28.0% 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: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) INFINITY)
   (/ (* (expm1 (* n (log1p (/ i n)))) 100.0) (/ i n))
   (* 100.0 (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333)))))))

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= ((double) INFINITY)) {
		tmp = (expm1((n * log1p((i / n)))) * 100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) <= Inf)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 31.9%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. pow-to-exp 29.1%

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

      4. expm1-def 37.9%

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

      5. add-log-exp 29.1%

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

      6. pow-to-exp 31.9%

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

      7. log-pow 37.9%

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

      8. log1p-udef 96.8%

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

   3. Applied egg-rr 96.8%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 80.3%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]

      2. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      3. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      6. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      7. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      8. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 2: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n)) INFINITY)
   (* n (/ (expm1 (* n (log1p (/ i n)))) (/ i 100.0)))
   (* 100.0 (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333)))))))

C

double code(double i, double n) {
	double tmp;
	if (((pow((1.0 + (i / n)), n) + -1.0) / (i / n)) <= ((double) INFINITY)) {
		tmp = n * (expm1((n * log1p((i / n)))) / (i / 100.0));
	} else {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n)) <= Inf)
		tmp = Float64(n * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / 100.0)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], Infinity], N[(n * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 31.9%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. pow-to-exp 29.1%

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

      4. expm1-def 37.9%

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

      5. add-log-exp 29.1%

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

      6. pow-to-exp 31.9%

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

      7. log-pow 37.9%

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

      8. log1p-udef 96.8%

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

   3. Applied egg-rr 96.8%

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

      1. associate-/r/ 96.1%

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

   5. Applied egg-rr 96.1%

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

      1. *-commutative 96.1%

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

      2. associate-/l* 96.1%

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

   7. Simplified 96.1%

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

   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. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 80.3%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in i around 0 100.0%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]

      2. associate-+r+ 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      3. +-commutative 100.0%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. *-commutative 100.0%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. fma-def 100.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      6. *-commutative 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      7. unpow2 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      8. associate-*l* 100.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 100.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \end{array} \]

Alternative 3: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -3.4 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.72 \cdot 10^{-267}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-67}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -3.4e-218)
     t_0
     (if (<= n 1.72e-267)
       (* 100.0 (/ 0.0 (/ i n)))
       (if (<= n 3e-67)
         (* 100.0 (/ n (+ (fma i -0.5 1.0) (* i (* i 0.08333333333333333)))))
         t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -3.4e-218) {
		tmp = t_0;
	} else if (n <= 1.72e-267) {
		tmp = 100.0 * (0.0 / (i / n));
	} else if (n <= 3e-67) {
		tmp = 100.0 * (n / (fma(i, -0.5, 1.0) + (i * (i * 0.08333333333333333))));
	} 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 <= -3.4e-218)
		tmp = t_0;
	elseif (n <= 1.72e-267)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	elseif (n <= 3e-67)
		tmp = Float64(100.0 * Float64(n / Float64(fma(i, -0.5, 1.0) + Float64(i * Float64(i * 0.08333333333333333)))));
	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, -3.4e-218], t$95$0, If[LessEqual[n, 1.72e-267], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3e-67], N[(100.0 * N[(n / N[(N[(i * -0.5 + 1.0), $MachinePrecision] + N[(i * N[(i * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.39999999999999986e-218 or 3.00000000000000032e-67 < n

   1. Initial program 20.8%

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

   2. Taylor expanded in n around inf 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-/l* 35.8%

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

      3. expm1-def 90.5%

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

   4. Simplified 90.5%

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

   if -3.39999999999999986e-218 < n < 1.72e-267

   1. Initial program 94.4%

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

   2. Taylor expanded in i around 0 100.0%

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

   if 1.72e-267 < n < 3.00000000000000032e-67

   1. Initial program 22.9%

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

   2. Taylor expanded in n around inf 6.6%

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

      1. *-commutative 6.6%

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

      2. associate-/l* 6.6%

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

      3. expm1-def 32.3%

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

   4. Simplified 32.3%

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

   5. Taylor expanded in i around 0 70.7%

      \[\leadsto \frac{n}{\color{blue}{1 + \left(0.08333333333333333 \cdot {i}^{2} + -0.5 \cdot i\right)}} \cdot 100 \]
   6. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \frac{n}{1 + \color{blue}{\left(-0.5 \cdot i + 0.08333333333333333 \cdot {i}^{2}\right)}} \cdot 100 \]

