Compound Interest

Percentage Accurate: 27.8% → 96.7%

Time: 18.7s

Alternatives: 18

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 27.8% 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: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 2e-173)
     (/ n (/ i (* (expm1 (* n (log1p (/ i n)))) 100.0)))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (/ 1.0 (/ (fma (* i 0.01) (+ -0.5 (/ 0.5 n)) 0.01) n))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 2e-173) {
		tmp = n / (i / (expm1((n * log1p((i / n)))) * 100.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 1.0 / (fma((i * 0.01), (-0.5 + (0.5 / n)), 0.01) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 2e-173)
		tmp = Float64(n / Float64(i / Float64(expm1(Float64(n * log1p(Float64(i / n)))) * 100.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(i * 0.01), Float64(-0.5 + Float64(0.5 / n)), 0.01) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-173], N[(n / N[(i / N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(i * 0.01), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + 0.01), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2.0000000000000001e-173

   1. Initial program 25.5%

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

      1. associate-*r/ 25.5%

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

      2. sub-neg 25.5%

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

      3. distribute-lft-in 25.5%

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

      4. metadata-eval 25.5%

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

      5. metadata-eval 25.5%

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

      6. fma-udef 25.5%

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

      7. associate-/r/ 25.5%

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

      8. *-commutative 25.5%

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

      9. clear-num 25.5%

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

      10. un-div-inv 25.5%

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

      11. fma-udef 25.5%

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

      12. metadata-eval 25.5%

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

      13. metadata-eval 25.5%

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

      14. distribute-lft-in 25.5%

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

      15. sub-neg 25.5%

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

      16. *-commutative 25.5%

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

   3. Applied egg-rr 97.6%

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

   if 2.0000000000000001e-173 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. fma-def 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. fma-udef 0.0%

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

      7. associate-/r/ 1.9%

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

      8. *-commutative 1.9%

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

      9. clear-num 1.9%

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

      10. un-div-inv 1.9%

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

      11. fma-udef 1.9%

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

      12. metadata-eval 1.9%

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

      13. metadata-eval 1.9%

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

      14. distribute-lft-in 1.9%

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

      15. sub-neg 1.9%

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

      16. *-commutative 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 99.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 99.9%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]
   7. Step-by-step derivation

      1. clear-num 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}}} \]

      2. inv-pow 99.9%

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}\right)}^{-1}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}}} \]

      2. fma-def 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) + 0.01}}{n}} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(\frac{\color{blue}{0.5 \cdot 1}}{n} + -0.5\right)\right) + 0.01}{n}} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(\color{blue}{0.5 \cdot \frac{1}{n}} + -0.5\right)\right) + 0.01}{n}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} + \color{blue}{\left(-0.5\right)}\right)\right) + 0.01}{n}} \]

      6. sub-neg 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} - 0.5\right)}\right) + 0.01}{n}} \]

      7. associate-*r* 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(0.01 \cdot i\right) \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)} + 0.01}{n}} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(i \cdot 0.01\right)} \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right) + 0.01}{n}} \]

      9. fma-def 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(i \cdot 0.01, 0.5 \cdot \frac{1}{n} - 0.5, 0.01\right)}}{n}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{0.5 \cdot \frac{1}{n} + \left(-0.5\right)}, 0.01\right)}{n}} \]

      11. associate-*r/ 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right), 0.01\right)}{n}} \]

      12. metadata-eval 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \frac{\color{blue}{0.5}}{n} + \left(-0.5\right), 0.01\right)}{n}} \]

      13. metadata-eval 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \frac{0.5}{n} + \color{blue}{-0.5}, 0.01\right)}{n}} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{-0.5 + \frac{0.5}{n}}, 0.01\right)}{n}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\frac{n}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot 100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}\\ \end{array} \]

Alternative 2: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 2e-173)
     (* (expm1 (* n (log1p (/ i n)))) (* 100.0 (/ n i)))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (/ 1.0 (/ (fma (* i 0.01) (+ -0.5 (/ 0.5 n)) 0.01) n))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 2e-173) {
		tmp = expm1((n * log1p((i / n)))) * (100.0 * (n / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 1.0 / (fma((i * 0.01), (-0.5 + (0.5 / n)), 0.01) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 2e-173)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(100.0 * Float64(n / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(i * 0.01), Float64(-0.5 + Float64(0.5 / n)), 0.01) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-173], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(i * 0.01), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + 0.01), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2.0000000000000001e-173

   1. Initial program 25.5%

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

      1. associate-*r/ 25.5%

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

      2. sub-neg 25.5%

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

      3. distribute-lft-in 25.5%

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

      4. metadata-eval 25.5%

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

      5. metadata-eval 25.5%

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

      6. fma-udef 25.5%

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

      7. associate-/r/ 25.5%

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

      8. *-commutative 25.5%

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

      9. expm1-log1p-u 23.0%

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

      10. expm1-udef 17.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\right)} - 1} \]

   3. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 68.7%

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

      2. expm1-log1p 95.7%

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

      3. *-commutative 95.7%

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

   5. Simplified 95.7%

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

   if 2.0000000000000001e-173 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. fma-def 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. fma-udef 0.0%

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

      7. associate-/r/ 1.9%

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

      8. *-commutative 1.9%

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

      9. clear-num 1.9%

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

      10. un-div-inv 1.9%

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

      11. fma-udef 1.9%

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

      12. metadata-eval 1.9%

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

      13. metadata-eval 1.9%

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

      14. distribute-lft-in 1.9%

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

      15. sub-neg 1.9%

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

      16. *-commutative 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 99.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 99.9%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]
   7. Step-by-step derivation

      1. clear-num 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}}} \]

      2. inv-pow 99.9%

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}\right)}^{-1}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}}} \]

      2. fma-def 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) + 0.01}}{n}} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(\frac{\color{blue}{0.5 \cdot 1}}{n} + -0.5\right)\right) + 0.01}{n}} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(\color{blue}{0.5 \cdot \frac{1}{n}} + -0.5\right)\right) + 0.01}{n}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} + \color{blue}{\left(-0.5\right)}\right)\right) + 0.01}{n}} \]

      6. sub-neg 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} - 0.5\right)}\right) + 0.01}{n}} \]

      7. associate-*r* 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(0.01 \cdot i\right) \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)} + 0.01}{n}} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(i \cdot 0.01\right)} \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right) + 0.01}{n}} \]

      9. fma-def 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(i \cdot 0.01, 0.5 \cdot \frac{1}{n} - 0.5, 0.01\right)}}{n}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{0.5 \cdot \frac{1}{n} + \left(-0.5\right)}, 0.01\right)}{n}} \]

      11. associate-*r/ 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right), 0.01\right)}{n}} \]

      12. metadata-eval 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \frac{\color{blue}{0.5}}{n} + \left(-0.5\right), 0.01\right)}{n}} \]

      13. metadata-eval 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \frac{0.5}{n} + \color{blue}{-0.5}, 0.01\right)}{n}} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{-0.5 + \frac{0.5}{n}}, 0.01\right)}{n}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(100 \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}\\ \end{array} \]

