Compound Interest

Percentage Accurate: 27.8% → 97.5%

Time: 28.1s

Alternatives: 17

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: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-254}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 5e-254)
     (* n (/ 100.0 (/ i (expm1 (* n (log1p (/ i n)))))))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-254) {
		tmp = n * (100.0 / (i / expm1((n * log1p((i / n))))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-254) {
		tmp = n * (100.0 / (i / Math.expm1((n * Math.log1p((i / n))))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 5e-254:
		tmp = n * (100.0 / (i / math.expm1((n * math.log1p((i / n))))))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 5e-254)
		tmp = Float64(n * Float64(100.0 / Float64(i / expm1(Float64(n * log1p(Float64(i / n)))))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-254], N[(n * N[(100.0 / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 5.0000000000000003e-254

   1. Initial program 30.8%

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

      1. associate-*r/ 30.8%

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

      2. sub-neg 30.8%

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

      3. distribute-lft-in 30.8%

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

      4. metadata-eval 30.8%

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

      5. metadata-eval 30.8%

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

      6. metadata-eval 30.8%

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

      7. fma-def 30.8%

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

      8. metadata-eval 30.8%

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

   3. Simplified 30.8%

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

      1. fma-udef 30.8%

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

      2. metadata-eval 30.8%

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

      3. distribute-lft-in 30.8%

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

      4. metadata-eval 30.8%

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

      5. sub-neg 30.8%

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

      6. associate-*r/ 30.8%

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

      7. associate-/r/ 30.6%

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

      8. sub-neg 30.6%

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

      9. metadata-eval 30.6%

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

      10. associate-*r* 30.6%

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

   6. Applied egg-rr 98.4%

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

      1. clear-num 98.4%

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

      2. un-div-inv 98.5%

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

   8. Applied egg-rr 98.5%

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

   if 5.0000000000000003e-254 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 97.3%

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

      1. associate-*r/ 97.3%

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

      2. sub-neg 97.3%

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

      3. distribute-lft-in 97.6%

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

      4. metadata-eval 97.6%

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

      5. metadata-eval 97.6%

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

      6. metadata-eval 97.6%

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

      7. fma-def 97.3%

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

      8. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

      1. fma-udef 97.6%

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

   6. Applied egg-rr 97.6%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 64.9%

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

   7. Simplified 64.9%

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

   8. Taylor expanded in i around 0 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-254}:\\ \;\;\;\;n \cdot \frac{100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-254}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 5e-254)
     (* 100.0 (* (expm1 (* n (log1p (/ i n)))) (/ n i)))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-254) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) * (n / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-254) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) * (n / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 5e-254:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) * (n / i))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 5e-254)
		tmp = Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) * Float64(n / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-254], N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 5.0000000000000003e-254

   1. Initial program 30.8%

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

      1. clear-num 30.8%

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

      2. associate-/r/ 30.8%

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

      3. clear-num 30.6%

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

      4. add-exp-log 30.6%

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

      5. expm1-def 30.6%

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

      6. log-pow 40.8%

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

      7. log1p-udef 97.9%

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

   4. Applied egg-rr 97.9%

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

   if 5.0000000000000003e-254 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 97.3%

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

      1. associate-*r/ 97.3%

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

      2. sub-neg 97.3%

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

      3. distribute-lft-in 97.6%

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

      4. metadata-eval 97.6%

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

      5. metadata-eval 97.6%

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

      6. metadata-eval 97.6%

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

      7. fma-def 97.3%

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

      8. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

      1. fma-udef 97.6%

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

   6. Applied egg-rr 97.6%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 64.9%

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

   7. Simplified 64.9%

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

   8. Taylor expanded in i around 0 99.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-254}:\\ \;\;\;\;100 \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{n}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-254}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{t_0 \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 5e-254)
     (* n (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) i)))
     (if (<= t_1 INFINITY)
       (/ (+ (* t_0 100.0) -100.0) (/ i n))
       (*
        100.0
        (/ n (+ 1.0 (+ (* i -0.5) (* 0.08333333333333333 (pow i 2.0))))))))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-254) {
		tmp = n * (100.0 * (expm1((n * log1p((i / n)))) / i));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * pow(i, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 5e-254) {
		tmp = n * (100.0 * (Math.expm1((n * Math.log1p((i / n)))) / i));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * 100.0) + -100.0) / (i / n);
	} else {
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * Math.pow(i, 2.0)))));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 5e-254:
		tmp = n * (100.0 * (math.expm1((n * math.log1p((i / n)))) / i))
	elif t_1 <= math.inf:
		tmp = ((t_0 * 100.0) + -100.0) / (i / n)
	else:
		tmp = 100.0 * (n / (1.0 + ((i * -0.5) + (0.08333333333333333 * math.pow(i, 2.0)))))
	return tmp

