Compound Interest

Percentage Accurate: 28.5% → 84.0%

Time: 23.9s

Alternatives: 15

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.5% 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: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(n \cdot t_0\right) \cdot 100}{i}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 -1e-169)
     (/ (* (* n t_0) 100.0) i)
     (if (<= t_1 0.0)
       (/ (* n 100.0) (/ i (expm1 i)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ t_0 i)))
         (/ n (+ 0.01 (* i -0.005))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -1e-169) {
		tmp = ((n * t_0) * 100.0) / i;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) / (i / expm1(i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -1e-169) {
		tmp = ((n * t_0) * 100.0) / i;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) / (i / Math.expm1(i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= -1e-169:
		tmp = ((n * t_0) * 100.0) / i
	elif t_1 <= 0.0:
		tmp = (n * 100.0) / (i / math.expm1(i))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * (t_0 / i))
	else:
		tmp = n / (0.01 + (i * -0.005))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-169)
		tmp = Float64(Float64(Float64(n * t_0) * 100.0) / i);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(t_0 / i)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-169], N[(N[(N[(n * t$95$0), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * -0.005), $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)) < -1.00000000000000002e-169

   1. Initial program 96.0%

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

      1. associate-/r/ 95.8%

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

      2. sub-neg 95.8%

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

      3. metadata-eval 95.8%

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

   3. Simplified 95.8%

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

      1. *-commutative 95.8%

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

      2. associate-*l/ 96.0%

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

      3. associate-*l/ 96.2%

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

      4. *-commutative 96.2%

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

   5. Applied egg-rr 96.2%

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

   if -1.00000000000000002e-169 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 15.6%

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

      1. associate-*r/ 15.6%

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

      2. associate-/r/ 15.6%

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

      3. *-commutative 15.6%

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

      4. sub-neg 15.6%

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

      5. distribute-lft-in 15.6%

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

      6. fma-def 15.6%

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

      7. metadata-eval 15.6%

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

      8. metadata-eval 15.6%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in n around inf 32.8%

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

      1. associate-/l* 32.9%

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

      2. fma-neg 32.8%

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

      3. metadata-eval 32.8%

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

   6. Simplified 32.8%

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

   7. Taylor expanded in i around inf 32.9%

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

      1. sub-neg 32.9%

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

      2. metadata-eval 32.9%

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

      3. metadata-eval 32.9%

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

      4. distribute-lft-in 32.8%

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

      5. metadata-eval 32.8%

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

      6. sub-neg 32.8%

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

      7. expm1-def 77.2%

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

   9. Simplified 77.2%

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

       1. frac-2neg 77.2%

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

       2. div-inv 77.3%

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

       3. metadata-eval 77.3%

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

       4. metadata-eval 77.3%

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

       5. distribute-neg-frac 77.3%

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

       6. frac-2neg 77.3%

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

       7. clear-num 77.2%

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

       8. associate-/l* 77.3%

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

       9. distribute-neg-frac 77.3%

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

       10. metadata-eval 77.3%

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

   11. Applied egg-rr 77.3%

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

       1. associate-*r/ 77.3%

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

       2. distribute-lft-neg-out 77.3%

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

       3. distribute-rgt-neg-in 77.3%

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

       4. metadata-eval 77.3%

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

       5. *-commutative 77.3%

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

   13. Simplified 77.3%

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

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

   1. Initial program 95.0%

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

      1. associate-/r/ 95.3%

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

      2. sub-neg 95.3%

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

      3. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. associate-/r/ 1.9%

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

      3. *-commutative 1.9%

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

      4. sub-neg 1.9%

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

      5. distribute-lft-in 1.9%

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

      6. fma-def 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. fma-neg 1.9%

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

      3. metadata-eval 1.9%

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

   6. Simplified 1.9%

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

   7. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(n \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} + -1\right)\right) \cdot 100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \end{array} \]

Alternative 2: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n} + -1\\ t_1 := \frac{t_0}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-169}:\\ \;\;\;\;t_1 \cdot 100\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{t_0}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ (pow (+ 1.0 (/ i n)) n) -1.0)) (t_1 (/ t_0 (/ i n))))
   (if (<= t_1 -1e-169)
     (* t_1 100.0)
     (if (<= t_1 0.0)
       (/ (* n 100.0) (/ i (expm1 i)))
       (if (<= t_1 INFINITY)
         (* 100.0 (* n (/ t_0 i)))
         (/ n (+ 0.01 (* i -0.005))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -1e-169) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) / (i / expm1(i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n) + -1.0;
	double t_1 = t_0 / (i / n);
	double tmp;
	if (t_1 <= -1e-169) {
		tmp = t_1 * 100.0;
	} else if (t_1 <= 0.0) {
		tmp = (n * 100.0) / (i / Math.expm1(i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 100.0 * (n * (t_0 / i));
	} else {
		tmp = n / (0.01 + (i * -0.005));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n) + -1.0
	t_1 = t_0 / (i / n)
	tmp = 0
	if t_1 <= -1e-169:
		tmp = t_1 * 100.0
	elif t_1 <= 0.0:
		tmp = (n * 100.0) / (i / math.expm1(i))
	elif t_1 <= math.inf:
		tmp = 100.0 * (n * (t_0 / i))
	else:
		tmp = n / (0.01 + (i * -0.005))
	return tmp

Julia

function code(i, n)
	t_0 = Float64((Float64(1.0 + Float64(i / n)) ^ n) + -1.0)
	t_1 = Float64(t_0 / Float64(i / n))
	tmp = 0.0
	if (t_1 <= -1e-169)
		tmp = Float64(t_1 * 100.0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(i)));
	elseif (t_1 <= Inf)
		tmp = Float64(100.0 * Float64(n * Float64(t_0 / i)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-169], N[(t$95$1 * 100.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(100.0 * N[(n * N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(i * -0.005), $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)) < -1.00000000000000002e-169