      2. associate-+r+ 70.7%

         \[\leadsto \frac{n}{\color{blue}{\left(1 + -0.5 \cdot i\right) + 0.08333333333333333 \cdot {i}^{2}}} \cdot 100 \]

      3. +-commutative 70.7%

         \[\leadsto \frac{n}{\color{blue}{\left(-0.5 \cdot i + 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      4. *-commutative 70.7%

         \[\leadsto \frac{n}{\left(\color{blue}{i \cdot -0.5} + 1\right) + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      5. fma-def 70.7%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right)} + 0.08333333333333333 \cdot {i}^{2}} \cdot 100 \]

      6. *-commutative 70.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{{i}^{2} \cdot 0.08333333333333333}} \cdot 100 \]

      7. unpow2 70.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{\left(i \cdot i\right)} \cdot 0.08333333333333333} \cdot 100 \]

      8. associate-*l* 70.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + \color{blue}{i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]

   7. Simplified 70.7%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}} \cdot 100 \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-218}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 1.72 \cdot 10^{-267}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-67}:\\ \;\;\;\;100 \cdot \frac{n}{\mathsf{fma}\left(i, -0.5, 1\right) + i \cdot \left(i \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 4: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;i \leq 2.3 \cdot 10^{+256}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -2.2e-8)
     t_0
     (if (<= i 6.8e-33)
       (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n))))
       (if (<= i 2.3e+256) t_0 (* 100.0 (/ n (+ 1.0 (* i -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -2.2e-8) {
		tmp = t_0;
	} else if (i <= 6.8e-33) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (i <= 2.3e+256) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -2.2e-8) {
		tmp = t_0;
	} else if (i <= 6.8e-33) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (i <= 2.3e+256) {
		tmp = t_0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -2.2e-8:
		tmp = t_0
	elif i <= 6.8e-33:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	elif i <= 2.3e+256:
		tmp = t_0
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -2.2e-8)
		tmp = t_0;
	elseif (i <= 6.8e-33)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * n))));
	elseif (i <= 2.3e+256)
		tmp = t_0;
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2.2e-8], t$95$0, If[LessEqual[i, 6.8e-33], N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.3e+256], t$95$0, N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.1999999999999998e-8 or 6.8000000000000001e-33 < i < 2.2999999999999999e256

   1. Initial program 50.4%

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

   2. Taylor expanded in n around inf 72.1%

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

      1. expm1-def 75.2%

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

   4. Simplified 75.2%

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

   if -2.1999999999999998e-8 < i < 6.8000000000000001e-33

   1. Initial program 8.6%

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

   2. Taylor expanded in i around 0 85.6%

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

      1. associate-*r* 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-*r/ 85.6%

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

      4. metadata-eval 85.6%

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

   4. Simplified 85.6%

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

   if 2.2999999999999999e256 < i

   1. Initial program 28.6%

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

   2. Taylor expanded in n around inf 15.5%

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

      1. *-commutative 15.5%

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

      2. associate-/l* 15.5%

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

      3. expm1-def 15.5%

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

   4. Simplified 15.5%

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

   5. Taylor expanded in i around 0 74.3%

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

      1. *-commutative 74.3%

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

   7. Simplified 74.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 6.8 \cdot 10^{-33}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;i \leq 2.3 \cdot 10^{+256}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-221} \lor \neg \left(n \leq 4.2 \cdot 10^{-131}\right):\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-221) (not (<= n 4.2e-131)))
   (* n (/ (expm1 i) (/ i 100.0)))
   (* 100.0 (/ 0.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-221) || !(n <= 4.2e-131)) {
		tmp = n * (expm1(i) / (i / 100.0));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-221) || !(n <= 4.2e-131)) {
		tmp = n * (Math.expm1(i) / (i / 100.0));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-221) or not (n <= 4.2e-131):
		tmp = n * (math.expm1(i) / (i / 100.0))
	else:
		tmp = 100.0 * (0.0 / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-221) || !(n <= 4.2e-131))
		tmp = Float64(n * Float64(expm1(i) / Float64(i / 100.0)));
	else
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-221], N[Not[LessEqual[n, 4.2e-131]], $MachinePrecision]], N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.99999999999999996e-221 or 4.19999999999999994e-131 < 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/ 20.7%