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n)) ^ n
	t_1 = Float64(Float64(t_0 + -1.0) / Float64(i / n))
	tmp = 0.0
	if (t_1 <= 5e-254)
		tmp = Float64(n * Float64(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / i)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * 100.0) + -100.0) / Float64(i / n));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(Float64(i * -0.5) + Float64(0.08333333333333333 * (i ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-254], N[(n * N[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(t$95$0 * 100.0), $MachinePrecision] + -100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(N[(i * -0.5), $MachinePrecision] + N[(0.08333333333333333 * N[Power[i, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 5.0000000000000003e-254

   1. Initial program 30.8%

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

      1. associate-*r/ 30.8%

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

      2. sub-neg 30.8%

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

      3. distribute-lft-in 30.8%

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

      4. metadata-eval 30.8%

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

      5. metadata-eval 30.8%

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

      6. metadata-eval 30.8%

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

      7. fma-def 30.8%

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

      8. metadata-eval 30.8%

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

   3. Simplified 30.8%

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

      1. fma-udef 30.8%

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

      2. metadata-eval 30.8%

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

      3. distribute-lft-in 30.8%

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

      4. metadata-eval 30.8%

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

      5. sub-neg 30.8%

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

      6. associate-*r/ 30.8%

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

      7. associate-/r/ 30.6%

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

      8. sub-neg 30.6%

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

      9. metadata-eval 30.6%

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

      10. associate-*r* 30.6%

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

   6. Applied egg-rr 98.4%

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

   if 5.0000000000000003e-254 < (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < +inf.0

   1. Initial program 97.3%

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

      1. associate-*r/ 97.3%

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

      2. sub-neg 97.3%

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

      3. distribute-lft-in 97.6%

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

      4. metadata-eval 97.6%

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

      5. metadata-eval 97.6%

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

      6. metadata-eval 97.6%

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

      7. fma-def 97.3%

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

      8. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

      1. fma-udef 97.6%

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

   6. Applied egg-rr 97.6%

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

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

   1. Initial program 0.0%

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

      1. associate-/r/ 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in n around inf 1.8%

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

      1. associate-/l* 1.8%

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

      2. expm1-def 64.9%

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

   7. Simplified 64.9%

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

   8. Taylor expanded in i around 0 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 5 \cdot 10^{-254}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{i}\right)\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} \cdot 100 + -100}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + \left(i \cdot -0.5 + 0.08333333333333333 \cdot {i}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-5} \lor \neg \left(n \leq 0.000155\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.5e-5) (not (<= n 0.000155)))
   (* 100.0 (/ n (/ i (expm1 i))))
   (/ n (+ 0.01 (* (* i 0.01) (+ -0.5 (/ 0.5 n)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.5e-5) || !(n <= 0.000155)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.5e-5) || !(n <= 0.000155)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.5e-5) or not (n <= 0.000155):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.5e-5) || !(n <= 0.000155))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(-0.5 + Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.5e-5], N[Not[LessEqual[n, 0.000155]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.50000000000000012e-5 or 1.55e-4 < n

   1. Initial program 25.8%

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

      1. associate-/r/ 26.2%

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

      2. sub-neg 26.2%

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

      3. metadata-eval 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in n around inf 44.9%