   1. Initial program 96.0%

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

   if -1.00000000000000002e-169 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 15.6%

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

      1. associate-*r/ 15.6%

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

      2. associate-/r/ 15.6%

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

      3. *-commutative 15.6%

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

      4. sub-neg 15.6%

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

      5. distribute-lft-in 15.6%

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

      6. fma-def 15.6%

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

      7. metadata-eval 15.6%

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

      8. metadata-eval 15.6%

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

   3. Simplified 15.6%

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

   4. Taylor expanded in n around inf 32.8%

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

      1. associate-/l* 32.9%

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

      2. fma-neg 32.8%

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

      3. metadata-eval 32.8%

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

   6. Simplified 32.8%

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

   7. Taylor expanded in i around inf 32.9%

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

      1. sub-neg 32.9%

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

      2. metadata-eval 32.9%

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

      3. metadata-eval 32.9%

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

      4. distribute-lft-in 32.8%

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

      5. metadata-eval 32.8%

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

      6. sub-neg 32.8%

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

      7. expm1-def 77.2%

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

   9. Simplified 77.2%

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

       1. frac-2neg 77.2%

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

       2. div-inv 77.3%

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

       3. metadata-eval 77.3%

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

       4. metadata-eval 77.3%

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

       5. distribute-neg-frac 77.3%

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

       6. frac-2neg 77.3%

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

       7. clear-num 77.2%

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

       8. associate-/l* 77.3%

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

       9. distribute-neg-frac 77.3%

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

       10. metadata-eval 77.3%

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

   11. Applied egg-rr 77.3%

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

       1. associate-*r/ 77.3%

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

       2. distribute-lft-neg-out 77.3%

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

       3. distribute-rgt-neg-in 77.3%

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

       4. metadata-eval 77.3%

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

       5. *-commutative 77.3%

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

   13. Simplified 77.3%

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

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

   1. Initial program 95.0%

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

      1. associate-/r/ 95.3%

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

      2. sub-neg 95.3%

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

      3. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. associate-/r/ 1.9%

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

      3. *-commutative 1.9%

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

      4. sub-neg 1.9%

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

      5. distribute-lft-in 1.9%

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

      6. fma-def 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in n around inf 1.9%

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

      1. associate-/l* 1.9%

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

      2. fma-neg 1.9%

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

      3. metadata-eval 1.9%

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

   6. Simplified 1.9%

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

   7. Taylor expanded in i around 0 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot -0.005}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq -1 \cdot 10^{-169}:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \end{array} \]

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{if}\;i \leq -2 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq -3.9 \cdot 10^{-166}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* (expm1 i) (/ n i)))))
   (if (<= i -2e-23)
     t_0
     (if (<= i -3.9e-166)
       (/ (* n (* i (+ 100.0 (* i 50.0)))) i)
       (if (<= i 9e-38) (* 100.0 (+ n (* i (* n (- 0.5 (/ 0.5 n)))))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) * (n / i));
	double tmp;
	if (i <= -2e-23) {
		tmp = t_0;
	} else if (i <= -3.9e-166) {
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	} else if (i <= 9e-38) {
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) * (n / i));
	double tmp;
	if (i <= -2e-23) {
		tmp = t_0;
	} else if (i <= -3.9e-166) {
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	} else if (i <= 9e-38) {
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) * (n / i))
	tmp = 0
	if i <= -2e-23:
		tmp = t_0
	elif i <= -3.9e-166:
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i
	elif i <= 9e-38:
		tmp = 100.0 * (n + (i * (n * (0.5 - (0.5 / n)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) * Float64(n / i)))
	tmp = 0.0
	if (i <= -2e-23)
		tmp = t_0;
	elseif (i <= -3.9e-166)
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * 50.0)))) / i);
	elseif (i <= 9e-38)
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(n * Float64(0.5 - Float64(0.5 / n))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2e-23], t$95$0, If[LessEqual[i, -3.9e-166], N[(N[(n * N[(i * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[i, 9e-38], N[(100.0 * N[(n + N[(i * N[(n * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.99999999999999992e-23 or 9.00000000000000018e-38 < i

   1. Initial program 43.4%

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

      1. associate-/r/ 43.6%

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

      2. sub-neg 43.6%

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

      3. metadata-eval 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in n around inf 54.6%

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

      1. associate-/l* 54.6%

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

      2. expm1-def 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in n around 0 54.6%

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

      1. expm1-def 57.8%

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

      2. associate-/l* 57.8%

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

      3. associate-/r/ 57.6%

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

   9. Simplified 57.6%

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

   if -1.99999999999999992e-23 < i < -3.89999999999999999e-166

   1. Initial program 15.7%

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

      1. associate-*r/ 15.7%

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

      2. associate-/r/ 16.1%

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

      3. *-commutative 16.1%

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

      4. sub-neg 16.1%

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

      5. distribute-lft-in 16.1%

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

      6. fma-def 16.1%

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

      7. metadata-eval 16.1%

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

      8. metadata-eval 16.1%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in i around 0 76.5%

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

      1. distribute-lft-out 76.5%

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

      2. unpow2 76.5%

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

      3. associate-*r/ 76.5%

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

      4. metadata-eval 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in n around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. associate-*r* 77.6%

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

   9. Simplified 77.6%

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

       1. associate-*r/ 89.4%

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

       2. distribute-rgt-in 89.4%

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

       3. associate-*l* 89.4%

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

       4. distribute-lft-out 89.4%

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

       5. associate-*l* 89.4%

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

       6. metadata-eval 89.4%

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

   11. Applied egg-rr 89.4%

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

   if -3.89999999999999999e-166 < i < 9.00000000000000018e-38

   1. Initial program 4.6%

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

      1. associate-/r/ 5.2%

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

      2. sub-neg 5.2%

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

      3. metadata-eval 5.2%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in i around 0 93.7%