Alternative 3: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 2e-173)
     (* (expm1 (* n (log1p (/ i n)))) (/ (* n 100.0) i))
     (if (<= t_1 INFINITY)
       (* n (/ (+ (* t_0 100.0) -100.0) i))
       (/ 1.0 (/ (fma (* i 0.01) (+ -0.5 (/ 0.5 n)) 0.01) n))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 2e-173) {
		tmp = expm1((n * log1p((i / n)))) * ((n * 100.0) / i);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = 1.0 / (fma((i * 0.01), (-0.5 + (0.5 / n)), 0.01) / n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 2e-173)
		tmp = Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(Float64(n * 100.0) / i));
	elseif (t_1 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(t_0 * 100.0) + -100.0) / i));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(i * 0.01), Float64(-0.5 + Float64(0.5 / n)), 0.01) / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-173], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(n * N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(i * 0.01), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] + 0.01), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 2.0000000000000001e-173

   1. Initial program 25.5%

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

      1. associate-*r/ 25.5%

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

      2. sub-neg 25.5%

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

      3. distribute-lft-in 25.5%

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

      4. metadata-eval 25.5%

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

      5. metadata-eval 25.5%

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

      6. fma-udef 25.5%

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

      7. associate-/r/ 25.5%

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

      8. *-commutative 25.5%

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

      9. expm1-log1p-u 23.0%

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

      10. expm1-udef 17.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\right)} - 1} \]

   3. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 68.7%

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

      2. expm1-log1p 95.7%

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

      3. associate-*l/ 95.9%

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

   5. Simplified 95.9%

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

   if 2.0000000000000001e-173 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 99.7%

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

      1. associate-/r/ 99.8%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. fma-def 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. sub-neg 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. metadata-eval 0.0%

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

      6. fma-udef 0.0%

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

      7. associate-/r/ 1.9%

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

      8. *-commutative 1.9%

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

      9. clear-num 1.9%

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

      10. un-div-inv 1.9%

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

      11. fma-udef 1.9%

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

      12. metadata-eval 1.9%

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

      13. metadata-eval 1.9%

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

      14. distribute-lft-in 1.9%

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

      15. sub-neg 1.9%

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

      16. *-commutative 1.9%

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

   3. Applied egg-rr 1.9%

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

   4. Taylor expanded in i around 0 99.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 99.9%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 99.9%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]
   7. Step-by-step derivation

      1. clear-num 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}}} \]

      2. inv-pow 99.9%

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}\right)}^{-1}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}{n}}} \]

      2. fma-def 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) + 0.01}}{n}} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(\frac{\color{blue}{0.5 \cdot 1}}{n} + -0.5\right)\right) + 0.01}{n}} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(\color{blue}{0.5 \cdot \frac{1}{n}} + -0.5\right)\right) + 0.01}{n}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} + \color{blue}{\left(-0.5\right)}\right)\right) + 0.01}{n}} \]

      6. sub-neg 99.9%

         \[\leadsto \frac{1}{\frac{0.01 \cdot \left(i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} - 0.5\right)}\right) + 0.01}{n}} \]

      7. associate-*r* 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(0.01 \cdot i\right) \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)} + 0.01}{n}} \]

      8. *-commutative 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(i \cdot 0.01\right)} \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right) + 0.01}{n}} \]

      9. fma-def 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(i \cdot 0.01, 0.5 \cdot \frac{1}{n} - 0.5, 0.01\right)}}{n}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{0.5 \cdot \frac{1}{n} + \left(-0.5\right)}, 0.01\right)}{n}} \]

      11. associate-*r/ 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right), 0.01\right)}{n}} \]

      12. metadata-eval 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \frac{\color{blue}{0.5}}{n} + \left(-0.5\right), 0.01\right)}{n}} \]

      13. metadata-eval 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \frac{0.5}{n} + \color{blue}{-0.5}, 0.01\right)}{n}} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, \color{blue}{-0.5 + \frac{0.5}{n}}, 0.01\right)}{n}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 2 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n \cdot 100}{i}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;n \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(i \cdot 0.01, -0.5 + \frac{0.5}{n}, 0.01\right)}{n}}\\ \end{array} \]

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{-5} \lor \neg \left(i \leq 1.5 \cdot 10^{-7}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\left(0.01 + \left(\frac{i}{n} \cdot 0.005 + i \cdot -0.01\right)\right) - i \cdot -0.005}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -5e-5) (not (<= i 1.5e-7)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (/ n (- (+ 0.01 (+ (* (/ i n) 0.005) (* i -0.01))) (* i -0.005)))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -5e-5) || !(i <= 1.5e-7)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -5e-5) || !(i <= 1.5e-7)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -5e-5) or not (i <= 1.5e-7):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -5e-5) || !(i <= 1.5e-7))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(n / Float64(Float64(0.01 + Float64(Float64(Float64(i / n) * 0.005) + Float64(i * -0.01))) - Float64(i * -0.005)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -5e-5], N[Not[LessEqual[i, 1.5e-7]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(0.01 + N[(N[(N[(i / n), $MachinePrecision] * 0.005), $MachinePrecision] + N[(i * -0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -5.00000000000000024e-5 or 1.4999999999999999e-7 < i

   1. Initial program 48.3%

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

   2. Taylor expanded in n around inf 72.8%

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

      1. expm1-def 73.2%

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

   4. Simplified 73.2%

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

   if -5.00000000000000024e-5 < i < 1.4999999999999999e-7

   1. Initial program 11.6%

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

      1. associate-*r/ 11.6%

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

      2. sub-neg 11.6%

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

      3. distribute-lft-in 11.6%

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

      4. metadata-eval 11.6%

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

      5. metadata-eval 11.6%

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

      6. fma-udef 11.6%

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

      7. associate-/r/ 11.9%

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

      8. *-commutative 11.9%

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

      9. clear-num 11.9%

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

      10. un-div-inv 11.9%

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

      11. fma-udef 11.9%

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

      12. metadata-eval 11.9%

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

      13. metadata-eval 11.9%

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

      14. distribute-lft-in 11.9%

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

      15. sub-neg 11.9%

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

      16. *-commutative 11.9%

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

   3. Applied egg-rr 81.3%

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

   4. Taylor expanded in i around 0 93.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 93.7%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 93.7%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 93.7%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 93.7%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) + 0.01}} \]

      2. flip-+ 93.7%

         \[\leadsto \frac{n}{\color{blue}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - 0.01 \cdot 0.01}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}}} \]

      3. metadata-eval 93.7%

         \[\leadsto \frac{n}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - \color{blue}{0.0001}}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}} \]

   8. Applied egg-rr 93.7%

      \[\leadsto \frac{n}{\color{blue}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - 0.0001}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}}} \]

   9. Taylor expanded in n around 0 93.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5 \cdot 10^{-5} \lor \neg \left(i \leq 1.5 \cdot 10^{-7}\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\left(0.01 + \left(\frac{i}{n} \cdot 0.005 + i \cdot -0.01\right)\right) - i \cdot -0.005}\\ \end{array} \]