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

      2. *-commutative 20.7%

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

      3. pow-to-exp 17.9%

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

      4. expm1-def 22.8%

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

      5. add-log-exp 17.9%

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

      6. pow-to-exp 20.7%

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

      7. log-pow 22.8%

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

      8. log1p-udef 79.2%

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

   3. Applied egg-rr 79.2%

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

      1. associate-/r/ 79.1%

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

   5. Applied egg-rr 79.1%

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

      1. *-commutative 79.1%

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

      2. associate-/l* 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in n around inf 88.4%

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

   if -4.99999999999999996e-221 < n < 4.19999999999999994e-131

   1. Initial program 55.9%

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

   2. Taylor expanded in i around 0 72.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-221} \lor \neg \left(n \leq 4.2 \cdot 10^{-131}\right):\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 6: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-217} \lor \neg \left(n \leq 2.4 \cdot 10^{-136}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.9e-217) (not (<= n 2.4e-136)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (* 100.0 (/ 0.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e-217) || !(n <= 2.4e-136)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e-217) || !(n <= 2.4e-136)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.9e-217) or not (n <= 2.4e-136):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 100.0 * (0.0 / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.9e-217) || !(n <= 2.4e-136))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.9e-217], N[Not[LessEqual[n, 2.4e-136]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.89999999999999993e-217 or 2.3999999999999999e-136 < n

   1. Initial program 20.7%

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

   2. Taylor expanded in n around inf 33.8%

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

      1. *-commutative 33.8%

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

      2. associate-/l* 33.8%

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

      3. expm1-def 88.5%

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

   4. Simplified 88.5%

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

   if -1.89999999999999993e-217 < n < 2.3999999999999999e-136

   1. Initial program 55.9%

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

   2. Taylor expanded in i around 0 72.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-217} \lor \neg \left(n \leq 2.4 \cdot 10^{-136}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 7: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-217}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-132}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.1e-217)
   (* n (/ (* 100.0 (expm1 i)) i))
   (if (<= n 3.1e-132)
     (* 100.0 (/ 0.0 (/ i n)))
     (* n (/ (expm1 i) (/ i 100.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-217) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else if (n <= 3.1e-132) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (expm1(i) / (i / 100.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.1e-217) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else if (n <= 3.1e-132) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (Math.expm1(i) / (i / 100.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.1e-217:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	elif n <= 3.1e-132:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = n * (math.expm1(i) / (i / 100.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.1e-217)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	elseif (n <= 3.1e-132)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(n * Float64(expm1(i) / Float64(i / 100.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.1e-217], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.1e-132], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.09999999999999991e-217

   1. Initial program 19.3%

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

      1. associate-/r/ 19.6%

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

      2. associate-*r* 19.6%

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

      3. *-commutative 19.6%

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

      4. associate-*r/ 19.7%

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

      5. sub-neg 19.7%

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

      6. distribute-lft-in 19.6%

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

      7. fma-def 19.7%

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

      8. metadata-eval 19.7%

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

      9. metadata-eval 19.7%

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

   3. Simplified 19.7%

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

      1. fma-udef 19.6%

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

      2. *-commutative 19.6%

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

   5. Applied egg-rr 19.6%

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

   6. Taylor expanded in n around inf 36.5%

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

      1. sub-neg 36.5%

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

      2. metadata-eval 36.5%

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

      3. metadata-eval 36.5%

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

      4. distribute-lft-in 36.4%

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

      5. metadata-eval 36.4%

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

      6. sub-neg 36.4%

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

      7. expm1-def 86.5%

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

   8. Simplified 86.5%

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

   if -1.09999999999999991e-217 < n < 3.10000000000000008e-132

   1. Initial program 55.9%

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

   2. Taylor expanded in i around 0 72.9%

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

   if 3.10000000000000008e-132 < n

   1. Initial program 22.2%

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

      1. associate-*r/ 22.2%

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

      2. *-commutative 22.2%

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

      3. pow-to-exp 19.2%

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

      4. expm1-def 22.0%

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

      5. add-log-exp 19.2%

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

      6. pow-to-exp 22.2%

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

      7. log-pow 22.0%

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

      8. log1p-udef 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. associate-/r/ 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-/l* 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in n around inf 90.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-217}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-132}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{100}}\\ \end{array} \]