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

      1. distribute-lft-out 93.7%

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

      2. associate-*r/ 93.7%

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

      3. metadata-eval 93.7%

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

   6. Simplified 93.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;i \leq -3.9 \cdot 10^{-166}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-38}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(i\right) \cdot \frac{n}{i}\right)\\ \end{array} \]

Alternative 4: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-188}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.19:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (* n (/ (expm1 i) i)))))
   (if (<= n -4.7e-198)
     t_0
     (if (<= n 1.75e-188)
       (/ 0.0 (/ i n))
       (if (<= n 0.19) (/ n (+ 0.01 (* i -0.005))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n * (expm1(i) / i));
	double tmp;
	if (n <= -4.7e-198) {
		tmp = t_0;
	} else if (n <= 1.75e-188) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.19) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n * (Math.expm1(i) / i));
	double tmp;
	if (n <= -4.7e-198) {
		tmp = t_0;
	} else if (n <= 1.75e-188) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.19) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n * (math.expm1(i) / i))
	tmp = 0
	if n <= -4.7e-198:
		tmp = t_0
	elif n <= 1.75e-188:
		tmp = 0.0 / (i / n)
	elif n <= 0.19:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n * Float64(expm1(i) / i)))
	tmp = 0.0
	if (n <= -4.7e-198)
		tmp = t_0;
	elseif (n <= 1.75e-188)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 0.19)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.7e-198], t$95$0, If[LessEqual[n, 1.75e-188], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.19], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.7e-198 or 0.19 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 22.9%

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

      2. sub-neg 22.9%

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

      3. metadata-eval 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in n around inf 34.2%

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

      1. expm1-def 86.4%

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

   6. Simplified 86.4%

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

   if -4.7e-198 < n < 1.75e-188

   1. Initial program 55.5%

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

      1. associate-*r/ 55.5%

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

      2. sub-neg 55.5%

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

      3. distribute-lft-in 55.5%

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

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in i around 0 72.1%

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

   if 1.75e-188 < n < 0.19

   1. Initial program 10.5%

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

      1. associate-*r/ 10.5%

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

      2. associate-/r/ 10.6%

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

      3. *-commutative 10.6%

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

      4. sub-neg 10.6%

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

      5. distribute-lft-in 10.6%

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

      6. fma-def 10.6%

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

      7. metadata-eval 10.6%

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

      8. metadata-eval 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in n around inf 3.3%

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

      1. associate-/l* 3.3%

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

      2. fma-neg 3.3%

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

      3. metadata-eval 3.3%

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

   6. Simplified 3.3%

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

   7. Taylor expanded in i around 0 67.4%

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

      1. *-commutative 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{-198}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-188}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.19:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]

Alternative 5: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -4.8 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.5:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -4.8e-198)
     t_0
     (if (<= n 4.4e-186)
       (/ 0.0 (/ i n))
       (if (<= n 0.5) (/ n (+ 0.01 (* i -0.005))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -4.8e-198) {
		tmp = t_0;
	} else if (n <= 4.4e-186) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.5) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -4.8e-198) {
		tmp = t_0;
	} else if (n <= 4.4e-186) {
		tmp = 0.0 / (i / n);
	} else if (n <= 0.5) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -4.8e-198:
		tmp = t_0
	elif n <= 4.4e-186:
		tmp = 0.0 / (i / n)
	elif n <= 0.5:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -4.8e-198)
		tmp = t_0;
	elseif (n <= 4.4e-186)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 0.5)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.8e-198], t$95$0, If[LessEqual[n, 4.4e-186], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.5], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.79999999999999973e-198 or 0.5 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 22.9%

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

      2. sub-neg 22.9%

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

      3. metadata-eval 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in n around inf 34.2%

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

      1. associate-/l* 34.2%

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

      2. expm1-def 86.4%

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

   6. Simplified 86.4%

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

   if -4.79999999999999973e-198 < n < 4.40000000000000026e-186

   1. Initial program 55.5%

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

      1. associate-*r/ 55.5%

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

      2. sub-neg 55.5%

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

      3. distribute-lft-in 55.5%

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

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in i around 0 72.1%

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

   if 4.40000000000000026e-186 < n < 0.5

   1. Initial program 10.5%

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

      1. associate-*r/ 10.5%

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

      2. associate-/r/ 10.6%

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

      3. *-commutative 10.6%

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

      4. sub-neg 10.6%

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

      5. distribute-lft-in 10.6%

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

      6. fma-def 10.6%

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

      7. metadata-eval 10.6%

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

      8. metadata-eval 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in n around inf 3.3%

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

      1. associate-/l* 3.3%

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

      2. fma-neg 3.3%

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

      3. metadata-eval 3.3%

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

   6. Simplified 3.3%

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

   7. Taylor expanded in i around 0 67.4%

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

      1. *-commutative 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-198}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 0.5:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i}{\mathsf{expm1}\left(i\right)}\\ \mathbf{if}\;n \leq -5.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{n \cdot 100}{t_0}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-183}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ i (expm1 i))))
   (if (<= n -5.9e-198)
     (/ (* n 100.0) t_0)
     (if (<= n 1.1e-183)
       (/ 0.0 (/ i n))
       (if (<= n 4.4e-5) (/ n (+ 0.01 (* i -0.005))) (* 100.0 (/ n t_0)))))))