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{-5} \lor \neg \left(i \leq 6.2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{n \cdot 100}{i} \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\left(0.01 + \left(\frac{i}{n} \cdot 0.005 + i \cdot -0.01\right)\right) - i \cdot -0.005}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -3.5e-5) (not (<= i 6.2e-7)))
   (* (/ (* n 100.0) i) (expm1 i))
   (/ n (- (+ 0.01 (+ (* (/ i n) 0.005) (* i -0.01))) (* i -0.005)))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -3.5e-5) || !(i <= 6.2e-7)) {
		tmp = ((n * 100.0) / i) * expm1(i);
	} else {
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -3.5e-5) || !(i <= 6.2e-7)) {
		tmp = ((n * 100.0) / i) * Math.expm1(i);
	} else {
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -3.5e-5) or not (i <= 6.2e-7):
		tmp = ((n * 100.0) / i) * math.expm1(i)
	else:
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -3.5e-5) || !(i <= 6.2e-7))
		tmp = Float64(Float64(Float64(n * 100.0) / i) * expm1(i));
	else
		tmp = Float64(n / Float64(Float64(0.01 + Float64(Float64(Float64(i / n) * 0.005) + Float64(i * -0.01))) - Float64(i * -0.005)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -3.5e-5], N[Not[LessEqual[i, 6.2e-7]], $MachinePrecision]], N[(N[(N[(n * 100.0), $MachinePrecision] / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(0.01 + N[(N[(N[(i / n), $MachinePrecision] * 0.005), $MachinePrecision] + N[(i * -0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -3.4999999999999997e-5 or 6.1999999999999999e-7 < i

   1. Initial program 48.3%

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

      1. associate-*r/ 48.4%

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

      2. sub-neg 48.4%

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

      3. distribute-lft-in 48.3%

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

      4. metadata-eval 48.3%

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

      5. metadata-eval 48.3%

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

      6. fma-udef 48.4%

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

      7. associate-/r/ 48.5%

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

      8. *-commutative 48.5%

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

      9. expm1-log1p-u 44.0%

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

      10. expm1-udef 29.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(n \cdot \frac{\mathsf{fma}\left(100, {\left(1 + \frac{i}{n}\right)}^{n}, -100\right)}{i}\right)} - 1} \]

   3. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \left(\frac{n}{i} \cdot 100\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 59.5%

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

      2. expm1-log1p 83.7%

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

      3. associate-*l/ 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in n around inf 73.4%

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

   if -3.4999999999999997e-5 < i < 6.1999999999999999e-7

   1. Initial program 11.6%

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

      1. associate-*r/ 11.6%

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

      2. sub-neg 11.6%

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

      3. distribute-lft-in 11.6%

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

      4. metadata-eval 11.6%

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

      5. metadata-eval 11.6%

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

      6. fma-udef 11.6%

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

      7. associate-/r/ 11.9%

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

      8. *-commutative 11.9%

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

      9. clear-num 11.9%

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

      10. un-div-inv 11.9%

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

      11. fma-udef 11.9%

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

      12. metadata-eval 11.9%

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

      13. metadata-eval 11.9%

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

      14. distribute-lft-in 11.9%

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

      15. sub-neg 11.9%

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

      16. *-commutative 11.9%

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

   3. Applied egg-rr 81.3%

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

   4. Taylor expanded in i around 0 93.7%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 93.7%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 93.7%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 93.7%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 93.7%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 93.7%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) + 0.01}} \]

      2. flip-+ 93.7%

         \[\leadsto \frac{n}{\color{blue}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - 0.01 \cdot 0.01}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}}} \]

      3. metadata-eval 93.7%

         \[\leadsto \frac{n}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - \color{blue}{0.0001}}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}} \]

   8. Applied egg-rr 93.7%

      \[\leadsto \frac{n}{\color{blue}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - 0.0001}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}}} \]

   9. Taylor expanded in n around 0 93.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -3.5 \cdot 10^{-5} \lor \neg \left(i \leq 6.2 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{n \cdot 100}{i} \cdot \mathsf{expm1}\left(i\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\left(0.01 + \left(\frac{i}{n} \cdot 0.005 + i \cdot -0.01\right)\right) - i \cdot -0.005}\\ \end{array} \]

Alternative 6: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.85 \cdot 10^{-5} \lor \neg \left(n \leq 1.62\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.85e-5) (not (<= n 1.62)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ n (+ 0.01 (* 0.01 (* i (+ -0.5 (/ 0.5 n))))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.85e-5) || !(n <= 1.62)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.85e-5) || !(n <= 1.62)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.85e-5) or not (n <= 1.62):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.85e-5) || !(n <= 1.62))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(-0.5 + Float64(0.5 / n))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.85e-5], N[Not[LessEqual[n, 1.62]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.8500000000000002e-5 or 1.6200000000000001 < n

   1. Initial program 23.5%

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

   2. Taylor expanded in n around inf 45.0%

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

      1. *-commutative 45.0%

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

      2. associate-/l* 45.0%

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

      3. expm1-def 94.9%

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

   4. Simplified 94.9%

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

   if -2.8500000000000002e-5 < n < 1.6200000000000001

   1. Initial program 34.0%

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

      1. associate-*r/ 34.0%

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

      2. sub-neg 34.0%

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

      3. distribute-lft-in 34.0%

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

      4. metadata-eval 34.0%

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

      5. metadata-eval 34.0%

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

      6. fma-udef 34.0%

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

      7. associate-/r/ 34.2%

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

      8. *-commutative 34.2%

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

      9. clear-num 34.2%

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

      10. un-div-inv 34.2%

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

      11. fma-udef 34.1%

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

      12. metadata-eval 34.1%

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

      13. metadata-eval 34.1%

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

      14. distribute-lft-in 34.2%

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

      15. sub-neg 34.2%

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

      16. *-commutative 34.2%

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

   3. Applied egg-rr 89.7%

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

   4. Taylor expanded in i around 0 83.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 83.9%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}\right)} \]

      2. associate-*r/ 83.9%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)\right)} \]

      3. metadata-eval 83.9%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)\right)} \]

      4. metadata-eval 83.9%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)} \]