Alternative 8: 63.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-219} \lor \neg \left(n \leq 4.1 \cdot 10^{-132}\right):\\ \;\;\;\;n \cdot \left(100 + \left(16.666666666666668 \cdot \left(i \cdot i\right) + i \cdot 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6.5e-219) (not (<= n 4.1e-132)))
   (* n (+ 100.0 (+ (* 16.666666666666668 (* i i)) (* i 50.0))))
   (* 100.0 (/ 0.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6.5e-219) || !(n <= 4.1e-132)) {
		tmp = n * (100.0 + ((16.666666666666668 * (i * i)) + (i * 50.0)));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-6.5d-219)) .or. (.not. (n <= 4.1d-132))) then
        tmp = n * (100.0d0 + ((16.666666666666668d0 * (i * i)) + (i * 50.0d0)))
    else
        tmp = 100.0d0 * (0.0d0 / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -6.5e-219) || !(n <= 4.1e-132)) {
		tmp = n * (100.0 + ((16.666666666666668 * (i * i)) + (i * 50.0)));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -6.5e-219) or not (n <= 4.1e-132):
		tmp = n * (100.0 + ((16.666666666666668 * (i * i)) + (i * 50.0)))
	else:
		tmp = 100.0 * (0.0 / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6.5e-219) || !(n <= 4.1e-132))
		tmp = Float64(n * Float64(100.0 + Float64(Float64(16.666666666666668 * Float64(i * i)) + Float64(i * 50.0))));
	else
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -6.5e-219) || ~((n <= 4.1e-132)))
		tmp = n * (100.0 + ((16.666666666666668 * (i * i)) + (i * 50.0)));
	else
		tmp = 100.0 * (0.0 / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6.5e-219], N[Not[LessEqual[n, 4.1e-132]], $MachinePrecision]], N[(n * N[(100.0 + N[(N[(16.666666666666668 * N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.49999999999999958e-219 or 4.10000000000000007e-132 < 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/ 21.0%

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

      2. associate-*r* 21.0%

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

      3. *-commutative 21.0%

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

      4. associate-*r/ 21.0%

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

      5. sub-neg 21.0%

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

      6. distribute-lft-in 21.0%

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

      7. fma-def 21.0%

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

      8. metadata-eval 21.0%

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

      9. metadata-eval 21.0%

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

   3. Simplified 21.0%

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

   4. Taylor expanded in i around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. distribute-lft-out 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in n around inf 71.6%

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

      1. distribute-lft-in 71.6%

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

      2. associate-*r* 71.6%

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

      3. metadata-eval 71.6%

         \[\leadsto n \cdot \left(100 + \left(\color{blue}{16.666666666666668} \cdot {i}^{2} + 100 \cdot \left(0.5 \cdot i\right)\right)\right) \]

      4. unpow2 71.6%

         \[\leadsto n \cdot \left(100 + \left(16.666666666666668 \cdot \color{blue}{\left(i \cdot i\right)} + 100 \cdot \left(0.5 \cdot i\right)\right)\right) \]

      5. associate-*r* 71.6%

         \[\leadsto n \cdot \left(100 + \left(16.666666666666668 \cdot \left(i \cdot i\right) + \color{blue}{\left(100 \cdot 0.5\right) \cdot i}\right)\right) \]

      6. metadata-eval 71.6%

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

   9. Simplified 71.6%

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

   if -6.49999999999999958e-219 < n < 4.10000000000000007e-132

   1. Initial program 55.9%

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

   2. Taylor expanded in i around 0 72.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-219} \lor \neg \left(n \leq 4.1 \cdot 10^{-132}\right):\\ \;\;\;\;n \cdot \left(100 + \left(16.666666666666668 \cdot \left(i \cdot i\right) + i \cdot 50\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 9: 58.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 3200:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+255}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ i (/ i n)))))
   (if (<= i -5e+36)
     t_0
     (if (<= i 3200.0) (* n 100.0) (if (<= i 7e+255) (* 50.0 (* i n)) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (i <= -5e+36) {
		tmp = t_0;
	} else if (i <= 3200.0) {
		tmp = n * 100.0;
	} else if (i <= 7e+255) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 * (i / (i / n))
    if (i <= (-5d+36)) then
        tmp = t_0
    else if (i <= 3200.0d0) then
        tmp = n * 100.0d0
    else if (i <= 7d+255) then
        tmp = 50.0d0 * (i * n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (i / (i / n));
	double tmp;
	if (i <= -5e+36) {
		tmp = t_0;
	} else if (i <= 3200.0) {
		tmp = n * 100.0;
	} else if (i <= 7e+255) {
		tmp = 50.0 * (i * n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (i / (i / n))
	tmp = 0
	if i <= -5e+36:
		tmp = t_0
	elif i <= 3200.0:
		tmp = n * 100.0
	elif i <= 7e+255:
		tmp = 50.0 * (i * n)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(i / Float64(i / n)))
	tmp = 0.0
	if (i <= -5e+36)
		tmp = t_0;
	elseif (i <= 3200.0)
		tmp = Float64(n * 100.0);
	elseif (i <= 7e+255)
		tmp = Float64(50.0 * Float64(i * n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 * (i / (i / n));
	tmp = 0.0;
	if (i <= -5e+36)
		tmp = t_0;
	elseif (i <= 3200.0)
		tmp = n * 100.0;
	elseif (i <= 7e+255)
		tmp = 50.0 * (i * n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5e+36], t$95$0, If[LessEqual[i, 3200.0], N[(n * 100.0), $MachinePrecision], If[LessEqual[i, 7e+255], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.99999999999999977e36 or 6.99999999999999971e255 < i

