Compound Interest

Percentage Accurate: 27.6% → 99.5%

Time: 20.8s

Alternatives: 12

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.6% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 (- INFINITY))
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) (/ i n))
       (if (<= t_1 INFINITY)
         (* n (/ (+ (* t_0 100.0) -100.0) i))
         (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * expm1((n * log1p((i / n))))) / (i / n);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (100.0 * Math.expm1((n * Math.log1p((i / n))))) / (i / n);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = (100.0 * math.expm1((n * math.log1p((i / n))))) / (i / n)
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / Float64(i / n));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 27.3%

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

      1. associate-*r/ 27.3%

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

      2. sub-neg 27.3%

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

      3. distribute-lft-in 27.2%

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

      4. fma-def 27.3%

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

      5. metadata-eval 27.3%

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

      6. metadata-eval 27.3%

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

   3. Simplified 27.3%

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

      1. fma-udef 27.2%

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

      2. metadata-eval 27.2%

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

      3. distribute-lft-in 27.3%

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

      4. metadata-eval 27.3%

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

      5. sub-neg 27.3%

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

      6. *-commutative 27.3%

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

      7. add-exp-log 27.3%

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

      8. expm1-def 27.3%

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

      9. log-pow 41.6%

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

      10. log1p-udef 99.6%

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

   5. Applied egg-rr 99.6%

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

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. fma-def 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.8%

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

      3. sub-neg 1.8%

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

      4. metadata-eval 1.8%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. clear-num 1.8%

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

      8. un-div-inv 1.8%

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

      9. metadata-eval 1.8%

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

      10. sub-neg 1.8%

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

      11. add-exp-log 1.8%

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

      12. expm1-def 1.8%

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

      13. log-pow 1.8%

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

      14. log1p-udef 1.8%

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

   3. Applied egg-rr 1.8%

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

   4. Taylor expanded in i around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.7%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-48}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 -5e-48)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (* n (/ (* 100.0 (expm1 (* n (log1p (/ i n))))) i))
       (if (<= t_1 INFINITY)
         (* n (/ (+ (* t_0 100.0) -100.0) i))
         (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5)))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-48) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * expm1((n * log1p((i / n))))) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= -5e-48) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = n * ((100.0 * Math.expm1((n * Math.log1p((i / n))))) / i);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= -5e-48:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = n * ((100.0 * math.expm1((n * math.log1p((i / n))))) / i)
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -5e-48)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(n * Float64(Float64(100.0 * expm1(Float64(n * log1p(Float64(i / n))))) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-48], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(n * N[(N[(100.0 * N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -4.9999999999999999e-48

   1. Initial program 100.0%

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

   if -4.9999999999999999e-48 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < -0.0

   1. Initial program 26.4%

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

      1. associate-/r/ 25.9%

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

      2. associate-*r* 25.9%

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

      3. *-commutative 25.9%

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

      4. associate-*r/ 25.9%

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

      5. sub-neg 25.9%

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

      6. distribute-lft-in 25.8%

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

      7. fma-def 25.9%

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

      8. metadata-eval 25.9%

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

      9. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

      1. fma-udef 26.3%

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

      2. metadata-eval 26.3%

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

      3. distribute-lft-in 26.4%

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

      4. metadata-eval 26.4%

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

      5. sub-neg 26.4%

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

      6. *-commutative 26.4%

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

      7. add-exp-log 26.4%

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

      8. expm1-def 26.4%

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

      9. log-pow 40.9%

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

      10. log1p-udef 99.6%

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

   5. Applied egg-rr 98.7%

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

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

   1. Initial program 99.6%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. fma-def 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-/r/ 1.8%