   6. Simplified 83.9%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.85 \cdot 10^{-5} \lor \neg \left(n \leq 1.62\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 7: 73.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+131} \lor \neg \left(n \leq 2\right):\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.25e+131) (not (<= n 2.0)))
   (*
    n
    (+
     100.0
     (*
      100.0
      (+
       (*
        (* i i)
        (+ (/ 0.3333333333333333 (* n n)) (+ 0.16666666666666666 (/ -0.5 n))))
       (* i (- 0.5 (/ 0.5 n)))))))
   (/ n (+ 0.01 (* 0.01 (* i (+ -0.5 (/ 0.5 n))))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.25e+131) || !(n <= 2.0)) {
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	} else {
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.25d+131)) .or. (.not. (n <= 2.0d0))) then
        tmp = n * (100.0d0 + (100.0d0 * (((i * i) * ((0.3333333333333333d0 / (n * n)) + (0.16666666666666666d0 + ((-0.5d0) / n)))) + (i * (0.5d0 - (0.5d0 / n))))))
    else
        tmp = n / (0.01d0 + (0.01d0 * (i * ((-0.5d0) + (0.5d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.25e+131) || !(n <= 2.0)) {
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	} else {
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.25e+131) or not (n <= 2.0):
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))))
	else:
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.25e+131) || !(n <= 2.0))
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(Float64(Float64(i * i) * Float64(Float64(0.3333333333333333 / Float64(n * n)) + Float64(0.16666666666666666 + Float64(-0.5 / n)))) + Float64(i * Float64(0.5 - Float64(0.5 / n)))))));
	else
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(-0.5 + Float64(0.5 / n))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.25e+131) || ~((n <= 2.0)))
		tmp = n * (100.0 + (100.0 * (((i * i) * ((0.3333333333333333 / (n * n)) + (0.16666666666666666 + (-0.5 / n)))) + (i * (0.5 - (0.5 / n))))));
	else
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.25e+131], N[Not[LessEqual[n, 2.0]], $MachinePrecision]], N[(n * N[(100.0 + N[(100.0 * N[(N[(N[(i * i), $MachinePrecision] * N[(N[(0.3333333333333333 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.24999999999999999e131 or 2 < n

   1. Initial program 22.3%

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

      1. associate-/r/ 22.6%

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

      2. associate-*r* 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-*r/ 22.7%

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

      5. sub-neg 22.7%

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

      6. distribute-lft-in 22.7%

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

      7. fma-def 22.7%

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

      8. metadata-eval 22.7%

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

      9. metadata-eval 22.7%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in i around 0 67.2%

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

      1. +-commutative 67.2%

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

      2. distribute-lft-out 67.2%

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

   6. Simplified 67.2%

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

   if -1.24999999999999999e131 < n < 2

   1. Initial program 32.0%

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

      1. associate-*r/ 32.1%

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

      2. sub-neg 32.1%

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

      3. distribute-lft-in 32.0%

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

      4. metadata-eval 32.0%

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

      5. metadata-eval 32.0%

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

      6. fma-udef 32.1%

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

      7. associate-/r/ 32.3%

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

      8. *-commutative 32.3%

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

      9. clear-num 32.3%

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

      10. un-div-inv 32.3%

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

      11. fma-udef 32.3%

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

      12. metadata-eval 32.3%

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

      13. metadata-eval 32.3%

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

      14. distribute-lft-in 32.3%

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

      15. sub-neg 32.3%

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

      16. *-commutative 32.3%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in i around 0 80.3%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 80.3%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}\right)} \]

      2. associate-*r/ 80.3%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)\right)} \]

      3. metadata-eval 80.3%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)\right)} \]

      4. metadata-eval 80.3%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)} \]

   6. Simplified 80.3%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+131} \lor \neg \left(n \leq 2\right):\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(\left(i \cdot i\right) \cdot \left(\frac{0.3333333333333333}{n \cdot n} + \left(0.16666666666666666 + \frac{-0.5}{n}\right)\right) + i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)}\\ \end{array} \]

Alternative 8: 69.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 7500000000000:\\ \;\;\;\;\frac{n}{\left(0.01 + \left(\frac{i}{n} \cdot 0.005 + i \cdot -0.01\right)\right) - i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 7500000000000.0)
   (/ n (- (+ 0.01 (+ (* (/ i n) 0.005) (* i -0.01))) (* i -0.005)))
   (* 200.0 (/ (* n n) i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 7500000000000.0) {
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005));
	} else {
		tmp = 200.0 * ((n * n) / 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 <= 7500000000000.0d0) then
        tmp = n / ((0.01d0 + (((i / n) * 0.005d0) + (i * (-0.01d0)))) - (i * (-0.005d0)))
    else
        tmp = 200.0d0 * ((n * n) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 7500000000000.0) {
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005));
	} else {
		tmp = 200.0 * ((n * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 7500000000000.0:
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005))
	else:
		tmp = 200.0 * ((n * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 7500000000000.0)
		tmp = Float64(n / Float64(Float64(0.01 + Float64(Float64(Float64(i / n) * 0.005) + Float64(i * -0.01))) - Float64(i * -0.005)));
	else
		tmp = Float64(200.0 * Float64(Float64(n * n) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 7500000000000.0)
		tmp = n / ((0.01 + (((i / n) * 0.005) + (i * -0.01))) - (i * -0.005));
	else
		tmp = 200.0 * ((n * n) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 7500000000000.0], N[(n / N[(N[(0.01 + N[(N[(N[(i / n), $MachinePrecision] * 0.005), $MachinePrecision] + N[(i * -0.01), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 7.5e12

   1. Initial program 23.6%

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

      1. associate-*r/ 23.6%

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

      2. sub-neg 23.6%

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

      3. distribute-lft-in 23.6%

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

      4. metadata-eval 23.6%

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

      5. metadata-eval 23.6%

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

      6. fma-udef 23.6%

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

      7. associate-/r/ 23.9%

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

      8. *-commutative 23.9%

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

      9. clear-num 23.9%

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

      10. un-div-inv 23.9%

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

      11. fma-udef 23.9%

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

      12. metadata-eval 23.9%

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

      13. metadata-eval 23.9%

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

      14. distribute-lft-in 23.9%

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

      15. sub-neg 23.9%

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

      16. *-commutative 23.9%

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

   3. Applied egg-rr 85.7%

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

   4. Taylor expanded in i around 0 77.4%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 77.4%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 77.4%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 77.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 77.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 77.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 77.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 77.4%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 77.4%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) + 0.01}} \]

      2. flip-+ 70.4%

         \[\leadsto \frac{n}{\color{blue}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - 0.01 \cdot 0.01}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}}} \]

      3. metadata-eval 70.4%

         \[\leadsto \frac{n}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - \color{blue}{0.0001}}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}} \]

   8. Applied egg-rr 70.4%

      \[\leadsto \frac{n}{\color{blue}{\frac{\left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) \cdot \left(0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)\right) - 0.0001}{0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right) - 0.01}}} \]

   9. Taylor expanded in n around 0 77.4%

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

   if 7.5e12 < i

   1. Initial program 43.0%

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

      1. associate-*r/ 43.1%

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

      2. sub-neg 43.1%

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

      3. distribute-lft-in 43.1%

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

      4. metadata-eval 43.1%

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

      5. metadata-eval 43.1%

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

      6. fma-udef 43.1%

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

      7. associate-/r/ 43.4%

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

      8. *-commutative 43.4%

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

      9. clear-num 43.4%

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

      10. un-div-inv 43.4%

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

      11. fma-udef 43.3%

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

      12. metadata-eval 43.3%

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

      13. metadata-eval 43.3%

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

      14. distribute-lft-in 43.4%

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

      15. sub-neg 43.4%

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

      16. *-commutative 43.4%

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

   3. Applied egg-rr 68.9%

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

   4. Taylor expanded in i around 0 27.2%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 27.2%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 27.2%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 27.2%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 43.5%