   1. Initial program 57.6%

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

   2. Taylor expanded in i around 0 33.8%

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

   if -4.99999999999999977e36 < i < 3200

   1. Initial program 9.3%

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

   2. Taylor expanded in i around 0 80.6%

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

      1. *-commutative 80.6%

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

   4. Simplified 80.6%

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

   if 3200 < i < 6.99999999999999971e255

   1. Initial program 51.7%

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

   2. Taylor expanded in i around 0 35.3%

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

      1. associate-*r* 35.3%

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

      2. *-commutative 35.3%

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

      3. associate-*r/ 35.3%

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

      4. metadata-eval 35.3%

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

   4. Simplified 35.3%

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

   5. Taylor expanded in n around inf 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-*r* 35.5%

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

   7. Simplified 35.5%

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

   8. Taylor expanded in i around inf 35.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+36}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 3200:\\ \;\;\;\;n \cdot 100\\ \mathbf{elif}\;i \leq 7 \cdot 10^{+255}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 10: 64.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+271}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -2.4 \cdot 10^{-220}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4e+271)
   (/ (* 100.0 (* i n)) i)
   (if (<= n -2.4e-220)
     (* 100.0 (/ n (+ 1.0 (* i -0.5))))
     (if (<= n 1.35e-134)
       (* 100.0 (/ 0.0 (/ i n)))
       (* n (+ 100.0 (* i 50.0)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4e+271) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -2.4e-220) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.35e-134) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4d+271)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= (-2.4d-220)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1.35d-134) then
        tmp = 100.0d0 * (0.0d0 / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4e+271) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= -2.4e-220) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.35e-134) {
		tmp = 100.0 * (0.0 / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4e+271:
		tmp = (100.0 * (i * n)) / i
	elif n <= -2.4e-220:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.35e-134:
		tmp = 100.0 * (0.0 / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4e+271)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= -2.4e-220)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.35e-134)
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4e+271)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= -2.4e-220)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1.35e-134)
		tmp = 100.0 * (0.0 / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4e+271], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, -2.4e-220], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.35e-134], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.99999999999999981e271

   1. Initial program 9.9%

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

      1. associate-*r/ 9.9%

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

      2. *-commutative 9.9%

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

      3. pow-to-exp 9.9%

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

      4. expm1-def 9.9%

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

      5. add-log-exp 9.9%

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

      6. pow-to-exp 9.9%

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

      7. log-pow 9.9%

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

      8. log1p-udef 91.5%

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

   3. Applied egg-rr 91.5%

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

   4. Taylor expanded in n around inf 91.7%

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

      1. associate-*r/ 83.9%

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

      2. expm1-def 92.1%

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

   6. Simplified 92.1%

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

   7. Taylor expanded in i around 0 52.5%

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

   if -3.99999999999999981e271 < n < -2.4000000000000001e-220

   1. Initial program 20.4%

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

   2. Taylor expanded in n around inf 29.8%

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

      1. *-commutative 29.8%

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

      2. associate-/l* 29.8%

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

      3. expm1-def 85.0%

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

   4. Simplified 85.0%

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

   5. Taylor expanded in i around 0 65.2%

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

      1. *-commutative 65.2%

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

   7. Simplified 65.2%

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

   if -2.4000000000000001e-220 < n < 1.3499999999999999e-134

   1. Initial program 55.9%

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

   2. Taylor expanded in i around 0 72.9%

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

   if 1.3499999999999999e-134 < n

   1. Initial program 22.2%

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

   2. Taylor expanded in i around 0 74.0%

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

      1. associate-*r* 74.0%

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

      2. *-commutative 74.0%

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

      3. associate-*r/ 74.0%

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

      4. metadata-eval 74.0%

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

   4. Simplified 74.0%

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

   5. Taylor expanded in n around inf 74.2%

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

      1. *-commutative 74.2%

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

      2. associate-*r* 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in n around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. *-commutative 74.2%

