Compound Interest

Percentage Accurate: 28.1% → 97.3%

Time: 21.9s

Alternatives: 14

Speedup: 16.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* (/ (expm1 (* n (log1p (/ i n)))) i) (* n 100.0))
     (if (<= t_0 INFINITY)
       (* t_0 100.0)
       (/ 1.0 (/ (+ 1.0 (* i -0.5)) (* n 100.0)))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (expm1((n * log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.expm1((n * Math.log1p((i / n)))) / i) * (n * 100.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.expm1((n * math.log1p((i / n)))) / i) * (n * 100.0)
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i) * Float64(n * 100.0));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / Float64(n * 100.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * N[(n * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(1.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 27.7%

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

      1. associate-*r/ 27.7%

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

      2. sub-neg 27.7%

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

      3. distribute-lft-in 27.7%

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

      4. metadata-eval 27.7%

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

      5. metadata-eval 27.7%

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

   3. Simplified 27.7%

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

      1. metadata-eval 27.7%

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

      2. metadata-eval 27.7%

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

      3. distribute-lft-in 27.7%

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

      4. sub-neg 27.7%

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

      5. *-commutative 27.7%

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

      6. associate-*l/ 27.7%

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

      7. associate-/r/ 27.7%

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

      8. associate-*l* 27.7%

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

      9. add-exp-log 27.7%

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

      10. expm1-def 27.7%

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

      11. log-pow 33.5%

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

      12. log1p-udef 96.0%

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

   5. Applied egg-rr 96.0%

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

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

   1. Initial program 98.6%

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

   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 82.9%

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

   4. Simplified 82.9%

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

      1. associate-*l/ 82.9%

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

      2. div-inv 82.8%

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

      3. associate-/r* 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in i around 0 20.3%

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

      1. clear-num 22.7%

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

      2. inv-pow 22.7%

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

      3. div-inv 22.7%

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

      4. sub-neg 22.7%

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

      5. metadata-eval 22.7%

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

      6. clear-num 22.7%

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

      7. *-commutative 22.7%

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

   9. Applied egg-rr 22.7%

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

       1. unpow-1 22.7%

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

       2. associate-*r/ 99.7%

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

       3. *-commutative 99.7%

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

       4. distribute-lft-in 99.7%

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

       5. rgt-mult-inverse 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot \left(n \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* n (* (/ (expm1 (* n (log1p (/ i n)))) i) 100.0))
     (if (<= t_0 INFINITY)
       (* t_0 100.0)
       (/ 1.0 (/ (+ 1.0 (* i -0.5)) (* n 100.0)))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = n * ((expm1((n * log1p((i / n)))) / i) * 100.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = n * ((Math.expm1((n * Math.log1p((i / n)))) / i) * 100.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = n * ((math.expm1((n * math.log1p((i / n)))) / i) * 100.0)
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = 1.0 / ((1.0 + (i * -0.5)) / (n * 100.0))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(n * Float64(Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i) * 100.0));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / Float64(n * 100.0)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(n * N[(N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(1.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 27.7%

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

      1. associate-*r/ 27.7%

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

      2. sub-neg 27.7%

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

      3. distribute-lft-in 27.7%

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

      4. metadata-eval 27.7%

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

      5. metadata-eval 27.7%

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

   3. Simplified 27.7%

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

      1. metadata-eval 27.7%

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

      2. metadata-eval 27.7%

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

      3. distribute-lft-in 27.7%

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

      4. sub-neg 27.7%

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

      5. associate-*r/ 27.7%

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

      6. associate-/r/ 27.7%

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

      7. associate-*r* 27.7%

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

      8. add-exp-log 27.7%

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

      9. expm1-def 27.7%

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

      10. log-pow 33.5%

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

      11. log1p-udef 95.9%

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

   5. Applied egg-rr 95.9%

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

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

   1. Initial program 98.6%

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

   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 82.9%

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

   4. Simplified 82.9%

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

      1. associate-*l/ 82.9%

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

      2. div-inv 82.8%

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

      3. associate-/r* 5.5%

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

   6. Applied egg-rr 5.5%

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

   7. Taylor expanded in i around 0 20.3%

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

      1. clear-num 22.7%

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

      2. inv-pow 22.7%

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

      3. div-inv 22.7%

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

      4. sub-neg 22.7%

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

      5. metadata-eval 22.7%

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

      6. clear-num 22.7%

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

      7. *-commutative 22.7%

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

   9. Applied egg-rr 22.7%

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

       1. unpow-1 22.7%

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

       2. associate-*r/ 99.7%

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

       3. *-commutative 99.7%

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

       4. distribute-lft-in 99.7%

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

       5. rgt-mult-inverse 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \left(\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i} \cdot 100\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + i \cdot -0.5}{n \cdot 100}}\\ \end{array} \]