C

double code(double i, double n) {
	double t_0 = i / expm1(i);
	double tmp;
	if (n <= -5.9e-198) {
		tmp = (n * 100.0) / t_0;
	} else if (n <= 1.1e-183) {
		tmp = 0.0 / (i / n);
	} else if (n <= 4.4e-5) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 100.0 * (n / t_0);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = i / Math.expm1(i);
	double tmp;
	if (n <= -5.9e-198) {
		tmp = (n * 100.0) / t_0;
	} else if (n <= 1.1e-183) {
		tmp = 0.0 / (i / n);
	} else if (n <= 4.4e-5) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 100.0 * (n / t_0);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = i / math.expm1(i)
	tmp = 0
	if n <= -5.9e-198:
		tmp = (n * 100.0) / t_0
	elif n <= 1.1e-183:
		tmp = 0.0 / (i / n)
	elif n <= 4.4e-5:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = 100.0 * (n / t_0)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(i / expm1(i))
	tmp = 0.0
	if (n <= -5.9e-198)
		tmp = Float64(Float64(n * 100.0) / t_0);
	elseif (n <= 1.1e-183)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 4.4e-5)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(100.0 * Float64(n / t_0));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -5.9e-198], N[(N[(n * 100.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[n, 1.1e-183], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.4e-5], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.89999999999999974e-198

   1. Initial program 22.4%

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

      1. associate-*r/ 22.4%

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

      2. associate-/r/ 22.8%

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

      3. *-commutative 22.8%

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

      4. sub-neg 22.8%

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

      5. distribute-lft-in 22.7%

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

      6. fma-def 22.8%

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

      7. metadata-eval 22.8%

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

      8. metadata-eval 22.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in n around inf 28.0%

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

      1. associate-/l* 28.0%

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

      2. fma-neg 28.0%

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

      3. metadata-eval 28.0%

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

   6. Simplified 28.0%

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

   7. Taylor expanded in i around inf 28.0%

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

      1. sub-neg 28.0%

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

      2. metadata-eval 28.0%

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

      3. metadata-eval 28.0%

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

      4. distribute-lft-in 28.0%

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

      5. metadata-eval 28.0%

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

      6. sub-neg 28.0%

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

      7. expm1-def 82.2%

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

   9. Simplified 82.2%

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

       1. frac-2neg 82.2%

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

       2. div-inv 82.3%

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

       3. metadata-eval 82.3%

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

       4. metadata-eval 82.3%

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

       5. distribute-neg-frac 82.3%

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

       6. frac-2neg 82.3%

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

       7. clear-num 82.3%

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

       8. associate-/l* 82.3%

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

       9. distribute-neg-frac 82.3%

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

       10. metadata-eval 82.3%

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

   11. Applied egg-rr 82.3%

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

       1. associate-*r/ 82.4%

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

       2. distribute-lft-neg-out 82.4%

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

       3. distribute-rgt-neg-in 82.4%

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

       4. metadata-eval 82.4%

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

       5. *-commutative 82.4%

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

   13. Simplified 82.4%

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

   if -5.89999999999999974e-198 < n < 1.1e-183

   1. Initial program 55.5%

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

      1. associate-*r/ 55.5%

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

      2. sub-neg 55.5%

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

      3. distribute-lft-in 55.5%

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

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in i around 0 72.1%

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

   if 1.1e-183 < n < 4.3999999999999999e-5

   1. Initial program 10.5%

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

      1. associate-*r/ 10.5%

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

      2. associate-/r/ 10.6%

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

      3. *-commutative 10.6%

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

      4. sub-neg 10.6%

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

      5. distribute-lft-in 10.6%

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

      6. fma-def 10.6%

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

      7. metadata-eval 10.6%

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

      8. metadata-eval 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in n around inf 3.3%

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

      1. associate-/l* 3.3%

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

      2. fma-neg 3.3%

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

      3. metadata-eval 3.3%

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

   6. Simplified 3.3%

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

   7. Taylor expanded in i around 0 67.4%

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

      1. *-commutative 67.4%

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

   9. Simplified 67.4%

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

   if 4.3999999999999999e-5 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 23.1%

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

      2. sub-neg 23.1%

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

      3. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in n around inf 43.9%

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

      1. associate-/l* 43.9%

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

      2. expm1-def 92.8%

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

   6. Simplified 92.8%

      \[\leadsto \color{blue}{100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-183}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 7: 68.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{if}\;n \leq -3.5 \cdot 10^{+210}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{n \cdot n} - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-188}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.3:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ n (+ 0.01 (* i -0.005)))))
   (if (<= n -3.5e+210)
     (*
      n
      (+
       100.0
       (*
        100.0
        (+
         (* i (- 0.5 (/ 0.5 n)))
         (*
          (* i i)
          (+
           0.16666666666666666
           (- (/ 0.3333333333333333 (* n n)) (/ 0.5 n))))))))
     (if (<= n -4.5e-198)
       t_0
       (if (<= n 1.75e-188)
         (/ 0.0 (/ i n))
         (if (<= n 3.3) t_0 (/ (* n (* i (+ 100.0 (* i 50.0)))) i)))))))