      \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
   8. Step-by-step derivation

      1. unpow2 43.5%

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

   9. Simplified 43.5%

      \[\leadsto \color{blue}{200 \cdot \frac{n \cdot n}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 7500000000000:\\ \;\;\;\;\frac{n}{\left(0.01 + \left(\frac{i}{n} \cdot 0.005 + i \cdot -0.01\right)\right) - i \cdot -0.005}\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \end{array} \]

Alternative 9: 69.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 7500000000000:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 7500000000000.0)
   (/ n (+ 0.01 (* 0.01 (* i (+ -0.5 (/ 0.5 n))))))
   (* 200.0 (/ (* n n) i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 7500000000000.0) {
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	} else {
		tmp = 200.0 * ((n * n) / 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 <= 7500000000000.0d0) then
        tmp = n / (0.01d0 + (0.01d0 * (i * ((-0.5d0) + (0.5d0 / n)))))
    else
        tmp = 200.0d0 * ((n * n) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 7500000000000.0) {
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	} else {
		tmp = 200.0 * ((n * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 7500000000000.0:
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))))
	else:
		tmp = 200.0 * ((n * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 7500000000000.0)
		tmp = Float64(n / Float64(0.01 + Float64(0.01 * Float64(i * Float64(-0.5 + Float64(0.5 / n))))));
	else
		tmp = Float64(200.0 * Float64(Float64(n * n) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 7500000000000.0)
		tmp = n / (0.01 + (0.01 * (i * (-0.5 + (0.5 / n)))));
	else
		tmp = 200.0 * ((n * n) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 7500000000000.0], N[(n / N[(0.01 + N[(0.01 * N[(i * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 7.5e12

   1. Initial program 23.6%

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

      1. associate-*r/ 23.6%

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

      2. sub-neg 23.6%

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

      3. distribute-lft-in 23.6%

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

      4. metadata-eval 23.6%

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

      5. metadata-eval 23.6%

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

      6. fma-udef 23.6%

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

      7. associate-/r/ 23.9%

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

      8. *-commutative 23.9%

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

      9. clear-num 23.9%

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

      10. un-div-inv 23.9%

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

      11. fma-udef 23.9%

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

      12. metadata-eval 23.9%

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

      13. metadata-eval 23.9%

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

      14. distribute-lft-in 23.9%

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

      15. sub-neg 23.9%

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

      16. *-commutative 23.9%

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

   3. Applied egg-rr 85.7%

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

   4. Taylor expanded in i around 0 77.4%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 77.4%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}\right)} \]

      2. associate-*r/ 77.4%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right)\right)} \]

      3. metadata-eval 77.4%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right)\right)} \]

      4. metadata-eval 77.4%

         \[\leadsto \frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right)\right)} \]

   6. Simplified 77.4%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(\frac{0.5}{n} + -0.5\right)\right)}} \]

   if 7.5e12 < i

   1. Initial program 43.0%

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

      1. associate-*r/ 43.1%

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

      2. sub-neg 43.1%

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

      3. distribute-lft-in 43.1%

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

      4. metadata-eval 43.1%

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

      5. metadata-eval 43.1%

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

      6. fma-udef 43.1%

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

      7. associate-/r/ 43.4%

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

      8. *-commutative 43.4%

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

      9. clear-num 43.4%

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

      10. un-div-inv 43.4%

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

      11. fma-udef 43.3%

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

      12. metadata-eval 43.3%

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

      13. metadata-eval 43.3%

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

      14. distribute-lft-in 43.4%

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

      15. sub-neg 43.4%

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

      16. *-commutative 43.4%

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

   3. Applied egg-rr 68.9%

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

   4. Taylor expanded in i around 0 27.2%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 27.2%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 27.2%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 27.2%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 27.2%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 43.5%

      \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
   8. Step-by-step derivation

      1. unpow2 43.5%

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

   9. Simplified 43.5%

      \[\leadsto \color{blue}{200 \cdot \frac{n \cdot n}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 7500000000000:\\ \;\;\;\;\frac{n}{0.01 + 0.01 \cdot \left(i \cdot \left(-0.5 + \frac{0.5}{n}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \end{array} \]

Alternative 10: 61.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.05 \cdot 10^{+20} \lor \neg \left(i \leq 92000\right):\\ \;\;\;\;200 \cdot \frac{n}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -1.05e+20) (not (<= i 92000.0)))
   (* 200.0 (/ n (/ i n)))
   (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -1.05e+20) || !(i <= 92000.0)) {
		tmp = 200.0 * (n / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((i <= (-1.05d+20)) .or. (.not. (i <= 92000.0d0))) then
        tmp = 200.0d0 * (n / (i / n))
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -1.05e+20) || !(i <= 92000.0)) {
		tmp = 200.0 * (n / (i / n));
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -1.05e+20) or not (i <= 92000.0):
		tmp = 200.0 * (n / (i / n))
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -1.05e+20) || !(i <= 92000.0))
		tmp = Float64(200.0 * Float64(n / Float64(i / n)));
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -1.05e+20) || ~((i <= 92000.0)))
		tmp = 200.0 * (n / (i / n));
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -1.05e+20], N[Not[LessEqual[i, 92000.0]], $MachinePrecision]], N[(200.0 * N[(n / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.05e20 or 92000 < i

   1. Initial program 52.4%

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

      1. associate-*r/ 52.5%

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

      2. sub-neg 52.5%

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

      3. distribute-lft-in 52.4%

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

      4. metadata-eval 52.4%

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

      5. metadata-eval 52.4%

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

      6. fma-udef 52.5%

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

      7. associate-/r/ 52.7%

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

      8. *-commutative 52.7%

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

      9. clear-num 52.7%

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

      10. un-div-inv 52.7%

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

      11. fma-udef 52.6%

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

      12. metadata-eval 52.6%

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

      13. metadata-eval 52.6%

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

      14. distribute-lft-in 52.7%

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

      15. sub-neg 52.7%

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

      16. *-commutative 52.7%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in i around 0 33.4%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 33.4%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 33.4%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 33.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 33.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 33.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 33.4%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 33.4%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 32.8%

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

      1. *-commutative 32.8%

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

   9. Simplified 32.8%

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

       1. expm1-log1p-u 32.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)\right)} \]

       2. expm1-udef 31.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)} - 1} \]

   11. Applied egg-rr 31.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 32.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)\right)} \]

       2. expm1-log1p 32.8%

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

       3. *-lft-identity 32.8%

          \[\leadsto \frac{\color{blue}{1 \cdot n}}{\frac{i}{n} \cdot 0.005} \]

       4. *-commutative 32.8%

          \[\leadsto \frac{1 \cdot n}{\color{blue}{0.005 \cdot \frac{i}{n}}} \]

       5. times-frac 32.8%

          \[\leadsto \color{blue}{\frac{1}{0.005} \cdot \frac{n}{\frac{i}{n}}} \]