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq -2.65 \cdot 10^{-166}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{i}{\frac{1}{n}}\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{-91}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -2e-15)
     t_0
     (if (<= i -2.65e-166)
       (* (/ 100.0 i) (/ i (/ 1.0 n)))
       (if (<= i 3.2e-91) (* 100.0 (+ n (* i -0.5))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -2e-15) {
		tmp = t_0;
	} else if (i <= -2.65e-166) {
		tmp = (100.0 / i) * (i / (1.0 / n));
	} else if (i <= 3.2e-91) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = t_0;
	}
	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 <= -2e-15) {
		tmp = t_0;
	} else if (i <= -2.65e-166) {
		tmp = (100.0 / i) * (i / (1.0 / n));
	} else if (i <= 3.2e-91) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -2e-15:
		tmp = t_0
	elif i <= -2.65e-166:
		tmp = (100.0 / i) * (i / (1.0 / n))
	elif i <= 3.2e-91:
		tmp = 100.0 * (n + (i * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -2e-15)
		tmp = t_0;
	elseif (i <= -2.65e-166)
		tmp = Float64(Float64(100.0 / i) * Float64(i / Float64(1.0 / n)));
	elseif (i <= 3.2e-91)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	else
		tmp = t_0;
	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, -2e-15], t$95$0, If[LessEqual[i, -2.65e-166], N[(N[(100.0 / i), $MachinePrecision] * N[(i / N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.2e-91], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.0000000000000002e-15 or 3.19999999999999996e-91 < i

   1. Initial program 46.6%

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

   2. Taylor expanded in n around inf 63.6%

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

      1. expm1-def 70.1%

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

   4. Simplified 70.1%

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

   if -2.0000000000000002e-15 < i < -2.64999999999999998e-166

   1. Initial program 17.3%

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

      1. associate-*r/ 17.3%

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

      2. sub-neg 17.3%

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

      3. distribute-lft-in 17.3%

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

      4. metadata-eval 17.3%

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

      5. metadata-eval 17.3%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in i around 0 17.8%

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

      1. *-commutative 17.8%

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

   6. Simplified 17.8%

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

   7. Taylor expanded in i around 0 51.5%

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

      1. div-inv 51.6%

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

      2. times-frac 86.3%

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

   9. Applied egg-rr 86.3%

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

   if -2.64999999999999998e-166 < i < 3.19999999999999996e-91

   1. Initial program 2.9%

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

   2. Taylor expanded in i around 0 97.4%

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

      1. associate-*r* 97.1%

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

      2. associate-*r/ 97.1%

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

      3. metadata-eval 97.1%

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

   4. Simplified 97.1%

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

   5. Taylor expanded in n around 0 97.4%

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

      1. *-commutative 97.4%

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

   7. Simplified 97.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{-15}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq -2.65 \cdot 10^{-166}:\\ \;\;\;\;\frac{100}{i} \cdot \frac{i}{\frac{1}{n}}\\ \mathbf{elif}\;i \leq 3.2 \cdot 10^{-91}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-163} \lor \neg \left(n \leq 1.35 \cdot 10^{-169}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.45e-163) (not (<= n 1.35e-169)))
   (* n (/ (* 100.0 (expm1 i)) i))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.45e-163) || !(n <= 1.35e-169)) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.45e-163) || !(n <= 1.35e-169)) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.45e-163) or not (n <= 1.35e-169):
		tmp = n * ((100.0 * math.expm1(i)) / i)
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.45e-163) || !(n <= 1.35e-169))
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.45e-163], N[Not[LessEqual[n, 1.35e-169]], $MachinePrecision]], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.4500000000000001e-163 or 1.3500000000000001e-169 < n