C

double code(double i, double n) {
	double t_0 = n / (0.01 + (i * -0.005));
	double tmp;
	if (n <= -3.5e+210) {
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * (0.16666666666666666 + ((0.3333333333333333 / (n * n)) - (0.5 / n)))))));
	} else if (n <= -4.5e-198) {
		tmp = t_0;
	} else if (n <= 1.75e-188) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.3) {
		tmp = t_0;
	} else {
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n / (0.01d0 + (i * (-0.005d0)))
    if (n <= (-3.5d+210)) then
        tmp = n * (100.0d0 + (100.0d0 * ((i * (0.5d0 - (0.5d0 / n))) + ((i * i) * (0.16666666666666666d0 + ((0.3333333333333333d0 / (n * n)) - (0.5d0 / n)))))))
    else if (n <= (-4.5d-198)) then
        tmp = t_0
    else if (n <= 1.75d-188) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 3.3d0) then
        tmp = t_0
    else
        tmp = (n * (i * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n / (0.01 + (i * -0.005));
	double tmp;
	if (n <= -3.5e+210) {
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * (0.16666666666666666 + ((0.3333333333333333 / (n * n)) - (0.5 / n)))))));
	} else if (n <= -4.5e-198) {
		tmp = t_0;
	} else if (n <= 1.75e-188) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.3) {
		tmp = t_0;
	} else {
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n / (0.01 + (i * -0.005))
	tmp = 0
	if n <= -3.5e+210:
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * (0.16666666666666666 + ((0.3333333333333333 / (n * n)) - (0.5 / n)))))))
	elif n <= -4.5e-198:
		tmp = t_0
	elif n <= 1.75e-188:
		tmp = 0.0 / (i / n)
	elif n <= 3.3:
		tmp = t_0
	else:
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n / Float64(0.01 + Float64(i * -0.005)))
	tmp = 0.0
	if (n <= -3.5e+210)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(Float64(i * Float64(0.5 - Float64(0.5 / n))) + Float64(Float64(i * i) * Float64(0.16666666666666666 + Float64(Float64(0.3333333333333333 / Float64(n * n)) - Float64(0.5 / n))))))));
	elseif (n <= -4.5e-198)
		tmp = t_0;
	elseif (n <= 1.75e-188)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 3.3)
		tmp = t_0;
	else
		tmp = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n / (0.01 + (i * -0.005));
	tmp = 0.0;
	if (n <= -3.5e+210)
		tmp = n * (100.0 + (100.0 * ((i * (0.5 - (0.5 / n))) + ((i * i) * (0.16666666666666666 + ((0.3333333333333333 / (n * n)) - (0.5 / n)))))));
	elseif (n <= -4.5e-198)
		tmp = t_0;
	elseif (n <= 1.75e-188)
		tmp = 0.0 / (i / n);
	elseif (n <= 3.3)
		tmp = t_0;
	else
		tmp = (n * (i * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.5e+210], N[(n * N[(100.0 + N[(100.0 * N[(N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -4.5e-198], t$95$0, If[LessEqual[n, 1.75e-188], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3], t$95$0, N[(N[(n * N[(i * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -3.5e210

   1. Initial program 15.5%

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

      1. associate-*r/ 15.5%

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

      2. associate-/r/ 16.3%

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

      3. *-commutative 16.3%

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

      4. sub-neg 16.3%

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

      5. distribute-lft-in 16.3%

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

      6. fma-def 16.3%

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

      7. metadata-eval 16.3%

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

      8. metadata-eval 16.3%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in i around 0 75.4%

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

      1. distribute-lft-out 75.4%

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

      2. associate-*r/ 75.4%

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

      3. metadata-eval 75.4%

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

      4. unpow2 75.4%

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

      5. associate--l+ 75.4%

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

      6. associate-*r/ 75.4%

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

      7. metadata-eval 75.4%

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

      8. unpow2 75.4%

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

      9. associate-*r/ 75.4%

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

      10. metadata-eval 75.4%

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

   6. Simplified 75.4%

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

   if -3.5e210 < n < -4.4999999999999998e-198 or 1.75e-188 < n < 3.2999999999999998

   1. Initial program 19.4%

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

      1. associate-*r/ 19.4%

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

      2. associate-/r/ 19.5%

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

      3. *-commutative 19.5%

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

      4. sub-neg 19.5%

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

      5. distribute-lft-in 19.5%

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

      6. fma-def 19.5%

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

      7. metadata-eval 19.5%

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

      8. metadata-eval 19.5%

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

   3. Simplified 19.5%

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

   4. Taylor expanded in n around inf 15.3%

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

      1. associate-/l* 15.3%

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

      2. fma-neg 15.3%

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

      3. metadata-eval 15.3%

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

   6. Simplified 15.3%

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

   7. Taylor expanded in i around 0 67.0%

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

      1. *-commutative 67.0%

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

   9. Simplified 67.0%

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

   if -4.4999999999999998e-198 < n < 1.75e-188

   1. Initial program 55.5%

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

      1. associate-*r/ 55.5%

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

      2. sub-neg 55.5%

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

      3. distribute-lft-in 55.5%

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

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in i around 0 72.1%

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

   if 3.2999999999999998 < n

   1. Initial program 22.7%

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

      1. associate-*r/ 22.8%

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

      2. associate-/r/ 23.3%

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

      3. *-commutative 23.3%

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

      4. sub-neg 23.3%

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

      5. distribute-lft-in 23.3%

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

      6. fma-def 23.3%

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

      7. metadata-eval 23.3%

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

      8. metadata-eval 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in i around 0 74.5%

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

      1. distribute-lft-out 74.5%

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

      2. unpow2 74.5%

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

      3. associate-*r/ 74.5%

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

      4. metadata-eval 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in n around inf 74.5%

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

      1. *-commutative 74.5%

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

      2. unpow2 74.5%

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

      3. associate-*r* 74.5%

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

   9. Simplified 74.5%

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

       1. associate-*r/ 78.3%

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

       2. distribute-rgt-in 78.3%

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

       3. associate-*l* 78.3%

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

       4. distribute-lft-out 78.3%

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

       5. associate-*l* 78.3%

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

       6. metadata-eval 78.3%

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

   11. Applied egg-rr 78.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.5 \cdot 10^{+210}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right) + \left(i \cdot i\right) \cdot \left(0.16666666666666666 + \left(\frac{0.3333333333333333}{n \cdot n} - \frac{0.5}{n}\right)\right)\right)\right)\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-188}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.3:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]

Alternative 8: 67.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{0.01 + i \cdot -0.005}\\ t_1 := \frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{if}\;n \leq -3.6 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-188}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.3:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ n (+ 0.01 (* i -0.005))))
        (t_1 (/ (* n (* i (+ 100.0 (* i 50.0)))) i)))
   (if (<= n -3.6e+210)
     t_1
     (if (<= n -5.2e-198)
       t_0
       (if (<= n 2.25e-188) (/ 0.0 (/ i n)) (if (<= n 3.3) t_0 t_1))))))