       6. metadata-eval 32.8%

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

   13. Simplified 32.8%

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

   if -1.05e20 < i < 92000

   1. Initial program 11.0%

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

   2. Taylor expanded in i around 0 80.5%

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

      1. *-commutative 80.5%

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

   4. Simplified 80.5%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.05 \cdot 10^{+20} \lor \neg \left(i \leq 92000\right):\\ \;\;\;\;200 \cdot \frac{n}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]

Alternative 11: 62.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-105} \lor \neg \left(n \leq 2.8 \cdot 10^{-132}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.1e-105) (not (<= n 2.8e-132)))
   (* n (+ 100.0 (* i 50.0)))
   (* 200.0 (/ n (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.1e-105) || !(n <= 2.8e-132)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 200.0 * (n / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.1d-105)) .or. (.not. (n <= 2.8d-132))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 200.0d0 * (n / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.1e-105) || !(n <= 2.8e-132)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 200.0 * (n / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.1e-105) or not (n <= 2.8e-132):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 200.0 * (n / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.1e-105) || !(n <= 2.8e-132))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(200.0 * Float64(n / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.1e-105) || ~((n <= 2.8e-132)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 200.0 * (n / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.1e-105], N[Not[LessEqual[n, 2.8e-132]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(200.0 * N[(n / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.10000000000000002e-105 or 2.80000000000000002e-132 < n

   1. Initial program 20.6%

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

      1. associate-/r/ 20.9%

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

      2. associate-*r* 20.9%

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

      3. *-commutative 20.9%

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

      4. associate-*r/ 20.9%

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

      5. sub-neg 20.9%

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

      6. distribute-lft-in 20.9%

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

      7. fma-def 20.9%

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

      8. metadata-eval 20.9%

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

      9. metadata-eval 20.9%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in i around 0 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*r/ 63.6%

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

      4. metadata-eval 63.6%

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

   6. Simplified 63.6%

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

   7. Taylor expanded in n around inf 63.5%

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

      1. *-commutative 63.5%

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

   9. Simplified 63.5%

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

   if -1.10000000000000002e-105 < n < 2.80000000000000002e-132

   1. Initial program 52.1%

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

      1. associate-*r/ 52.1%

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

      2. sub-neg 52.1%

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

      3. distribute-lft-in 52.1%

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

      4. metadata-eval 52.1%

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

      5. metadata-eval 52.1%

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

      6. fma-udef 52.1%

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

      7. associate-/r/ 52.4%

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

      8. *-commutative 52.4%

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

      9. clear-num 52.4%

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

      10. un-div-inv 52.4%

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

      11. fma-udef 52.4%

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

      12. metadata-eval 52.4%

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

      13. metadata-eval 52.4%

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

      14. distribute-lft-in 52.4%

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

      15. sub-neg 52.4%

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

      16. *-commutative 52.4%

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

   3. Applied egg-rr 82.3%

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

   4. Taylor expanded in i around 0 86.5%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 86.5%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 86.5%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 86.5%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 71.3%

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

      1. *-commutative 71.3%

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

   9. Simplified 71.3%

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

       1. expm1-log1p-u 71.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)\right)} \]

       2. expm1-udef 69.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)} - 1} \]

   11. Applied egg-rr 69.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 71.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)\right)} \]

       2. expm1-log1p 71.3%

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

       3. *-lft-identity 71.3%

          \[\leadsto \frac{\color{blue}{1 \cdot n}}{\frac{i}{n} \cdot 0.005} \]

       4. *-commutative 71.3%

          \[\leadsto \frac{1 \cdot n}{\color{blue}{0.005 \cdot \frac{i}{n}}} \]

       5. times-frac 71.3%

          \[\leadsto \color{blue}{\frac{1}{0.005} \cdot \frac{n}{\frac{i}{n}}} \]

       6. metadata-eval 71.3%

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

   13. Simplified 71.3%

       \[\leadsto \color{blue}{200 \cdot \frac{n}{\frac{i}{n}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-105} \lor \neg \left(n \leq 2.8 \cdot 10^{-132}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n}{\frac{i}{n}}\\ \end{array} \]

Alternative 12: 62.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-105} \lor \neg \left(n \leq 1.15 \cdot 10^{-137}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{n} \cdot 0.005}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.1e-105) (not (<= n 1.15e-137)))
   (* n (+ 100.0 (* i 50.0)))
   (/ n (* (/ i n) 0.005))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.1e-105) || !(n <= 1.15e-137)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = n / ((i / n) * 0.005);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.1d-105)) .or. (.not. (n <= 1.15d-137))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = n / ((i / n) * 0.005d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.1e-105) || !(n <= 1.15e-137)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = n / ((i / n) * 0.005);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.1e-105) or not (n <= 1.15e-137):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = n / ((i / n) * 0.005)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.1e-105) || !(n <= 1.15e-137))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(n / Float64(Float64(i / n) * 0.005));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.1e-105) || ~((n <= 1.15e-137)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = n / ((i / n) * 0.005);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.1e-105], N[Not[LessEqual[n, 1.15e-137]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(N[(i / n), $MachinePrecision] * 0.005), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.10000000000000002e-105 or 1.15000000000000004e-137 < n

   1. Initial program 20.6%

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

      1. associate-/r/ 20.9%

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

      2. associate-*r* 20.9%

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

      3. *-commutative 20.9%

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

      4. associate-*r/ 20.9%

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

      5. sub-neg 20.9%

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

      6. distribute-lft-in 20.9%

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

      7. fma-def 20.9%

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

      8. metadata-eval 20.9%

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

      9. metadata-eval 20.9%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in i around 0 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*r/ 63.6%

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

      4. metadata-eval 63.6%

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

   6. Simplified 63.6%

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

   7. Taylor expanded in n around inf 63.5%

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

      1. *-commutative 63.5%

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

   9. Simplified 63.5%

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

   if -1.10000000000000002e-105 < n < 1.15000000000000004e-137

   1. Initial program 52.1%

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

      1. associate-*r/ 52.1%

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

      2. sub-neg 52.1%

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

      3. distribute-lft-in 52.1%

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

      4. metadata-eval 52.1%

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

      5. metadata-eval 52.1%

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

      6. fma-udef 52.1%

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

      7. associate-/r/ 52.4%

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

      8. *-commutative 52.4%

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

      9. clear-num 52.4%

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

      10. un-div-inv 52.4%

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

      11. fma-udef 52.4%

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

      12. metadata-eval 52.4%

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

      13. metadata-eval 52.4%

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

      14. distribute-lft-in 52.4%

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

      15. sub-neg 52.4%

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

      16. *-commutative 52.4%

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

   3. Applied egg-rr 82.3%

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

   4. Taylor expanded in i around 0 86.5%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 86.5%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 86.5%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 86.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 86.5%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 71.3%

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

      1. *-commutative 71.3%

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

   9. Simplified 71.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{-105} \lor \neg \left(n \leq 1.15 \cdot 10^{-137}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\frac{i}{n} \cdot 0.005}\\ \end{array} \]