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

      3. sub-neg 1.8%

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

      4. metadata-eval 1.8%

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

      5. associate-*r* 1.8%

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

      6. *-commutative 1.8%

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

      7. clear-num 1.8%

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

      8. un-div-inv 1.8%

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

      9. metadata-eval 1.8%

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

      10. sub-neg 1.8%

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

      11. add-exp-log 1.8%

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

      12. expm1-def 1.8%

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

      13. log-pow 1.8%

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

      14. log1p-udef 1.8%

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

   3. Applied egg-rr 1.8%

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

   4. Taylor expanded in i around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. metadata-eval 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \frac{n \cdot 100}{\color{blue}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -5 \cdot 10^{-48}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 3: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-5} \lor \neg \left(n \leq 2.7 \cdot 10^{-22}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -4.2e-5) (not (<= n 2.7e-22)))
   (* n (/ (* 100.0 (expm1 i)) i))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -4.2e-5) || !(n <= 2.7e-22)) {
		tmp = n * ((100.0 * expm1(i)) / i);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -4.2e-5) || !(n <= 2.7e-22)) {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -4.2e-5) or not (n <= 2.7e-22):
		tmp = n * ((100.0 * math.expm1(i)) / i)
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -4.2e-5) || !(n <= 2.7e-22))
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -4.2e-5], N[Not[LessEqual[n, 2.7e-22]], $MachinePrecision]], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.19999999999999977e-5 or 2.7000000000000002e-22 < n

   1. Initial program 29.8%

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

      1. associate-/r/ 30.3%

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

      2. associate-*r* 30.3%

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

      3. *-commutative 30.3%

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

      4. associate-*r/ 30.3%

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

      5. sub-neg 30.3%

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

      6. distribute-lft-in 30.3%

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

      7. fma-def 30.3%

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

      8. metadata-eval 30.3%

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

      9. metadata-eval 30.3%

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

   3. Simplified 30.3%

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

      1. fma-udef 29.9%

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

      2. metadata-eval 29.9%

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

      3. distribute-lft-in 29.9%

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

      4. metadata-eval 29.9%

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

      5. sub-neg 29.9%

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

      6. *-commutative 29.9%

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

      7. add-exp-log 29.9%

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

      8. expm1-def 29.9%

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

      9. log-pow 18.9%

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

      10. log1p-udef 66.7%

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

   5. Applied egg-rr 66.6%

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

   6. Taylor expanded in n around inf 44.2%

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

      1. *-commutative 44.2%

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

      2. expm1-def 88.4%

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

      3. associate-*l/ 88.3%

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

   8. Simplified 88.3%

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

   if -4.19999999999999977e-5 < n < 2.7000000000000002e-22

   1. Initial program 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-/r/ 36.5%

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

      3. sub-neg 36.5%

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

      4. metadata-eval 36.5%

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

      5. associate-*r* 36.5%

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

      6. *-commutative 36.5%

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

      7. clear-num 36.5%

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

      8. un-div-inv 36.5%

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

      9. metadata-eval 36.5%

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

      10. sub-neg 36.5%

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

      11. add-exp-log 36.5%

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

      12. expm1-def 36.5%

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

      13. log-pow 63.4%

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

      14. log1p-udef 88.2%

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

   3. Applied egg-rr 88.2%

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

   4. Taylor expanded in i around 0 74.3%

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

      1. sub-neg 74.3%

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

      2. associate-*r/ 74.3%

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

      3. metadata-eval 74.3%

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

      4. metadata-eval 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{-5} \lor \neg \left(n \leq 2.7 \cdot 10^{-22}\right):\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 4: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-5} \lor \neg \left(n \leq 2.7 \cdot 10^{-22}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.9e-5) (not (<= n 2.7e-22)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.9e-5) || !(n <= 2.7e-22)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.9e-5) || !(n <= 2.7e-22)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.9e-5) or not (n <= 2.7e-22):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.9e-5) || !(n <= 2.7e-22))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.9e-5], N[Not[LessEqual[n, 2.7e-22]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.9e-5 or 2.7000000000000002e-22 < n

   1. Initial program 29.8%

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

   2. Taylor expanded in n around inf 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-/l* 44.2%

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

      3. expm1-def 88.4%

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

   4. Simplified 88.4%

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

   if -2.9e-5 < n < 2.7000000000000002e-22

   1. Initial program 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-/r/ 36.5%

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

      3. sub-neg 36.5%

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

      4. metadata-eval 36.5%

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

      5. associate-*r* 36.5%

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

      6. *-commutative 36.5%

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

      7. clear-num 36.5%

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

      8. un-div-inv 36.5%

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

      9. metadata-eval 36.5%

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

      10. sub-neg 36.5%

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

      11. add-exp-log 36.5%

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

      12. expm1-def 36.5%

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

      13. log-pow 63.4%

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

      14. log1p-udef 88.2%

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

   3. Applied egg-rr 88.2%

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

   4. Taylor expanded in i around 0 74.3%

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

      1. sub-neg 74.3%

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

      2. associate-*r/ 74.3%

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

      3. metadata-eval 74.3%

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

      4. metadata-eval 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-5} \lor \neg \left(n \leq 2.7 \cdot 10^{-22}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \end{array} \]