C

double code(double i, double n) {
	double t_0 = n / (0.01 + (i * -0.005));
	double t_1 = (n * (i * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -3.6e+210) {
		tmp = t_1;
	} else if (n <= -5.2e-198) {
		tmp = t_0;
	} else if (n <= 2.25e-188) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.3) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = n / (0.01d0 + (i * (-0.005d0)))
    t_1 = (n * (i * (100.0d0 + (i * 50.0d0)))) / i
    if (n <= (-3.6d+210)) then
        tmp = t_1
    else if (n <= (-5.2d-198)) then
        tmp = t_0
    else if (n <= 2.25d-188) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 3.3d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n / (0.01 + (i * -0.005));
	double t_1 = (n * (i * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -3.6e+210) {
		tmp = t_1;
	} else if (n <= -5.2e-198) {
		tmp = t_0;
	} else if (n <= 2.25e-188) {
		tmp = 0.0 / (i / n);
	} else if (n <= 3.3) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n / (0.01 + (i * -0.005))
	t_1 = (n * (i * (100.0 + (i * 50.0)))) / i
	tmp = 0
	if n <= -3.6e+210:
		tmp = t_1
	elif n <= -5.2e-198:
		tmp = t_0
	elif n <= 2.25e-188:
		tmp = 0.0 / (i / n)
	elif n <= 3.3:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n / Float64(0.01 + Float64(i * -0.005)))
	t_1 = Float64(Float64(n * Float64(i * Float64(100.0 + Float64(i * 50.0)))) / i)
	tmp = 0.0
	if (n <= -3.6e+210)
		tmp = t_1;
	elseif (n <= -5.2e-198)
		tmp = t_0;
	elseif (n <= 2.25e-188)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 3.3)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n / (0.01 + (i * -0.005));
	t_1 = (n * (i * (100.0 + (i * 50.0)))) / i;
	tmp = 0.0;
	if (n <= -3.6e+210)
		tmp = t_1;
	elseif (n <= -5.2e-198)
		tmp = t_0;
	elseif (n <= 2.25e-188)
		tmp = 0.0 / (i / n);
	elseif (n <= 3.3)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(n * N[(i * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -3.6e+210], t$95$1, If[LessEqual[n, -5.2e-198], t$95$0, If[LessEqual[n, 2.25e-188], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.3], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.6000000000000003e210 or 3.2999999999999998 < n

   1. Initial program 20.7%

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

      1. associate-*r/ 20.7%

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

      2. associate-/r/ 21.4%

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

      3. *-commutative 21.4%

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

      4. sub-neg 21.4%

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

      5. distribute-lft-in 21.4%

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

      6. fma-def 21.4%

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

      7. metadata-eval 21.4%

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

      8. metadata-eval 21.4%

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

   3. Simplified 21.4%

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

   4. Taylor expanded in i around 0 72.5%

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

      1. distribute-lft-out 72.5%

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

      2. unpow2 72.5%

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

      3. associate-*r/ 72.5%

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

      4. metadata-eval 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in n around inf 72.5%

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

      1. *-commutative 72.5%

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

      2. unpow2 72.5%

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

      3. associate-*r* 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r/ 77.1%

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

       2. distribute-rgt-in 77.1%

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

       3. associate-*l* 77.1%

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

       4. distribute-lft-out 77.1%

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

       5. associate-*l* 77.1%

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

       6. metadata-eval 77.1%

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

   11. Applied egg-rr 77.1%

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

   if -3.6000000000000003e210 < n < -5.20000000000000014e-198 or 2.24999999999999997e-188 < n < 3.2999999999999998

   1. Initial program 19.4%

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

      1. associate-*r/ 19.4%

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

      2. associate-/r/ 19.5%

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

      3. *-commutative 19.5%

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

      4. sub-neg 19.5%

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

      5. distribute-lft-in 19.5%

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

      6. fma-def 19.5%

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

      7. metadata-eval 19.5%

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

      8. metadata-eval 19.5%

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

   3. Simplified 19.5%

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

   4. Taylor expanded in n around inf 15.3%

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

      1. associate-/l* 15.3%

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

      2. fma-neg 15.3%

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

      3. metadata-eval 15.3%

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

   6. Simplified 15.3%

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

   7. Taylor expanded in i around 0 67.0%

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

      1. *-commutative 67.0%

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

   9. Simplified 67.0%

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

   if -5.20000000000000014e-198 < n < 2.24999999999999997e-188

   1. Initial program 55.5%

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

      1. associate-*r/ 55.5%

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

      2. sub-neg 55.5%

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

      3. distribute-lft-in 55.5%

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

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in i around 0 72.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.6 \cdot 10^{+210}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-188}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 3.3:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(i \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]

Alternative 9: 64.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{if}\;n \leq -4.5 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-187}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ n (+ 0.01 (* i -0.005)))))
   (if (<= n -4.5e-198)
     t_0
     (if (<= n 1.1e-187)
       (/ 0.0 (/ i n))
       (if (<= n 5.8e-6) t_0 (+ (* n 100.0) (* 50.0 (* i n))))))))

C

double code(double i, double n) {
	double t_0 = n / (0.01 + (i * -0.005));
	double tmp;
	if (n <= -4.5e-198) {
		tmp = t_0;
	} else if (n <= 1.1e-187) {
		tmp = 0.0 / (i / n);
	} else if (n <= 5.8e-6) {
		tmp = t_0;
	} else {
		tmp = (n * 100.0) + (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) :: t_0
    real(8) :: tmp
    t_0 = n / (0.01d0 + (i * (-0.005d0)))
    if (n <= (-4.5d-198)) then
        tmp = t_0
    else if (n <= 1.1d-187) then
        tmp = 0.0d0 / (i / n)
    else if (n <= 5.8d-6) then
        tmp = t_0
    else
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n / (0.01 + (i * -0.005));
	double tmp;
	if (n <= -4.5e-198) {
		tmp = t_0;
	} else if (n <= 1.1e-187) {
		tmp = 0.0 / (i / n);
	} else if (n <= 5.8e-6) {
		tmp = t_0;
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n / (0.01 + (i * -0.005))
	tmp = 0
	if n <= -4.5e-198:
		tmp = t_0
	elif n <= 1.1e-187:
		tmp = 0.0 / (i / n)
	elif n <= 5.8e-6:
		tmp = t_0
	else:
		tmp = (n * 100.0) + (50.0 * (i * n))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n / Float64(0.01 + Float64(i * -0.005)))
	tmp = 0.0
	if (n <= -4.5e-198)
		tmp = t_0;
	elseif (n <= 1.1e-187)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 5.8e-6)
		tmp = t_0;
	else
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n / (0.01 + (i * -0.005));
	tmp = 0.0;
	if (n <= -4.5e-198)
		tmp = t_0;
	elseif (n <= 1.1e-187)
		tmp = 0.0 / (i / n);
	elseif (n <= 5.8e-6)
		tmp = t_0;
	else
		tmp = (n * 100.0) + (50.0 * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.5e-198], t$95$0, If[LessEqual[n, 1.1e-187], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-6], t$95$0, N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.4999999999999998e-198 or 1.10000000000000004e-187 < n < 5.8000000000000004e-6