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

      1. associate-/l* 44.9%

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

      2. expm1-def 93.6%

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

   7. Simplified 93.6%

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

   if -2.50000000000000012e-5 < n < 1.55e-4

   1. Initial program 34.1%

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

      1. associate-/r/ 34.0%

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

      2. associate-*r* 34.0%

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

      3. *-commutative 34.0%

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

      4. associate-*r/ 34.0%

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

      5. sub-neg 34.0%

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

      6. distribute-lft-in 34.1%

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

      7. metadata-eval 34.1%

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

      8. metadata-eval 34.1%

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

      9. metadata-eval 34.1%

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

      10. fma-def 34.0%

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

      11. metadata-eval 34.0%

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

   3. Simplified 34.0%

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

      1. clear-num 34.0%

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

      2. un-div-inv 34.0%

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

      3. fma-udef 34.1%

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

      4. metadata-eval 34.1%

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

      5. distribute-lft-in 34.0%

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

      6. metadata-eval 34.0%

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

      7. sub-neg 34.0%

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

      8. add-exp-log 34.0%

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

      9. expm1-def 34.0%

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

      10. log-pow 53.8%

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

      11. log1p-udef 85.5%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in i around 0 80.8%

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

      1. associate-*r* 80.8%

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

      2. *-commutative 80.8%

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

      3. sub-neg 80.8%

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

      4. associate-*r/ 80.8%

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

      5. metadata-eval 80.8%

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

      6. metadata-eval 80.8%

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

   9. Simplified 80.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-5} \lor \neg \left(n \leq 0.000155\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-5} \lor \neg \left(n \leq 0.000155\right):\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6.4e-5) (not (<= n 0.000155)))
   (/ (* n 100.0) (/ i (expm1 i)))
   (/ n (+ 0.01 (* (* i 0.01) (+ -0.5 (/ 0.5 n)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6.4e-5) || !(n <= 0.000155)) {
		tmp = (n * 100.0) / (i / expm1(i));
	} else {
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -6.4e-5) || !(n <= 0.000155)) {
		tmp = (n * 100.0) / (i / Math.expm1(i));
	} else {
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -6.4e-5) or not (n <= 0.000155):
		tmp = (n * 100.0) / (i / math.expm1(i))
	else:
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6.4e-5) || !(n <= 0.000155))
		tmp = Float64(Float64(n * 100.0) / Float64(i / expm1(i)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(-0.5 + Float64(0.5 / n)))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6.4e-5], N[Not[LessEqual[n, 0.000155]], $MachinePrecision]], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.39999999999999971e-5 or 1.55e-4 < n

   1. Initial program 25.8%

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

      1. associate-/r/ 26.2%

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

      2. sub-neg 26.2%

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

      3. metadata-eval 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in n around inf 44.9%

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

      1. associate-/l* 44.9%

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

      2. expm1-def 93.6%

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

   7. Simplified 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-*l/ 93.7%

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

   9. Applied egg-rr 93.7%

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

   if -6.39999999999999971e-5 < n < 1.55e-4

   1. Initial program 34.1%

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

      1. associate-/r/ 34.0%

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

      2. associate-*r* 34.0%

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

      3. *-commutative 34.0%

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

      4. associate-*r/ 34.0%

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

      5. sub-neg 34.0%

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

      6. distribute-lft-in 34.1%

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

      7. metadata-eval 34.1%

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

      8. metadata-eval 34.1%

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

      9. metadata-eval 34.1%

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

      10. fma-def 34.0%

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

      11. metadata-eval 34.0%

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

   3. Simplified 34.0%

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

      1. clear-num 34.0%

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

      2. un-div-inv 34.0%

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

      3. fma-udef 34.1%

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

      4. metadata-eval 34.1%

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

      5. distribute-lft-in 34.0%

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

      6. metadata-eval 34.0%

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

      7. sub-neg 34.0%

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

      8. add-exp-log 34.0%

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

      9. expm1-def 34.0%

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

      10. log-pow 53.8%

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

      11. log1p-udef 85.5%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in i around 0 80.8%