Alternative 5: 70.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 2.7e-22)
   (/ (* n 100.0) (+ 1.0 (* i (+ (/ 0.5 n) -0.5))))
   (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 2.7e-22) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} 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 <= 2.7d-22) then
        tmp = (n * 100.0d0) / (1.0d0 + (i * ((0.5d0 / n) + (-0.5d0))))
    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 <= 2.7e-22) {
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 2.7e-22:
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 2.7e-22)
		tmp = Float64(Float64(n * 100.0) / Float64(1.0 + Float64(i * Float64(Float64(0.5 / n) + -0.5))));
	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 <= 2.7e-22)
		tmp = (n * 100.0) / (1.0 + (i * ((0.5 / n) + -0.5)));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 2.7e-22], N[(N[(n * 100.0), $MachinePrecision] / N[(1.0 + N[(i * N[(N[(0.5 / n), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.7000000000000002e-22

   1. Initial program 35.3%

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

      1. *-commutative 35.3%

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

      2. associate-/r/ 35.1%

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

      3. sub-neg 35.1%

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

      4. metadata-eval 35.1%

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

      5. associate-*r* 35.1%

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

      6. *-commutative 35.1%

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

      7. clear-num 35.1%

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

      8. un-div-inv 35.1%

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

      9. metadata-eval 35.1%

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

      10. sub-neg 35.1%

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

      11. add-exp-log 35.1%

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

      12. expm1-def 35.1%

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

      13. log-pow 42.1%

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

      14. log1p-udef 76.3%

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

   3. Applied egg-rr 76.3%

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

   4. Taylor expanded in i around 0 64.2%

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

      1. sub-neg 64.2%

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

      2. associate-*r/ 64.2%

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

      3. metadata-eval 64.2%

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

      4. metadata-eval 64.2%

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

   6. Simplified 64.2%

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

   if 2.7000000000000002e-22 < n

   1. Initial program 27.0%

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

      1. associate-*r/ 27.0%

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

      2. sub-neg 27.0%

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

      3. distribute-lft-in 27.0%

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

      4. fma-def 27.0%

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

      5. metadata-eval 27.0%

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

      6. metadata-eval 27.0%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in n around inf 45.5%

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

   5. Taylor expanded in i around 0 68.8%

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

      1. metadata-eval 68.8%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{1}} \cdot n \]

      2. associate-/r/ 68.7%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{\frac{1}{n}}} \]

      3. +-commutative 68.7%

         \[\leadsto \color{blue}{\frac{100}{\frac{1}{n}} + 50 \cdot \left(i \cdot n\right)} \]

      4. associate-/r/ 68.8%

         \[\leadsto \color{blue}{\frac{100}{1} \cdot n} + 50 \cdot \left(i \cdot n\right) \]

      5. metadata-eval 68.8%

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

      6. fma-def 68.9%

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

      7. *-commutative 68.9%

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

      8. *-commutative 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in n around 0 68.9%

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

      1. distribute-lft-in 68.8%

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

      2. *-commutative 68.8%

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

      3. distribute-lft-in 68.9%

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

   10. Simplified 68.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{n \cdot 100}{1 + i \cdot \left(\frac{0.5}{n} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 6: 62.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -50000000000 \lor \neg \left(n \leq 2.7 \cdot 10^{-22}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -50000000000.0) (not (<= n 2.7e-22)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -50000000000.0) || !(n <= 2.7e-22)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-50000000000.0d0)) .or. (.not. (n <= 2.7d-22))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -50000000000.0) || !(n <= 2.7e-22)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -50000000000.0) or not (n <= 2.7e-22):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -50000000000.0) || !(n <= 2.7e-22))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -50000000000.0) || ~((n <= 2.7e-22)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -50000000000.0], N[Not[LessEqual[n, 2.7e-22]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5e10 or 2.7000000000000002e-22 < n