   1. Initial program 18.8%

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

      1. associate-*r/ 18.8%

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

      2. associate-/r/ 19.1%

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

      3. *-commutative 19.1%

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

      4. sub-neg 19.1%

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

      5. distribute-lft-in 19.1%

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

      6. fma-def 19.1%

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

      7. metadata-eval 19.1%

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

      8. metadata-eval 19.1%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in n around inf 20.6%

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

      1. associate-/l* 20.6%

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

      2. fma-neg 20.5%

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

      3. metadata-eval 20.5%

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

   6. Simplified 20.5%

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

   7. Taylor expanded in i around 0 65.0%

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

      1. *-commutative 65.0%

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

   9. Simplified 65.0%

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

   if -4.4999999999999998e-198 < n < 1.10000000000000004e-187

   1. Initial program 55.5%

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

      1. associate-*r/ 55.5%

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

      2. sub-neg 55.5%

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

      3. distribute-lft-in 55.5%

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

      4. metadata-eval 55.5%

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

      5. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in i around 0 72.1%

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

   if 5.8000000000000004e-6 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 23.1%

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

      2. sub-neg 23.1%

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

      3. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in n around inf 43.9%

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

      1. associate-/l* 43.9%

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

      2. expm1-def 92.8%

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

   6. Simplified 92.8%

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

   7. Taylor expanded in i around 0 72.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{-187}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 10: 61.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.75:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{50 \cdot \left(i \cdot i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.75) (/ n (+ 0.01 (* i -0.005))) (* n (/ (* 50.0 (* i i)) i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.75) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * ((50.0 * (i * i)) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.75d0) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = n * ((50.0d0 * (i * i)) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.75) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = n * ((50.0 * (i * i)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.75:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = n * ((50.0 * (i * i)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.75)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(n * Float64(Float64(50.0 * Float64(i * i)) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.75)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = n * ((50.0 * (i * i)) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.75], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(50.0 * N[(i * i), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.75

   1. Initial program 17.4%

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

      1. associate-*r/ 17.5%

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

      2. associate-/r/ 17.9%

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

      3. *-commutative 17.9%

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

      4. sub-neg 17.9%

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

      5. distribute-lft-in 17.8%

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

      6. fma-def 17.9%

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

      7. metadata-eval 17.9%

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

      8. metadata-eval 17.9%

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

   3. Simplified 17.9%

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

   4. Taylor expanded in n around inf 24.8%

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

      1. associate-/l* 24.8%

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

      2. fma-neg 24.8%

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

      3. metadata-eval 24.8%

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

   6. Simplified 24.8%

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

   7. Taylor expanded in i around 0 72.5%

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

      1. *-commutative 72.5%

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

   9. Simplified 72.5%

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

   if 1.75 < i

   1. Initial program 44.9%

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

      1. associate-*r/ 44.9%

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

      2. associate-/r/ 45.2%

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

      3. *-commutative 45.2%

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

      4. sub-neg 45.2%

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

      5. distribute-lft-in 45.1%

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

      6. fma-def 45.2%

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

      7. metadata-eval 45.2%

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

      8. metadata-eval 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in i around 0 32.3%

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

      1. distribute-lft-out 32.3%

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

      2. unpow2 32.3%

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

      3. associate-*r/ 32.3%

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

      4. metadata-eval 32.3%

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

   6. Simplified 32.3%

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

   7. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. unpow2 32.8%

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

      3. associate-*r* 32.8%

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

   9. Simplified 32.8%

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

   10. Taylor expanded in i around inf 32.8%

       \[\leadsto n \cdot \frac{\color{blue}{50 \cdot {i}^{2}}}{i} \]
   11. Step-by-step derivation

       1. unpow2 32.8%

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

   12. Simplified 32.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.75:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{50 \cdot \left(i \cdot i\right)}{i}\\ \end{array} \]

Alternative 11: 62.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.0082:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 0.0082)
   (/ n (+ 0.01 (* i -0.005)))
   (+ (* n 100.0) (* 50.0 (* i n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 0.0082) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = (n * 100.0) + (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 (n <= 0.0082d0) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else
        tmp = (n * 100.0d0) + (50.0d0 * (i * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 0.0082) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = (n * 100.0) + (50.0 * (i * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 0.0082:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = (n * 100.0) + (50.0 * (i * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 0.0082)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(Float64(n * 100.0) + Float64(50.0 * Float64(i * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 0.0082)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = (n * 100.0) + (50.0 * (i * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 0.0082], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] + N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.00820000000000000069

   1. Initial program 24.7%

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

      1. associate-*r/ 24.7%

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

      2. associate-/r/ 25.0%

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

      3. *-commutative 25.0%

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

      4. sub-neg 25.0%

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

      5. distribute-lft-in 25.0%

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

      6. fma-def 25.0%

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

      7. metadata-eval 25.0%

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

      8. metadata-eval 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in n around inf 24.2%