   1. Initial program 21.8%

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

      1. associate-/r/ 22.2%

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

      2. associate-*r* 22.1%

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

      3. *-commutative 22.1%

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

      4. associate-*r/ 22.2%

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

      5. sub-neg 22.2%

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

      6. distribute-lft-in 22.2%

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

      7. metadata-eval 22.2%

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

      8. metadata-eval 22.2%

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

      9. metadata-eval 22.2%

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

      10. fma-def 22.2%

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

      11. metadata-eval 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in n around inf 37.4%

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

      1. sub-neg 37.4%

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

      2. metadata-eval 37.4%

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

      3. +-commutative 37.4%

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

   6. Applied egg-rr 37.4%

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

      1. +-commutative 37.4%

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

      2. metadata-eval 37.4%

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

      3. distribute-lft-in 37.5%

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

      4. metadata-eval 37.5%

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

      5. sub-neg 37.5%

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

      6. expm1-def 88.8%

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

   8. Simplified 88.8%

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

   if -1.4500000000000001e-163 < n < 1.3500000000000001e-169

   1. Initial program 58.2%

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

      1. associate-*r/ 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-lft-in 58.2%

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

      4. metadata-eval 58.2%

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

      5. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in i around 0 76.8%

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

   5. Taylor expanded in i around 0 76.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{-163} \lor \neg \left(n \leq 1.35 \cdot 10^{-169}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-157} \lor \neg \left(n \leq 10^{-170}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.05e-157) (not (<= n 1e-170)))
   (* 100.0 (/ n (/ i (expm1 i))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.05e-157) || !(n <= 1e-170)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.05e-157) || !(n <= 1e-170)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.05e-157) or not (n <= 1e-170):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.05e-157) || !(n <= 1e-170))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.05e-157], N[Not[LessEqual[n, 1e-170]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.05e-157 or 9.99999999999999983e-171 < n

   1. Initial program 21.8%

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

   2. Taylor expanded in n around inf 37.4%

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

      1. *-commutative 37.4%

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

      2. associate-/l* 37.5%

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

      3. expm1-def 88.9%

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

   4. Simplified 88.9%

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

   if -1.05e-157 < n < 9.99999999999999983e-171

   1. Initial program 58.2%

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

      1. associate-*r/ 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-lft-in 58.2%

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

      4. metadata-eval 58.2%

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

      5. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in i around 0 76.8%

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

   5. Taylor expanded in i around 0 76.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-157} \lor \neg \left(n \leq 10^{-170}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 63.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-162}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-174}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.8e-162)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 5.2e-174) 0.0 (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.8e-162) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 5.2e-174) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (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 <= (-3.8d-162)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 5.2d-174) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n + ((0.5d0 - (0.5d0 / n)) * (i * n)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.8e-162) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 5.2e-174) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.8e-162:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 5.2e-174:
		tmp = 0.0
	else:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.8e-162)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 5.2e-174)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.8e-162)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 5.2e-174)
		tmp = 0.0;
	else
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.8e-162], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.2e-174], 0.0, N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.80000000000000005e-162

   1. Initial program 25.0%

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

   2. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 38.6%

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

      3. expm1-def 87.9%

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

   4. Simplified 87.9%

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

   5. Taylor expanded in i around 0 62.7%

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

      1. *-commutative 62.7%

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

   7. Simplified 62.7%

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

   if -3.80000000000000005e-162 < n < 5.2000000000000004e-174

   1. Initial program 58.2%

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

      1. associate-*r/ 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-lft-in 58.2%

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

      4. metadata-eval 58.2%

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

      5. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in i around 0 76.8%

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

   5. Taylor expanded in i around 0 76.8%

      \[\leadsto \color{blue}{0} \]

   if 5.2000000000000004e-174 < n

   1. Initial program 17.9%

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

   2. Taylor expanded in i around 0 66.9%

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

      1. associate-*r* 66.5%

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

      2. associate-*r/ 66.5%

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

      3. metadata-eval 66.5%

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

   4. Simplified 66.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{-162}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{-174}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \end{array} \]

Alternative 7: 62.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-161}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-174}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.4e-161)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 6.6e-174) 0.0 (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n)))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-161) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 6.6e-174) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.4d-161)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 6.6d-174) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.4e-161) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 6.6e-174) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.4e-161:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 6.6e-174:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.4e-161)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 6.6e-174)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.4e-161)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 6.6e-174)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.4e-161], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.6e-174], 0.0, N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.39999999999999996e-161