   1. Initial program 29.3%

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

      1. associate-*r/ 29.4%

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

      2. sub-neg 29.4%

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

      3. distribute-lft-in 29.4%

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

      4. fma-def 29.4%

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

      5. metadata-eval 29.4%

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

      6. metadata-eval 29.4%

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

   3. Simplified 29.4%

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

   4. Taylor expanded in n around inf 44.1%

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

   5. Taylor expanded in i around 0 58.9%

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

      1. metadata-eval 58.9%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{1}} \cdot n \]

      2. associate-/r/ 58.7%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{\frac{1}{n}}} \]

      3. +-commutative 58.7%

         \[\leadsto \color{blue}{\frac{100}{\frac{1}{n}} + 50 \cdot \left(i \cdot n\right)} \]

      4. associate-/r/ 58.9%

         \[\leadsto \color{blue}{\frac{100}{1} \cdot n} + 50 \cdot \left(i \cdot n\right) \]

      5. metadata-eval 58.9%

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

      6. fma-def 58.9%

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

      7. *-commutative 58.9%

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

      8. *-commutative 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in n around 0 58.9%

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

      1. distribute-lft-in 58.9%

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

      2. *-commutative 58.9%

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

      3. distribute-lft-in 58.9%

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

   10. Simplified 58.9%

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

   if -5e10 < n < 2.7000000000000002e-22

   1. Initial program 37.8%

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

   2. Taylor expanded in i around 0 52.7%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -50000000000 \lor \neg \left(n \leq 2.7 \cdot 10^{-22}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]

Alternative 7: 63.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-201}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 6.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.25e-201)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 6.4e-94) (/ 0.0 (/ i n)) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.25e-201) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 6.4e-94) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.25d-201)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 6.4d-94) then
        tmp = 0.0d0 / (i / n)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.25e-201) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 6.4e-94) {
		tmp = 0.0 / (i / n);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.25e-201:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 6.4e-94:
		tmp = 0.0 / (i / n)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.25e-201)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 6.4e-94)
		tmp = Float64(0.0 / Float64(i / n));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.25e-201)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 6.4e-94)
		tmp = 0.0 / (i / n);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.25e-201], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6.4e-94], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.25e-201

   1. Initial program 31.9%

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

      1. associate-/r/ 32.2%

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

      2. associate-*r* 32.2%

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

      3. *-commutative 32.2%

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

      4. associate-*r/ 32.2%

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

      5. sub-neg 32.2%

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

      6. distribute-lft-in 32.1%

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

      7. fma-def 32.2%

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

      8. metadata-eval 32.2%

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

      9. metadata-eval 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in n around inf 33.1%

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

      1. associate-/l* 33.8%

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

      2. *-commutative 33.8%

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

      3. fma-neg 33.9%

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

      4. metadata-eval 33.9%

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

   6. Simplified 33.9%

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

   7. Taylor expanded in i around 0 53.3%

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

      1. *-commutative 53.3%

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

   9. Simplified 53.3%

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

   if -1.25e-201 < n < 6.39999999999999993e-94

   1. Initial program 46.6%

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

      1. associate-*r/ 46.6%

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

      2. sub-neg 46.6%

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

      3. distribute-lft-in 46.6%

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

      4. fma-def 46.6%

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

      5. metadata-eval 46.6%

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

      6. metadata-eval 46.6%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in n around inf 32.5%

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

   5. Taylor expanded in i around 0 66.6%

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

   if 6.39999999999999993e-94 < n

   1. Initial program 25.9%

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

      1. associate-*r/ 25.9%

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

      2. sub-neg 25.9%

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

      3. distribute-lft-in 25.9%

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

      4. fma-def 25.9%

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

      5. metadata-eval 25.9%

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

      6. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   4. Taylor expanded in n around inf 41.4%

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

   5. Taylor expanded in i around 0 68.8%

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

      1. metadata-eval 68.8%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{1}} \cdot n \]

      2. associate-/r/ 68.6%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{\frac{1}{n}}} \]