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

      1. associate-*r* 80.8%

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

      2. *-commutative 80.8%

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

      3. sub-neg 80.8%

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

      4. associate-*r/ 80.8%

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

      5. metadata-eval 80.8%

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

      6. metadata-eval 80.8%

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

   9. Simplified 80.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.4 \cdot 10^{-5} \lor \neg \left(n \leq 0.000155\right):\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 0.000155:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -7.2e-5)
   (* (* n 100.0) (/ (expm1 i) i))
   (if (<= n 0.000155)
     (/ n (+ 0.01 (* (* i 0.01) (+ -0.5 (/ 0.5 n)))))
     (* 100.0 (/ n (/ i (expm1 i)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -7.2e-5) {
		tmp = (n * 100.0) * (expm1(i) / i);
	} else if (n <= 0.000155) {
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	} else {
		tmp = 100.0 * (n / (i / expm1(i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -7.2e-5) {
		tmp = (n * 100.0) * (Math.expm1(i) / i);
	} else if (n <= 0.000155) {
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	} else {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -7.2e-5:
		tmp = (n * 100.0) * (math.expm1(i) / i)
	elif n <= 0.000155:
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))))
	else:
		tmp = 100.0 * (n / (i / math.expm1(i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -7.2e-5)
		tmp = Float64(Float64(n * 100.0) * Float64(expm1(i) / i));
	elseif (n <= 0.000155)
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(-0.5 + Float64(0.5 / n)))));
	else
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -7.2e-5], N[(N[(n * 100.0), $MachinePrecision] * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 0.000155], N[(n / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.20000000000000018e-5

   1. Initial program 34.2%

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

      1. *-commutative 34.2%

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

      2. associate-/r/ 34.6%

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

      3. associate-*l* 34.5%

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

      4. sub-neg 34.5%

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

      5. metadata-eval 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in n around inf 48.7%

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

      1. expm1-def 90.4%

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

   7. Simplified 90.4%

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

   if -7.20000000000000018e-5 < n < 1.55e-4

   1. Initial program 34.1%

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

      1. associate-/r/ 34.0%

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

      2. associate-*r* 34.0%

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

      3. *-commutative 34.0%

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

      4. associate-*r/ 34.0%

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

      5. sub-neg 34.0%

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

      6. distribute-lft-in 34.1%

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

      7. metadata-eval 34.1%

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

      8. metadata-eval 34.1%

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

      9. metadata-eval 34.1%

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

      10. fma-def 34.0%

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

      11. metadata-eval 34.0%

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

   3. Simplified 34.0%

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

      1. clear-num 34.0%

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

      2. un-div-inv 34.0%

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

      3. fma-udef 34.1%

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

      4. metadata-eval 34.1%

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

      5. distribute-lft-in 34.0%

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

      6. metadata-eval 34.0%

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

      7. sub-neg 34.0%

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

      8. add-exp-log 34.0%

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

      9. expm1-def 34.0%

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

      10. log-pow 53.8%

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

      11. log1p-udef 85.5%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in i around 0 80.8%

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

      1. associate-*r* 80.8%

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

      2. *-commutative 80.8%

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

      3. sub-neg 80.8%

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

      4. associate-*r/ 80.8%

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

      5. metadata-eval 80.8%

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

      6. metadata-eval 80.8%

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

   9. Simplified 80.8%

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

   if 1.55e-4 < n

   1. Initial program 17.2%

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

      1. associate-/r/ 17.6%

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

      2. sub-neg 17.6%

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

      3. metadata-eval 17.6%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in n around inf 40.9%

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

      1. associate-/l* 40.9%

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

      2. expm1-def 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 0.000155:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{+231} \lor \neg \left(n \leq 0.000155\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.65e+231) (not (<= n 0.000155)))
   (* n (+ 100.0 (* i 50.0)))
   (/ n (+ 0.01 (* (* i 0.01) (+ -0.5 (/ 0.5 n)))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.65e+231) || !(n <= 0.000155)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = n / (0.01 + ((i * 0.01) * (-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 <= (-2.65d+231)) .or. (.not. (n <= 0.000155d0))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = n / (0.01d0 + ((i * 0.01d0) * ((-0.5d0) + (0.5d0 / n))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.65e+231) || !(n <= 0.000155)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.65e+231) or not (n <= 0.000155):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.65e+231) || !(n <= 0.000155))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(n / Float64(0.01 + Float64(Float64(i * 0.01) * Float64(-0.5 + Float64(0.5 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.65e+231) || ~((n <= 0.000155)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = n / (0.01 + ((i * 0.01) * (-0.5 + (0.5 / n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.65e+231], N[Not[LessEqual[n, 0.000155]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n / N[(0.01 + N[(N[(i * 0.01), $MachinePrecision] * N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.6499999999999999e231 or 1.55e-4 < n