   1. Initial program 25.0%

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

   2. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 38.6%

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

      3. expm1-def 87.9%

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

   4. Simplified 87.9%

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

   5. Taylor expanded in i around 0 62.7%

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

      1. *-commutative 62.7%

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

   7. Simplified 62.7%

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

   if -1.39999999999999996e-161 < n < 6.6000000000000002e-174

   1. Initial program 58.2%

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

      1. associate-*r/ 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-lft-in 58.2%

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

      4. metadata-eval 58.2%

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

      5. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in i around 0 76.8%

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

   5. Taylor expanded in i around 0 76.8%

      \[\leadsto \color{blue}{0} \]

   if 6.6000000000000002e-174 < n

   1. Initial program 17.9%

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

      1. associate-/r/ 18.3%

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

      2. associate-*r* 18.3%

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

      3. *-commutative 18.3%

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

      4. associate-*r/ 18.3%

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

      5. sub-neg 18.3%

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

      6. distribute-lft-in 18.3%

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

      7. metadata-eval 18.3%

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

      8. metadata-eval 18.3%

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

      9. metadata-eval 18.3%

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

      10. fma-def 18.3%

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

      11. metadata-eval 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in i around 0 66.9%

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

      1. associate-*r/ 66.9%

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

      2. metadata-eval 66.9%

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

   6. Simplified 66.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-161}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{-174}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \end{array} \]

Alternative 8: 61.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-164} \lor \neg \left(n \leq 1.12 \cdot 10^{-170}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.9e-164) (not (<= n 1.12e-170)))
   (* n (+ 100.0 (* i 50.0)))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e-164) || !(n <= 1.12e-170)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.9d-164)) .or. (.not. (n <= 1.12d-170))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.9e-164) || !(n <= 1.12e-170)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.9e-164) or not (n <= 1.12e-170):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.9e-164) || !(n <= 1.12e-170))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.9e-164) || ~((n <= 1.12e-170)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.9e-164], N[Not[LessEqual[n, 1.12e-170]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.89999999999999995e-164 or 1.12000000000000009e-170 < n

   1. Initial program 21.8%

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

      1. associate-/r/ 22.2%

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

      2. associate-*r* 22.1%

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

      3. *-commutative 22.1%

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

      4. associate-*r/ 22.2%

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

      5. sub-neg 22.2%

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

      6. distribute-lft-in 22.2%

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

      7. metadata-eval 22.2%

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

      8. metadata-eval 22.2%

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

      9. metadata-eval 22.2%

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

      10. fma-def 22.2%

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

      11. metadata-eval 22.2%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in n around inf 37.4%

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

   5. Taylor expanded in i around 0 61.6%

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

      1. *-commutative 61.6%

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

   7. Simplified 61.6%

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

   if -1.89999999999999995e-164 < n < 1.12000000000000009e-170

   1. Initial program 58.2%

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

      1. associate-*r/ 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-lft-in 58.2%

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

      4. metadata-eval 58.2%

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

      5. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in i around 0 76.8%

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

   5. Taylor expanded in i around 0 76.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{-164} \lor \neg \left(n \leq 1.12 \cdot 10^{-170}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 59.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.5:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.5)
   (* -200.0 (/ n i))
   (if (<= i 2.05e-11) (* 100.0 (+ n (* i -0.5))) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.5) {
		tmp = -200.0 * (n / i);
	} else if (i <= 2.05e-11) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.5d0)) then
        tmp = (-200.0d0) * (n / i)
    else if (i <= 2.05d-11) then
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.5) {
		tmp = -200.0 * (n / i);
	} else if (i <= 2.05e-11) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.5:
		tmp = -200.0 * (n / i)
	elif i <= 2.05e-11:
		tmp = 100.0 * (n + (i * -0.5))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.5)
		tmp = Float64(-200.0 * Float64(n / i));
	elseif (i <= 2.05e-11)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.5)
		tmp = -200.0 * (n / i);
	elseif (i <= 2.05e-11)
		tmp = 100.0 * (n + (i * -0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.5], N[(-200.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.05e-11], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if i < -1.5