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

      1. associate-/l* 24.2%

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

      2. fma-neg 24.1%

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

      3. metadata-eval 24.1%

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

   6. Simplified 24.1%

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

   7. Taylor expanded in i around 0 62.0%

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

      1. *-commutative 62.0%

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

   9. Simplified 62.0%

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

   if 0.00820000000000000069 < n

   1. Initial program 22.4%

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

      1. associate-/r/ 23.1%

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

      2. sub-neg 23.1%

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

      3. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in n around inf 43.9%

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

      1. associate-/l* 43.9%

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

      2. expm1-def 92.8%

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

   6. Simplified 92.8%

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

   7. Taylor expanded in i around 0 72.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.0082:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + 50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 12: 60.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.75:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.75) (/ n (+ 0.01 (* i -0.005))) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.75) {
		tmp = n / (0.01 + (i * -0.005));
	} 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 <= 1.75d0) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    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 <= 1.75) {
		tmp = n / (0.01 + (i * -0.005));
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.75:
		tmp = n / (0.01 + (i * -0.005))
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.75)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	else
		tmp = Float64(50.0 * Float64(i * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.75)
		tmp = n / (0.01 + (i * -0.005));
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.75], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.75

   1. Initial program 17.4%

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

      1. associate-*r/ 17.5%

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

      2. associate-/r/ 17.9%

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

      3. *-commutative 17.9%

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

      4. sub-neg 17.9%

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

      5. distribute-lft-in 17.8%

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

      6. fma-def 17.9%

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

      7. metadata-eval 17.9%

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

      8. metadata-eval 17.9%

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

   3. Simplified 17.9%

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

   4. Taylor expanded in n around inf 24.8%

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

      1. associate-/l* 24.8%

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

      2. fma-neg 24.8%

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

      3. metadata-eval 24.8%

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

   6. Simplified 24.8%

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

   7. Taylor expanded in i around 0 72.5%

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

      1. *-commutative 72.5%

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

   9. Simplified 72.5%

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

   if 1.75 < i

   1. Initial program 44.9%

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

      1. associate-*r/ 44.9%

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

      2. associate-/r/ 45.2%

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

      3. *-commutative 45.2%

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

      4. sub-neg 45.2%

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

      5. distribute-lft-in 45.1%

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

      6. fma-def 45.2%

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

      7. metadata-eval 45.2%

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

      8. metadata-eval 45.2%

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

   3. Simplified 45.2%

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

   4. Taylor expanded in i around 0 32.3%

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

      1. distribute-lft-out 32.3%

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

      2. unpow2 32.3%

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

      3. associate-*r/ 32.3%

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

      4. metadata-eval 32.3%

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

   6. Simplified 32.3%

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

   7. Taylor expanded in n around inf 32.8%

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

      1. *-commutative 32.8%

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

      2. unpow2 32.8%

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

      3. associate-*r* 32.8%

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

   9. Simplified 32.8%

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

   10. Taylor expanded in i around inf 31.5%

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

       1. *-commutative 31.5%

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

   12. Simplified 31.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.75:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 13: 54.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5.2 \cdot 10^{+25}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 5.2e+25) (* n 100.0) (* 50.0 (* i n))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 5.2e+25) {
		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 <= 5.2d+25) 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 <= 5.2e+25) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (i * n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 5.2e+25:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (i * n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 5.2e+25)
		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 <= 5.2e+25)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (i * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 5.2e+25], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(i * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.1999999999999997e25

   1. Initial program 17.6%

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

      1. associate-/r/ 18.0%

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

      2. sub-neg 18.0%

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

      3. metadata-eval 18.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in i around 0 66.7%

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

      1. *-commutative 66.7%

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

   6. Simplified 66.7%

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

   if 5.1999999999999997e25 < i

   1. Initial program 46.9%

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

      1. associate-*r/ 46.9%

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

      2. associate-/r/ 47.1%

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

      3. *-commutative 47.1%

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

      4. sub-neg 47.1%

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

      5. distribute-lft-in 47.1%

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

      6. fma-def 47.1%

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

      7. metadata-eval 47.1%

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

      8. metadata-eval 47.1%

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

   3. Simplified 47.1%

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

   4. Taylor expanded in i around 0 34.8%

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

      1. distribute-lft-out 34.9%

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

      2. unpow2 34.9%

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

      3. associate-*r/ 34.9%

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

      4. metadata-eval 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in n around inf 35.2%

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

      1. *-commutative 35.2%

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

      2. unpow2 35.2%

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

      3. associate-*r* 35.2%

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

   9. Simplified 35.2%

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

   10. Taylor expanded in i around inf 33.8%

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

       1. *-commutative 33.8%

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

   12. Simplified 33.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5.2 \cdot 10^{+25}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(i \cdot n\right)\\ \end{array} \]

Alternative 14: 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 24.1%

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

   1. associate-*r/ 24.1%

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

   2. associate-/r/ 24.5%

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

   3. *-commutative 24.5%

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

   4. sub-neg 24.5%

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

   5. distribute-lft-in 24.5%

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

   6. fma-def 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

3. Simplified 24.5%

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

4. Taylor expanded in i around 0 59.3%

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

   1. distribute-lft-out 59.3%

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

   2. unpow2 59.3%

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

   3. associate-*r/ 59.3%

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

   4. metadata-eval 59.3%

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

6. Simplified 59.3%

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

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 15: 49.5% 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 24.1%

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

   1. associate-/r/ 24.5%

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

   2. sub-neg 24.5%

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

   3. metadata-eval 24.5%

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

3. Simplified 24.5%

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

4. Taylor expanded in i around 0 53.0%

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

   1. *-commutative 53.0%

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

6. Simplified 53.0%

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

7. Final simplification 53.0%

   \[\leadsto n \cdot 100 \]

Developer target: 34.5% 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 2023297 
(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))))