Alternative 13: 63.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.3 \cdot 10^{+20}:\\ \;\;\;\;200 \cdot \frac{n}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 92000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.3e+20)
   (* 200.0 (/ n (/ i n)))
   (if (<= i 92000.0) (* n 100.0) (* 200.0 (/ (* n n) i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.3e+20) {
		tmp = 200.0 * (n / (i / n));
	} else if (i <= 92000.0) {
		tmp = n * 100.0;
	} else {
		tmp = 200.0 * ((n * n) / 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 <= (-1.3d+20)) then
        tmp = 200.0d0 * (n / (i / n))
    else if (i <= 92000.0d0) then
        tmp = n * 100.0d0
    else
        tmp = 200.0d0 * ((n * n) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.3e+20) {
		tmp = 200.0 * (n / (i / n));
	} else if (i <= 92000.0) {
		tmp = n * 100.0;
	} else {
		tmp = 200.0 * ((n * n) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.3e+20:
		tmp = 200.0 * (n / (i / n))
	elif i <= 92000.0:
		tmp = n * 100.0
	else:
		tmp = 200.0 * ((n * n) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.3e+20)
		tmp = Float64(200.0 * Float64(n / Float64(i / n)));
	elseif (i <= 92000.0)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(200.0 * Float64(Float64(n * n) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.3e+20)
		tmp = 200.0 * (n / (i / n));
	elseif (i <= 92000.0)
		tmp = n * 100.0;
	else
		tmp = 200.0 * ((n * n) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.3e+20], N[(200.0 * N[(n / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 92000.0], N[(n * 100.0), $MachinePrecision], N[(200.0 * N[(N[(n * n), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.3e20

   1. Initial program 64.2%

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

      1. associate-*r/ 64.2%

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

      2. sub-neg 64.2%

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

      3. distribute-lft-in 64.2%

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

      4. metadata-eval 64.2%

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

      5. metadata-eval 64.2%

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

      6. fma-udef 64.2%

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

      7. associate-/r/ 64.3%

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

      8. *-commutative 64.3%

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

      9. clear-num 64.3%

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

      10. un-div-inv 64.4%

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

      11. fma-udef 64.4%

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

      12. metadata-eval 64.4%

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

      13. metadata-eval 64.4%

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

      14. distribute-lft-in 64.4%

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

      15. sub-neg 64.4%

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

      16. *-commutative 64.4%

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

   3. Applied egg-rr 95.7%

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

   4. Taylor expanded in i around 0 41.0%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 41.0%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 41.0%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 41.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 41.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 41.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 41.0%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 41.0%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 32.2%

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

      1. *-commutative 32.2%

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

   9. Simplified 32.2%

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

       1. expm1-log1p-u 30.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)\right)} \]

       2. expm1-udef 29.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)} - 1} \]

   11. Applied egg-rr 29.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 30.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{\frac{i}{n} \cdot 0.005}\right)\right)} \]

       2. expm1-log1p 32.2%

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

       3. *-lft-identity 32.2%

          \[\leadsto \frac{\color{blue}{1 \cdot n}}{\frac{i}{n} \cdot 0.005} \]

       4. *-commutative 32.2%

          \[\leadsto \frac{1 \cdot n}{\color{blue}{0.005 \cdot \frac{i}{n}}} \]

       5. times-frac 32.2%

          \[\leadsto \color{blue}{\frac{1}{0.005} \cdot \frac{n}{\frac{i}{n}}} \]

       6. metadata-eval 32.2%

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

   13. Simplified 32.2%

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

   if -1.3e20 < i < 92000

   1. Initial program 11.0%

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

   2. Taylor expanded in i around 0 80.5%

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

      1. *-commutative 80.5%

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

   4. Simplified 80.5%

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

   if 92000 < i

   1. Initial program 41.5%

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

      1. associate-*r/ 41.6%

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

      2. sub-neg 41.6%

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

      3. distribute-lft-in 41.5%

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

      4. metadata-eval 41.5%

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

      5. metadata-eval 41.5%

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

      6. fma-udef 41.6%

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

      7. associate-/r/ 41.9%

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

      8. *-commutative 41.9%

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

      9. clear-num 41.9%

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

      10. un-div-inv 41.8%

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

      11. fma-udef 41.8%

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

      12. metadata-eval 41.8%

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

      13. metadata-eval 41.8%

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

      14. distribute-lft-in 41.8%

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

      15. sub-neg 41.8%

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

      16. *-commutative 41.8%

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

   3. Applied egg-rr 70.0%

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

   4. Taylor expanded in i around 0 26.5%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 26.5%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 26.5%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 26.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 26.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 26.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 26.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 26.5%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 42.2%

      \[\leadsto \color{blue}{200 \cdot \frac{{n}^{2}}{i}} \]
   8. Step-by-step derivation

      1. unpow2 42.2%

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

   9. Simplified 42.2%

      \[\leadsto \color{blue}{200 \cdot \frac{n \cdot n}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.3 \cdot 10^{+20}:\\ \;\;\;\;200 \cdot \frac{n}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 92000:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;200 \cdot \frac{n \cdot n}{i}\\ \end{array} \]

Alternative 14: 63.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-135}:\\ \;\;\;\;\frac{n}{\frac{i}{n} \cdot 0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.8e-203)
   (/ n (+ 0.01 (* i -0.005)))
   (if (<= n 1.3e-135) (/ n (* (/ i n) 0.005)) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.8e-203) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.3e-135) {
		tmp = n / ((i / n) * 0.005);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.8d-203)) then
        tmp = n / (0.01d0 + (i * (-0.005d0)))
    else if (n <= 1.3d-135) then
        tmp = n / ((i / n) * 0.005d0)
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.8e-203) {
		tmp = n / (0.01 + (i * -0.005));
	} else if (n <= 1.3e-135) {
		tmp = n / ((i / n) * 0.005);
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.8e-203:
		tmp = n / (0.01 + (i * -0.005))
	elif n <= 1.3e-135:
		tmp = n / ((i / n) * 0.005)
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.8e-203)
		tmp = Float64(n / Float64(0.01 + Float64(i * -0.005)));
	elseif (n <= 1.3e-135)
		tmp = Float64(n / Float64(Float64(i / n) * 0.005));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.8e-203)
		tmp = n / (0.01 + (i * -0.005));
	elseif (n <= 1.3e-135)
		tmp = n / ((i / n) * 0.005);
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.8e-203], N[(n / N[(0.01 + N[(i * -0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.3e-135], N[(n / N[(N[(i / n), $MachinePrecision] * 0.005), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.7999999999999997e-203

   1. Initial program 25.6%

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

      1. associate-*r/ 25.7%

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

      2. sub-neg 25.7%

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

      3. distribute-lft-in 25.7%

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

      4. metadata-eval 25.7%

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

      5. metadata-eval 25.7%

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

      6. fma-udef 25.7%

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

      7. associate-/r/ 25.9%

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

      8. *-commutative 25.9%

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

      9. clear-num 25.9%

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

      10. un-div-inv 25.9%

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

      11. fma-udef 25.9%

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

      12. metadata-eval 25.9%

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

      13. metadata-eval 25.9%

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

      14. distribute-lft-in 25.9%

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

      15. sub-neg 25.9%

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

      16. *-commutative 25.9%

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

   3. Applied egg-rr 83.4%

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

   4. Taylor expanded in i around 0 62.5%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 62.5%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 62.5%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 62.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 62.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 62.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 62.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 62.5%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around inf 60.0%