   1. Initial program 58.4%

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

   2. Taylor expanded in n around inf 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-/l* 85.9%

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

      3. expm1-def 85.9%

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

   4. Simplified 85.9%

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

      1. associate-*l/ 85.7%

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

      2. div-inv 85.7%

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

      3. associate-/r* 85.7%

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

   6. Applied egg-rr 85.7%

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

   7. Taylor expanded in i around 0 30.2%

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

   8. Taylor expanded in i around inf 30.2%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]

   if -1.5 < i < 2.05e-11

   1. Initial program 8.2%

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

   2. Taylor expanded in i around 0 88.1%

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

      1. associate-*r* 88.0%

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

      2. associate-*r/ 88.0%

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

      3. metadata-eval 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in n around 0 87.3%

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

      1. *-commutative 87.3%

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

   7. Simplified 87.3%

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

   if 2.05e-11 < i

   1. Initial program 44.1%

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

      1. associate-*r/ 44.1%

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

      2. sub-neg 44.1%

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

      3. distribute-lft-in 44.0%

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

      4. metadata-eval 44.0%

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

      5. metadata-eval 44.0%

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

   3. Simplified 44.0%

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

   4. Taylor expanded in i around 0 29.2%

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

   5. Taylor expanded in i around 0 29.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.5:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{-11}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 63.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-173}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e-165)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 3.1e-173) 0.0 (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e-165) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.1e-173) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d-165)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 3.1d-173) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e-165) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 3.1e-173) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e-165:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 3.1e-173:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e-165)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 3.1e-173)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e-165)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 3.1e-173)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e-165], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.1e-173], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.4e-165

   1. Initial program 25.0%

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

   2. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 38.6%

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

      3. expm1-def 87.9%

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

   4. Simplified 87.9%

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

      1. associate-*l/ 87.9%

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

      2. associate-/l* 87.9%

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

   6. Applied egg-rr 87.9%

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

   7. Taylor expanded in i around 0 62.7%

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

      1. *-commutative 62.7%

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

   9. Simplified 62.7%

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

   if -3.4e-165 < n < 3.10000000000000005e-173

   1. Initial program 58.2%

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

      1. associate-*r/ 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-lft-in 58.2%

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

      4. metadata-eval 58.2%

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

      5. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in i around 0 76.8%

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

   5. Taylor expanded in i around 0 76.8%

      \[\leadsto \color{blue}{0} \]

   if 3.10000000000000005e-173 < n

   1. Initial program 17.9%

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

      1. associate-/r/ 18.3%

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

      2. associate-*r* 18.3%

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

      3. *-commutative 18.3%

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

      4. associate-*r/ 18.3%

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

      5. sub-neg 18.3%

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

      6. distribute-lft-in 18.3%

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

      7. metadata-eval 18.3%

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

      8. metadata-eval 18.3%

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

      9. metadata-eval 18.3%

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

      10. fma-def 18.3%

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

      11. metadata-eval 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in n around inf 36.0%

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

   5. Taylor expanded in i around 0 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-173}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 11: 63.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-174}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -3.4e-165)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (if (<= n 3.2e-174) 0.0 (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -3.4e-165) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 3.2e-174) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.4d-165)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 3.2d-174) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -3.4e-165) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 3.2e-174) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -3.4e-165:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 3.2e-174:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -3.4e-165)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 3.2e-174)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -3.4e-165)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 3.2e-174)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -3.4e-165], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.2e-174], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.4e-165

   1. Initial program 25.0%

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

   2. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

      2. associate-/l* 38.6%

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

      3. expm1-def 87.9%

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

   4. Simplified 87.9%

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

   5. Taylor expanded in i around 0 62.7%

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

      1. *-commutative 62.7%

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

   7. Simplified 62.7%

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

   if -3.4e-165 < n < 3.2e-174

   1. Initial program 58.2%

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

      1. associate-*r/ 58.2%

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

      2. sub-neg 58.2%

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

      3. distribute-lft-in 58.2%

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

      4. metadata-eval 58.2%

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

      5. metadata-eval 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in i around 0 76.8%