   1. Initial program 15.7%

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

      1. associate-*r/ 15.7%

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

      2. sub-neg 15.7%

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

      3. distribute-lft-in 15.7%

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

      4. metadata-eval 15.7%

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

      5. metadata-eval 15.7%

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

      6. metadata-eval 15.7%

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

      7. fma-def 15.7%

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

      8. metadata-eval 15.7%

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

   3. Simplified 15.7%

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

      1. fma-udef 15.7%

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

      2. metadata-eval 15.7%

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

      3. distribute-lft-in 15.7%

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

      4. metadata-eval 15.7%

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

      5. sub-neg 15.7%

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

      6. associate-*r/ 15.7%

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

      7. associate-/r/ 16.1%

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

      8. sub-neg 16.1%

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

      9. metadata-eval 16.1%

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

      10. associate-*r* 16.1%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in i around 0 69.7%

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

      1. associate-*r/ 69.7%

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

      2. metadata-eval 69.7%

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

   9. Simplified 69.7%

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

   10. Taylor expanded in n around inf 69.7%

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

       1. *-commutative 69.7%

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

   12. Simplified 69.7%

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

   if -2.6499999999999999e231 < n < 1.55e-4

   1. Initial program 36.3%

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

      1. associate-/r/ 36.3%

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

      2. associate-*r* 36.3%

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

      3. *-commutative 36.3%

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

      4. associate-*r/ 36.4%

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

      5. sub-neg 36.4%

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

      6. distribute-lft-in 36.4%

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

      7. metadata-eval 36.4%

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

      8. metadata-eval 36.4%

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

      9. metadata-eval 36.4%

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

      10. fma-def 36.4%

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

      11. metadata-eval 36.4%

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

   3. Simplified 36.4%

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

      1. clear-num 36.3%

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

      2. un-div-inv 36.3%

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

      3. fma-udef 36.3%

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

      4. metadata-eval 36.3%

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

      5. distribute-lft-in 36.3%

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

      6. metadata-eval 36.3%

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

      7. sub-neg 36.3%

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

      8. add-exp-log 36.3%

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

      9. expm1-def 36.3%

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

      10. log-pow 46.0%

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

      11. log1p-udef 80.5%

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

   6. Applied egg-rr 80.5%

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

   7. Taylor expanded in i around 0 72.9%

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

      1. associate-*r* 72.9%

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

      2. *-commutative 72.9%

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

      3. sub-neg 72.9%

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

      4. associate-*r/ 72.9%

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

      5. metadata-eval 72.9%

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

      6. metadata-eval 72.9%

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

   9. Simplified 72.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{+231} \lor \neg \left(n \leq 0.000155\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{0.01 + \left(i \cdot 0.01\right) \cdot \left(-0.5 + \frac{0.5}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -1.8 \cdot 10^{+231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.45 \cdot 10^{-221}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -1.8e+231)
     t_0
     (if (<= n -1.45e-221)
       (* 100.0 (/ n (+ 1.0 (* i -0.5))))
       (if (<= n 1.3e-178) (* (* n 100.0) (/ 0.0 i)) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.8e+231) {
		tmp = t_0;
	} else if (n <= -1.45e-221) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.3e-178) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-1.8d+231)) then
        tmp = t_0
    else if (n <= (-1.45d-221)) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else if (n <= 1.3d-178) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.8e+231) {
		tmp = t_0;
	} else if (n <= -1.45e-221) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else if (n <= 1.3e-178) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -1.8e+231:
		tmp = t_0
	elif n <= -1.45e-221:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	elif n <= 1.3e-178:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -1.8e+231)
		tmp = t_0;
	elseif (n <= -1.45e-221)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	elseif (n <= 1.3e-178)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -1.8e+231)
		tmp = t_0;
	elseif (n <= -1.45e-221)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	elseif (n <= 1.3e-178)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.8e+231], t$95$0, If[LessEqual[n, -1.45e-221], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.3e-178], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.8e231 or 1.29999999999999999e-178 < n