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

      1. *-commutative 60.0%

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

   9. Simplified 60.0%

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

   if -4.7999999999999997e-203 < n < 1.30000000000000002e-135

   1. Initial program 54.2%

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

      1. associate-*r/ 54.2%

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

      2. sub-neg 54.2%

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

      3. distribute-lft-in 54.2%

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

      4. metadata-eval 54.2%

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

      5. metadata-eval 54.2%

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

      6. fma-udef 54.2%

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

      7. associate-/r/ 54.6%

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

      8. *-commutative 54.6%

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

      9. clear-num 54.6%

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

      10. un-div-inv 54.6%

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

      11. fma-udef 54.6%

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

      12. metadata-eval 54.6%

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

      13. metadata-eval 54.6%

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

      14. distribute-lft-in 54.6%

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

      15. sub-neg 54.6%

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

      16. *-commutative 54.6%

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

   3. Applied egg-rr 77.0%

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

   4. Taylor expanded in i around 0 89.5%

      \[\leadsto \frac{n}{\color{blue}{0.01 + 0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 89.5%

         \[\leadsto \frac{n}{\color{blue}{0.01 \cdot \left(i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right)\right) + 0.01}} \]

      2. fma-def 89.5%

         \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(0.5 \cdot \frac{1}{n} - 0.5\right), 0.01\right)}} \]

      3. sub-neg 89.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \color{blue}{\left(0.5 \cdot \frac{1}{n} + \left(-0.5\right)\right)}, 0.01\right)} \]

      4. associate-*r/ 89.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{n}} + \left(-0.5\right)\right), 0.01\right)} \]

      5. metadata-eval 89.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{\color{blue}{0.5}}{n} + \left(-0.5\right)\right), 0.01\right)} \]

      6. metadata-eval 89.5%

         \[\leadsto \frac{n}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + \color{blue}{-0.5}\right), 0.01\right)} \]

   6. Simplified 89.5%

      \[\leadsto \frac{n}{\color{blue}{\mathsf{fma}\left(0.01, i \cdot \left(\frac{0.5}{n} + -0.5\right), 0.01\right)}} \]

   7. Taylor expanded in n around 0 78.5%

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

      1. *-commutative 78.5%

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

   9. Simplified 78.5%

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

   if 1.30000000000000002e-135 < n

   1. Initial program 18.3%

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

      1. associate-/r/ 18.5%

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

      2. associate-*r* 18.6%

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

      3. *-commutative 18.6%

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

      4. associate-*r/ 18.6%

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

      5. sub-neg 18.6%

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

      6. distribute-lft-in 18.5%

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

      7. fma-def 18.6%

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

      8. metadata-eval 18.6%

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

      9. metadata-eval 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in i around 0 67.2%

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

      1. associate-*r* 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*r/ 67.2%

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

      4. metadata-eval 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in n around inf 67.2%

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

      1. *-commutative 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{n}{0.01 + i \cdot -0.005}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-135}:\\ \;\;\;\;\frac{n}{\frac{i}{n} \cdot 0.005}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]

Alternative 15: 58.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2e+30)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 9.5e+28) (* n 100.0) (* n (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2e+30) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 9.5e+28) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2d+30)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 9.5d+28) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2e+30) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 9.5e+28) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2e+30:
		tmp = 100.0 * (i / (i / n))
	elif i <= 9.5e+28:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2e+30)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 9.5e+28)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2e+30)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 9.5e+28)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2e+30], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 9.5e+28], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2e30

   1. Initial program 66.7%

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

   2. Taylor expanded in i around 0 30.6%

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

   if -2e30 < i < 9.49999999999999927e28

   1. Initial program 12.1%

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

   2. Taylor expanded in i around 0 74.7%

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

      1. *-commutative 74.7%

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

   4. Simplified 74.7%

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

   if 9.49999999999999927e28 < i

   1. Initial program 46.3%

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

   2. Taylor expanded in i around 0 35.3%

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

      1. unpow2 35.3%

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

      2. associate-*r/ 35.3%

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

      3. metadata-eval 35.3%

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

   4. Simplified 35.3%

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

   5. Taylor expanded in n around inf 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

   7. Simplified 39.8%

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

   8. Taylor expanded in i around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-*l* 28.2%

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

   10. Simplified 28.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+30}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]

Alternative 16: 55.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 9.5e+28) (* n 100.0) (* n (* i 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 9.5e+28) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 9.5d+28) then
        tmp = n * 100.0d0
    else
        tmp = n * (i * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 9.5e+28) {
		tmp = n * 100.0;
	} else {
		tmp = n * (i * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 9.5e+28:
		tmp = n * 100.0
	else:
		tmp = n * (i * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 9.5e+28)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(i * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 9.5e+28)
		tmp = n * 100.0;
	else
		tmp = n * (i * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 9.5e+28], N[(n * 100.0), $MachinePrecision], N[(n * N[(i * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 9.49999999999999927e28

   1. Initial program 23.5%

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

   2. Taylor expanded in i around 0 60.2%

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

      1. *-commutative 60.2%

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

   4. Simplified 60.2%

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

   if 9.49999999999999927e28 < i

   1. Initial program 46.3%

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

   2. Taylor expanded in i around 0 35.3%

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

      1. unpow2 35.3%

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

      2. associate-*r/ 35.3%

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

      3. metadata-eval 35.3%

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

   4. Simplified 35.3%

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

   5. Taylor expanded in n around inf 39.8%

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

      1. *-commutative 39.8%

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

      2. unpow2 39.8%

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

   7. Simplified 39.8%

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

   8. Taylor expanded in i around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-*l* 28.2%

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

   10. Simplified 28.2%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(i \cdot 50\right)\\ \end{array} \]

Alternative 17: 2.8% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

   1. associate-/r/ 27.8%

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

   2. associate-*r* 27.8%

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

   3. *-commutative 27.8%

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

   4. associate-*r/ 27.8%

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

   5. sub-neg 27.8%

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

   6. distribute-lft-in 27.8%

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

   7. fma-def 27.8%

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

   8. metadata-eval 27.8%

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

   9. metadata-eval 27.8%

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

3. Simplified 27.8%

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

4. Taylor expanded in i around 0 53.7%

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

   1. associate-*r* 53.7%

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

   2. *-commutative 53.7%

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

   3. associate-*r/ 53.7%

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

   4. metadata-eval 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in n around 0 2.9%

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

   1. *-commutative 2.9%

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

9. Simplified 2.9%

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

10. Final simplification 2.9%

    \[\leadsto i \cdot -50 \]

Alternative 18: 49.4% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

2. Taylor expanded in i around 0 50.5%

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

   1. *-commutative 50.5%

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

4. Simplified 50.5%

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

5. Final simplification 50.5%

   \[\leadsto n \cdot 100 \]

Developer target: 34.4% 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 2023200 
(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))))