      3. +-commutative 68.6%

         \[\leadsto \color{blue}{\frac{100}{\frac{1}{n}} + 50 \cdot \left(i \cdot n\right)} \]

      4. associate-/r/ 68.8%

         \[\leadsto \color{blue}{\frac{100}{1} \cdot n} + 50 \cdot \left(i \cdot n\right) \]

      5. metadata-eval 68.8%

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

      6. fma-def 68.8%

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

      7. *-commutative 68.8%

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

      8. *-commutative 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in n around 0 68.8%

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

      1. distribute-lft-in 68.8%

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

      2. *-commutative 68.8%

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

      3. distribute-lft-in 68.8%

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

   10. Simplified 68.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{-201}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 6.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 8: 58.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+47}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+106}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -5e+47)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 5.8e+106) (* n 100.0) (* 50.0 (* i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -5e+47) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 5.8e+106) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-5d+47)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 5.8d+106) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -5e+47) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 5.8e+106) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -5e+47:
		tmp = 100.0 * (i / (i / n))
	elif i <= 5.8e+106:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -5e+47)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 5.8e+106)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -5e+47)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 5.8e+106)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -5e+47], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 5.8e+106], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -5.00000000000000022e47

   1. Initial program 62.5%

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

   2. Taylor expanded in i around 0 24.0%

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

   if -5.00000000000000022e47 < i < 5.8000000000000004e106

   1. Initial program 11.7%

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

   2. Taylor expanded in i around 0 67.3%

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

      1. *-commutative 67.3%

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

   4. Simplified 67.3%

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

   if 5.8000000000000004e106 < i

   1. Initial program 65.8%

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

      1. associate-*r/ 65.9%

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

      2. sub-neg 65.9%

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

      3. distribute-lft-in 65.9%

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

      4. fma-def 65.9%

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

      5. metadata-eval 65.9%

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

      6. metadata-eval 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in n around inf 51.4%

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

   5. Taylor expanded in i around 0 36.7%

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

      1. metadata-eval 36.7%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{1}} \cdot n \]

      2. associate-/r/ 36.7%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{\frac{1}{n}}} \]

      3. +-commutative 36.7%

         \[\leadsto \color{blue}{\frac{100}{\frac{1}{n}} + 50 \cdot \left(i \cdot n\right)} \]

      4. associate-/r/ 36.7%

         \[\leadsto \color{blue}{\frac{100}{1} \cdot n} + 50 \cdot \left(i \cdot n\right) \]

      5. metadata-eval 36.7%

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

      6. fma-def 36.7%

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

      7. *-commutative 36.7%

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

      8. *-commutative 36.7%

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

   7. Simplified 36.7%

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

      1. fma-udef 36.7%

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

      2. *-commutative 36.7%

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

      3. *-commutative 36.7%

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

      4. associate-*l* 36.7%

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

   9. Applied egg-rr 36.7%

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

   10. Taylor expanded in i around inf 36.7%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{+47}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{+106}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 9: 62.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 4.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 4.9e-40) (/ n (+ 0.01 (* i -0.005))) (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 4.9e-40) {
		tmp = n / (0.01 + (i * -0.005));
	} 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 <= 4.9d-40) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    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 <= 4.9e-40) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 4.9e-40:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 4.9e-40)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	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 <= 4.9e-40)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 4.9e-40], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 4.8999999999999997e-40

   1. Initial program 36.1%

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

      1. associate-/r/ 35.9%

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

      2. associate-*r* 35.8%

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

      3. *-commutative 35.8%

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

      4. associate-*r/ 35.9%

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

      5. sub-neg 35.9%

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

      6. distribute-lft-in 35.8%

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

      7. fma-def 35.9%

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

      8. metadata-eval 35.9%

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

      9. metadata-eval 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in n around inf 31.6%