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

      3. distribute-lft-in 74.2%

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

      4. *-rgt-identity 74.2%

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

      5. distribute-rgt-in 74.2%

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

      6. +-commutative 74.2%

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

      7. associate-*l* 74.2%

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

      8. distribute-lft-out 74.2%

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

      9. associate-*l* 74.2%

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

      10. metadata-eval 74.2%

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

   10. Simplified 74.2%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+271}:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq -2.4 \cdot 10^{-220}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-134}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 11: 61.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{+52} \lor \neg \left(n \leq 2.92 \cdot 10^{-67}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.7e+52) (not (<= n 2.92e-67)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.7e+52) || !(n <= 2.92e-67)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.7d+52)) .or. (.not. (n <= 2.92d-67))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.7e+52) || !(n <= 2.92e-67)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.7e+52) or not (n <= 2.92e-67):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.7e+52) || !(n <= 2.92e-67))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.7e+52) || ~((n <= 2.92e-67)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.7e+52], N[Not[LessEqual[n, 2.92e-67]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.7e52 or 2.92000000000000004e-67 < n

   1. Initial program 19.7%

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

   2. Taylor expanded in i around 0 68.1%

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

      1. associate-*r* 68.1%

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

      2. *-commutative 68.1%

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

      3. associate-*r/ 68.1%

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

      4. metadata-eval 68.1%

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

   4. Simplified 68.1%

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

   5. Taylor expanded in n around inf 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*r* 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in n around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. *-commutative 68.2%

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

      3. distribute-lft-in 68.2%

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

      4. *-rgt-identity 68.2%

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

      5. distribute-rgt-in 68.2%

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

      6. +-commutative 68.2%

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

      7. associate-*l* 68.2%

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

      8. distribute-lft-out 68.2%

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

      9. associate-*l* 68.2%

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

      10. metadata-eval 68.2%

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

   10. Simplified 68.2%

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

   if -1.7e52 < n < 2.92000000000000004e-67

   1. Initial program 35.9%

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

   2. Taylor expanded in i around 0 60.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.7 \cdot 10^{+52} \lor \neg \left(n \leq 2.92 \cdot 10^{-67}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 12: 61.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-216} \lor \neg \left(n \leq 1.35 \cdot 10^{-134}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.3e-216) (not (<= n 1.35e-134)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ 0.0 (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-216) || !(n <= 1.35e-134)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.3d-216)) .or. (.not. (n <= 1.35d-134))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (0.0d0 / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e-216) || !(n <= 1.35e-134)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (0.0 / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.3e-216) or not (n <= 1.35e-134):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (0.0 / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.3e-216) || !(n <= 1.35e-134))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(0.0 / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.3e-216) || ~((n <= 1.35e-134)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (0.0 / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.3e-216], N[Not[LessEqual[n, 1.35e-134]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2999999999999999e-216 or 1.3499999999999999e-134 < n

   1. Initial program 20.7%

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

   2. Taylor expanded in i around 0 66.3%

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

      1. associate-*r* 66.4%

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

      2. *-commutative 66.4%

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

      3. associate-*r/ 66.4%

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

      4. metadata-eval 66.4%

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

   4. Simplified 66.4%

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

   5. Taylor expanded in n around inf 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*r* 66.3%

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

   7. Simplified 66.3%

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

   8. Taylor expanded in n around 0 66.3%

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

      1. +-commutative 66.3%

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

      2. *-commutative 66.3%

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

      3. distribute-lft-in 66.3%

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

      4. *-rgt-identity 66.3%

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

      5. distribute-rgt-in 66.3%

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

      6. +-commutative 66.3%

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

      7. associate-*l* 66.3%

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

      8. distribute-lft-out 66.3%

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

      9. associate-*l* 66.3%

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

      10. metadata-eval 66.3%

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

   10. Simplified 66.3%

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

   if -1.2999999999999999e-216 < n < 1.3499999999999999e-134

   1. Initial program 55.9%

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

   2. Taylor expanded in i around 0 72.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{1} - 1}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{-216} \lor \neg \left(n \leq 1.35 \cdot 10^{-134}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{0}{\frac{i}{n}}\\ \end{array} \]