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

   5. Taylor expanded in i around 0 76.8%

      \[\leadsto \color{blue}{0} \]

   if 3.2e-174 < n

   1. Initial program 17.9%

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

      1. associate-/r/ 18.3%

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

      2. associate-*r* 18.3%

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

      3. *-commutative 18.3%

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

      4. associate-*r/ 18.3%

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

      5. sub-neg 18.3%

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

      6. distribute-lft-in 18.3%

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

      7. metadata-eval 18.3%

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

      8. metadata-eval 18.3%

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

      9. metadata-eval 18.3%

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

      10. fma-def 18.3%

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

      11. metadata-eval 18.3%

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

   3. Simplified 18.3%

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

   4. Taylor expanded in n around inf 36.0%

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

   5. Taylor expanded in i around 0 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{-165}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 3.2 \cdot 10^{-174}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 12: 59.0% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -27:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{-11}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -27.0) 0.0 (if (<= i 2.05e-11) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -27.0) {
		tmp = 0.0;
	} else if (i <= 2.05e-11) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-27.0d0)) then
        tmp = 0.0d0
    else if (i <= 2.05d-11) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -27.0) {
		tmp = 0.0;
	} else if (i <= 2.05e-11) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -27.0:
		tmp = 0.0
	elif i <= 2.05e-11:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -27.0)
		tmp = 0.0;
	elseif (i <= 2.05e-11)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -27.0)
		tmp = 0.0;
	elseif (i <= 2.05e-11)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -27.0], 0.0, If[LessEqual[i, 2.05e-11], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -27 or 2.05e-11 < i

   1. Initial program 51.6%

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

      1. associate-*r/ 51.6%

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

      2. sub-neg 51.6%

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

      3. distribute-lft-in 51.6%

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

      4. metadata-eval 51.6%

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

      5. metadata-eval 51.6%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in i around 0 28.9%

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

   5. Taylor expanded in i around 0 28.9%

      \[\leadsto \color{blue}{0} \]

   if -27 < i < 2.05e-11

   1. Initial program 8.2%

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

   2. Taylor expanded in i around 0 87.1%

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

      1. *-commutative 87.1%

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

   4. Simplified 87.1%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -27:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{-11}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 59.3% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{-11}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0) (* -200.0 (/ n i)) (if (<= i 2.05e-11) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 * (n / i);
	} else if (i <= 2.05e-11) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2.0d0)) then
        tmp = (-200.0d0) * (n / i)
    else if (i <= 2.05d-11) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 * (n / i);
	} else if (i <= 2.05e-11) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = -200.0 * (n / i)
	elif i <= 2.05e-11:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(-200.0 * Float64(n / i));
	elseif (i <= 2.05e-11)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = -200.0 * (n / i);
	elseif (i <= 2.05e-11)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(-200.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.05e-11], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 58.4%

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

   2. Taylor expanded in n around inf 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-/l* 85.9%

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

      3. expm1-def 85.9%

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

   4. Simplified 85.9%

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

      1. associate-*l/ 85.7%

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

      2. div-inv 85.7%

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

      3. associate-/r* 85.7%

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

   6. Applied egg-rr 85.7%

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

   7. Taylor expanded in i around 0 30.2%

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

   8. Taylor expanded in i around inf 30.2%

      \[\leadsto \color{blue}{-200 \cdot \frac{n}{i}} \]

   if -2 < i < 2.05e-11

   1. Initial program 8.2%

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

   2. Taylor expanded in i around 0 87.1%

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

      1. *-commutative 87.1%

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

   4. Simplified 87.1%

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

   if 2.05e-11 < i

   1. Initial program 44.1%

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

      1. associate-*r/ 44.1%

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

      2. sub-neg 44.1%

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

      3. distribute-lft-in 44.0%

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

      4. metadata-eval 44.0%

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

      5. metadata-eval 44.0%

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

   3. Simplified 44.0%

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

   4. Taylor expanded in i around 0 29.2%

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

   5. Taylor expanded in i around 0 29.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 2.05 \cdot 10^{-11}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 18.3% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.9%

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

   1. associate-*r/ 27.9%

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

   2. sub-neg 27.9%

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

   3. distribute-lft-in 27.9%

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

   4. metadata-eval 27.9%

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

   5. metadata-eval 27.9%

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

3. Simplified 27.9%

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

4. Taylor expanded in i around 0 17.6%

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

5. Taylor expanded in i around 0 17.9%

   \[\leadsto \color{blue}{0} \]

6. Final simplification 17.9%

   \[\leadsto 0 \]

Developer target: 35.0% 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 2023315 
(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))))