   1. Initial program 16.1%

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

      1. associate-*r/ 16.1%

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

      2. sub-neg 16.1%

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

      3. distribute-lft-in 16.2%

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

      4. metadata-eval 16.2%

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

      5. metadata-eval 16.2%

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

      6. metadata-eval 16.2%

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

      7. fma-def 16.1%

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

      8. metadata-eval 16.1%

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

   3. Simplified 16.1%

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

      1. fma-udef 16.2%

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

      2. metadata-eval 16.2%

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

      3. distribute-lft-in 16.1%

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

      4. metadata-eval 16.1%

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

      5. sub-neg 16.1%

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

      6. associate-*r/ 16.1%

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

      7. associate-/r/ 16.5%

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

      8. sub-neg 16.5%

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

      9. metadata-eval 16.5%

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

      10. associate-*r* 16.5%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in i around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. metadata-eval 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in n around inf 65.6%

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

       1. *-commutative 65.6%

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

   12. Simplified 65.6%

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

   if -1.8e231 < n < -1.44999999999999997e-221

   1. Initial program 35.1%

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

      1. associate-/r/ 35.0%

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

      2. sub-neg 35.0%

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

      3. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in n around inf 32.6%

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

      1. associate-/l* 32.6%

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

      2. expm1-def 78.5%

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

   7. Simplified 78.5%

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

   8. Taylor expanded in i around 0 60.8%

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

      1. *-commutative 60.8%

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

   10. Simplified 60.8%

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

   if -1.44999999999999997e-221 < n < 1.29999999999999999e-178

   1. Initial program 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-/r/ 55.0%

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

      3. associate-*l* 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in i around 0 87.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{+231}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -1.45 \cdot 10^{-221}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-178}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -1.5 \cdot 10^{+231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.42 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{0.01 \cdot \frac{1 + i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-177}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -1.5e+231)
     t_0
     (if (<= n -1.42e-223)
       (/ 1.0 (* 0.01 (/ (+ 1.0 (* i -0.5)) n)))
       (if (<= n 1.75e-177) (* (* n 100.0) (/ 0.0 i)) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.5e+231) {
		tmp = t_0;
	} else if (n <= -1.42e-223) {
		tmp = 1.0 / (0.01 * ((1.0 + (i * -0.5)) / n));
	} else if (n <= 1.75e-177) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-1.5d+231)) then
        tmp = t_0
    else if (n <= (-1.42d-223)) then
        tmp = 1.0d0 / (0.01d0 * ((1.0d0 + (i * (-0.5d0))) / n))
    else if (n <= 1.75d-177) then
        tmp = (n * 100.0d0) * (0.0d0 / i)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.5e+231) {
		tmp = t_0;
	} else if (n <= -1.42e-223) {
		tmp = 1.0 / (0.01 * ((1.0 + (i * -0.5)) / n));
	} else if (n <= 1.75e-177) {
		tmp = (n * 100.0) * (0.0 / i);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -1.5e+231:
		tmp = t_0
	elif n <= -1.42e-223:
		tmp = 1.0 / (0.01 * ((1.0 + (i * -0.5)) / n))
	elif n <= 1.75e-177:
		tmp = (n * 100.0) * (0.0 / i)
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -1.5e+231)
		tmp = t_0;
	elseif (n <= -1.42e-223)
		tmp = Float64(1.0 / Float64(0.01 * Float64(Float64(1.0 + Float64(i * -0.5)) / n)));
	elseif (n <= 1.75e-177)
		tmp = Float64(Float64(n * 100.0) * Float64(0.0 / i));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -1.5e+231)
		tmp = t_0;
	elseif (n <= -1.42e-223)
		tmp = 1.0 / (0.01 * ((1.0 + (i * -0.5)) / n));
	elseif (n <= 1.75e-177)
		tmp = (n * 100.0) * (0.0 / i);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.5e+231], t$95$0, If[LessEqual[n, -1.42e-223], N[(1.0 / N[(0.01 * N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.75e-177], N[(N[(n * 100.0), $MachinePrecision] * N[(0.0 / i), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.5000000000000001e231 or 1.7500000000000001e-177 < n