Alternative 13: 62.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -400000:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-67}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -400000.0)
   (/ (* 100.0 (* i n)) i)
   (if (<= n 3e-67) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -400000.0) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 3e-67) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-400000.0d0)) then
        tmp = (100.0d0 * (i * n)) / i
    else if (n <= 3d-67) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -400000.0) {
		tmp = (100.0 * (i * n)) / i;
	} else if (n <= 3e-67) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -400000.0:
		tmp = (100.0 * (i * n)) / i
	elif n <= 3e-67:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -400000.0)
		tmp = Float64(Float64(100.0 * Float64(i * n)) / i);
	elseif (n <= 3e-67)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -400000.0)
		tmp = (100.0 * (i * n)) / i;
	elseif (n <= 3e-67)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -400000.0], N[(N[(100.0 * N[(i * n), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 3e-67], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4e5

   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. *-commutative 17.5%

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

      3. pow-to-exp 13.5%

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

      4. expm1-def 13.5%

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

      5. add-log-exp 13.5%

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

      6. pow-to-exp 17.5%

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

      7. log-pow 13.5%

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

      8. log1p-udef 72.7%

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

   3. Applied egg-rr 72.7%

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

   4. Taylor expanded in n around inf 48.1%

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

      1. associate-*r/ 46.8%

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

      2. expm1-def 94.3%

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

   6. Simplified 94.3%

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

   7. Taylor expanded in i around 0 57.7%

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

   if -4e5 < n < 3.00000000000000032e-67

   1. Initial program 36.7%

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

   2. Taylor expanded in i around 0 60.1%

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

   if 3.00000000000000032e-67 < n

   1. Initial program 22.8%

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

   2. Taylor expanded in i around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. *-commutative 76.6%

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

      3. associate-*r/ 76.6%

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

      4. metadata-eval 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in n around inf 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*r* 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in n around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. *-commutative 76.7%

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

      3. distribute-lft-in 76.7%

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

      4. *-rgt-identity 76.7%

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

      5. distribute-rgt-in 76.7%

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

      6. +-commutative 76.7%

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

      7. associate-*l* 76.7%

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

      8. distribute-lft-out 76.7%

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

      9. associate-*l* 76.7%

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

      10. metadata-eval 76.7%

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

   10. Simplified 76.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -400000:\\ \;\;\;\;\frac{100 \cdot \left(i \cdot n\right)}{i}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-67}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 14: 54.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 3100:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 3100.0) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 3100.0) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 3100.0d0) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 3100.0) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 3100.0:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 3100.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 3100.0)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 3100.0], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 3100

   1. Initial program 20.3%

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

   2. Taylor expanded in i around 0 64.9%

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

      1. *-commutative 64.9%

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

   4. Simplified 64.9%

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

   if 3100 < i

   1. Initial program 48.7%

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

   2. Taylor expanded in i around 0 31.1%

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

      1. associate-*r* 31.2%

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

      2. *-commutative 31.2%

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

      3. associate-*r/ 31.2%

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

      4. metadata-eval 31.2%

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

   4. Simplified 31.2%

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

   5. Taylor expanded in n around inf 31.5%

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

      1. *-commutative 31.5%

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

      2. associate-*r* 31.5%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in i around inf 31.5%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3100:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 15: 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 26.3%

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

2. Taylor expanded in i around 0 57.6%

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

   1. associate-*r* 57.7%

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

   2. *-commutative 57.7%

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

   3. associate-*r/ 57.7%

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

   4. metadata-eval 57.7%

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

4. Simplified 57.7%

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

5. Taylor expanded in n around 0 2.8%

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

   1. *-commutative 2.8%

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

7. Simplified 2.8%

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

8. Final simplification 2.8%

   \[\leadsto i \cdot -50 \]

Alternative 16: 49.1% 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 26.3%

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

2. Taylor expanded in i around 0 52.3%

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

   1. *-commutative 52.3%

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

4. Simplified 52.3%

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

5. Final simplification 52.3%

   \[\leadsto n \cdot 100 \]

Developer target: 34.2% 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 2023195 
(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))))