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

      1. associate-/l* 32.1%

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

      2. *-commutative 32.1%

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

      3. fma-neg 32.1%

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

      4. metadata-eval 32.1%

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

   6. Simplified 32.1%

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

   7. Taylor expanded in i around 0 51.6%

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

      1. *-commutative 51.6%

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

   9. Simplified 51.6%

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

   if 4.8999999999999997e-40 < n

   1. Initial program 25.9%

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

      1. associate-*r/ 25.9%

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

      2. sub-neg 25.9%

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

      3. distribute-lft-in 25.9%

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

      4. fma-def 25.9%

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

      5. metadata-eval 25.9%

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

      6. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   4. Taylor expanded in n around inf 43.6%

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

   5. Taylor expanded in i around 0 69.2%

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

      1. metadata-eval 69.2%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{1}} \cdot n \]

      2. associate-/r/ 69.0%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{\frac{1}{n}}} \]

      3. +-commutative 69.0%

         \[\leadsto \color{blue}{\frac{100}{\frac{1}{n}} + 50 \cdot \left(i \cdot n\right)} \]

      4. associate-/r/ 69.2%

         \[\leadsto \color{blue}{\frac{100}{1} \cdot n} + 50 \cdot \left(i \cdot n\right) \]

      5. metadata-eval 69.2%

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

      6. fma-def 69.3%

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

      7. *-commutative 69.3%

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

      8. *-commutative 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in n around 0 69.2%

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

      1. distribute-lft-in 69.2%

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

      2. *-commutative 69.2%

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

      3. distribute-lft-in 69.2%

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

   10. Simplified 69.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 10: 54.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5.8 \cdot 10^{+106}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5.8e+106) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5.8e+106) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 5.8d+106) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (i * n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 5.8e+106) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5.8e+106:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5.8e+106)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 5.8e+106)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5.8e+106], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.8000000000000004e106

   1. Initial program 24.7%

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

   2. Taylor expanded in i around 0 51.3%

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

      1. *-commutative 51.3%

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

   4. Simplified 51.3%

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

   if 5.8000000000000004e106 < i

   1. Initial program 65.8%

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

      1. associate-*r/ 65.9%

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

      2. sub-neg 65.9%

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

      3. distribute-lft-in 65.9%

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

      4. fma-def 65.9%

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

      5. metadata-eval 65.9%

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

      6. metadata-eval 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in n around inf 51.4%

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

   5. Taylor expanded in i around 0 36.7%

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

      1. metadata-eval 36.7%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{1}} \cdot n \]

      2. associate-/r/ 36.7%

         \[\leadsto 50 \cdot \left(i \cdot n\right) + \color{blue}{\frac{100}{\frac{1}{n}}} \]

      3. +-commutative 36.7%

         \[\leadsto \color{blue}{\frac{100}{\frac{1}{n}} + 50 \cdot \left(i \cdot n\right)} \]

      4. associate-/r/ 36.7%

         \[\leadsto \color{blue}{\frac{100}{1} \cdot n} + 50 \cdot \left(i \cdot n\right) \]

      5. metadata-eval 36.7%

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

      6. fma-def 36.7%

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

      7. *-commutative 36.7%

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

      8. *-commutative 36.7%

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

   7. Simplified 36.7%

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

      1. fma-udef 36.7%

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

      2. *-commutative 36.7%

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

      3. *-commutative 36.7%

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

      4. associate-*l* 36.7%

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

   9. Applied egg-rr 36.7%

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

   10. Taylor expanded in i around inf 36.7%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5.8 \cdot 10^{+106}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 11: 2.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

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

   1. *-commutative 32.5%

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

   2. associate-/r/ 32.6%

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

   3. associate-*l* 32.6%

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

   4. sub-neg 32.6%

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

   5. metadata-eval 32.6%

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

3. Simplified 32.6%

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

4. Taylor expanded in i around 0 48.2%

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

   1. associate-*r/ 48.2%

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

   2. metadata-eval 48.2%

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

6. Simplified 48.2%

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

7. Taylor expanded in n around 0 2.7%

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

   1. *-commutative 2.7%

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

9. Simplified 2.7%

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

10. Final simplification 2.7%

    \[\leadsto i \cdot -50 \]

Alternative 12: 50.3% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

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

2. Taylor expanded in i around 0 42.3%

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

   1. *-commutative 42.3%

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

4. Simplified 42.3%

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

5. Final simplification 42.3%

   \[\leadsto n \cdot 100 \]

Developer target: 33.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2023333 
(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))))