   1. Initial program 16.0%

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

      1. associate-*r/ 16.0%

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

      2. sub-neg 16.0%

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

      3. distribute-lft-in 16.0%

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

      4. metadata-eval 16.0%

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

      5. metadata-eval 16.0%

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

      6. metadata-eval 16.0%

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

      7. fma-def 16.0%

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

      8. metadata-eval 16.0%

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

   3. Simplified 16.0%

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

      1. fma-udef 16.0%

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

      2. metadata-eval 16.0%

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

      3. distribute-lft-in 16.0%

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

      4. metadata-eval 16.0%

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

      5. sub-neg 16.0%

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

      6. associate-*r/ 16.0%

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

      7. associate-/r/ 16.3%

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

      8. sub-neg 16.3%

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

      9. metadata-eval 16.3%

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

      10. associate-*r* 16.3%

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

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in i around 0 65.7%

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

      1. associate-*r/ 65.7%

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

      2. metadata-eval 65.7%

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

   9. Simplified 65.7%

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

   10. Taylor expanded in n around inf 65.9%

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

       1. *-commutative 65.9%

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

   12. Simplified 65.9%

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

   if -1.5000000000000001e231 < n < -1.4199999999999999e-223

   1. Initial program 35.4%

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

      1. associate-/r/ 35.3%

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

      2. sub-neg 35.3%

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

      3. metadata-eval 35.3%

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

   3. Simplified 35.3%

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

   5. Taylor expanded in n around inf 32.9%

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

      1. associate-/l* 32.9%

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

      2. expm1-def 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in i around 0 60.4%

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

      1. *-commutative 60.4%

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

   10. Simplified 60.4%

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

       1. associate-*r/ 60.4%

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

       2. *-commutative 60.4%

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

       3. clear-num 60.7%

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

       4. +-commutative 60.7%

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

       5. fma-def 60.7%

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

       6. *-commutative 60.7%

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

   12. Applied egg-rr 60.7%

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

   13. Taylor expanded in n around 0 60.5%

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

   if -1.4199999999999999e-223 < n < 1.7500000000000001e-177

   1. Initial program 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-/r/ 55.0%

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

      3. associate-*l* 55.0%

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

      4. sub-neg 55.0%

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

      5. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in i around 0 87.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+231}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq -1.42 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{0.01 \cdot \frac{1 + i \cdot -0.5}{n}}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-177}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \frac{0}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.05 \cdot 10^{+232} \lor \neg \left(n \leq 4.2 \cdot 10^{-55}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.05e+232) (not (<= n 4.2e-55)))
   (* n (+ 100.0 (* i 50.0)))
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.05e+232) || !(n <= 4.2e-55)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.05d+232)) .or. (.not. (n <= 4.2d-55))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.05e+232) || !(n <= 4.2e-55)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.05e+232) or not (n <= 4.2e-55):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.05e+232) || !(n <= 4.2e-55))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.05e+232) || ~((n <= 4.2e-55)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.05e+232], N[Not[LessEqual[n, 4.2e-55]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.05000000000000001e232 or 4.2000000000000003e-55 < n

   1. Initial program 15.9%

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

      1. associate-*r/ 15.9%

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

      2. sub-neg 15.9%

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

      3. distribute-lft-in 15.9%

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

      4. metadata-eval 15.9%

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

      5. metadata-eval 15.9%

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

      6. metadata-eval 15.9%

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

      7. fma-def 15.9%

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

      8. metadata-eval 15.9%

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

   3. Simplified 15.9%

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

      1. fma-udef 15.9%

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

      2. metadata-eval 15.9%

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

      3. distribute-lft-in 15.9%

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

      4. metadata-eval 15.9%

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

      5. sub-neg 15.9%

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

      6. associate-*r/ 15.9%

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

      7. associate-/r/ 16.2%

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

      8. sub-neg 16.2%

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

      9. metadata-eval 16.2%

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

      10. associate-*r* 16.2%

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

   6. Applied egg-rr 78.4%

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

   7. Taylor expanded in i around 0 69.1%

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

      1. associate-*r/ 69.1%

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

      2. metadata-eval 69.1%

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

   9. Simplified 69.1%

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

   10. Taylor expanded in n around inf 69.2%

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

       1. *-commutative 69.2%

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

   12. Simplified 69.2%

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

   if -2.05000000000000001e232 < n < 4.2000000000000003e-55

   1. Initial program 37.2%

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

      1. associate-/r/ 37.3%

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

      2. sub-neg 37.3%

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

      3. metadata-eval 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in n around inf 32.1%

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

      1. associate-/l* 32.2%

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

      2. expm1-def 61.1%

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

   7. Simplified 61.1%

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

   8. Taylor expanded in i around 0 59.8%

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

      1. *-commutative 59.8%

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

   10. Simplified 59.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.05 \cdot 10^{+232} \lor \neg \left(n \leq 4.2 \cdot 10^{-55}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{+114} \lor \neg \left(n \leq 1.25 \cdot 10^{-90}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -6.2e+114) (not (<= n 1.25e-90)))
   (* n (+ 100.0 (* i 50.0)))
   (/ (* i 100.0) (/ i n))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -6.2e+114) || !(n <= 1.25e-90)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (i * 100.0) / (i / n);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-6.2d+114)) .or. (.not. (n <= 1.25d-90))) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = (i * 100.0d0) / (i / n)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -6.2e+114) || !(n <= 1.25e-90)) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (i * 100.0) / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -6.2e+114) or not (n <= 1.25e-90):
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = (i * 100.0) / (i / n)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -6.2e+114) || !(n <= 1.25e-90))
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(Float64(i * 100.0) / Float64(i / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -6.2e+114) || ~((n <= 1.25e-90)))
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = (i * 100.0) / (i / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -6.2e+114], N[Not[LessEqual[n, 1.25e-90]], $MachinePrecision]], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * 100.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6.2000000000000001e114 or 1.25000000000000005e-90 < n

   1. Initial program 17.9%

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

      1. associate-*r/ 17.9%

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

      2. sub-neg 17.9%

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

      3. distribute-lft-in 17.9%

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

      4. metadata-eval 17.9%

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

      5. metadata-eval 17.9%

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

      6. metadata-eval 17.9%

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

      7. fma-def 17.9%

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

      8. metadata-eval 17.9%

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

   3. Simplified 17.9%

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

      1. fma-udef 17.9%

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

      2. metadata-eval 17.9%

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

      3. distribute-lft-in 17.9%

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

      4. metadata-eval 17.9%

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

      5. sub-neg 17.9%

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

      6. associate-*r/ 17.9%

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

      7. associate-/r/ 18.4%

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

      8. sub-neg 18.4%

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

      9. metadata-eval 18.4%

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

      10. associate-*r* 18.3%

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

   6. Applied egg-rr 73.8%

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

   7. Taylor expanded in i around 0 65.7%

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

      1. associate-*r/ 65.7%

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

      2. metadata-eval 65.7%

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

   9. Simplified 65.7%

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

   10. Taylor expanded in n around inf 65.8%

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

       1. *-commutative 65.8%

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

   12. Simplified 65.8%

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

   if -6.2000000000000001e114 < n < 1.25000000000000005e-90

   1. Initial program 41.3%

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

      1. associate-*r/ 41.3%

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

      2. sub-neg 41.3%

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

      3. distribute-lft-in 41.3%

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

      4. metadata-eval 41.3%

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

      5. metadata-eval 41.3%

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

      6. metadata-eval 41.3%

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

      7. fma-def 41.3%

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

      8. metadata-eval 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in i around 0 56.5%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{+114} \lor \neg \left(n \leq 1.25 \cdot 10^{-90}\right):\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 100}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.6:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 4.6 \cdot 10^{+218}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.6)
   (/ -200.0 (/ i n))
   (if (<= i 4.6e+218) (* n (+ 100.0 (* i 50.0))) (* (/ n i) -200.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.6) {
		tmp = -200.0 / (i / n);
	} else if (i <= 4.6e+218) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.6d0)) then
        tmp = (-200.0d0) / (i / n)
    else if (i <= 4.6d+218) then
        tmp = n * (100.0d0 + (i * 50.0d0))
    else
        tmp = (n / i) * (-200.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.6) {
		tmp = -200.0 / (i / n);
	} else if (i <= 4.6e+218) {
		tmp = n * (100.0 + (i * 50.0));
	} else {
		tmp = (n / i) * -200.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.6:
		tmp = -200.0 / (i / n)
	elif i <= 4.6e+218:
		tmp = n * (100.0 + (i * 50.0))
	else:
		tmp = (n / i) * -200.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.6)
		tmp = Float64(-200.0 / Float64(i / n));
	elseif (i <= 4.6e+218)
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	else
		tmp = Float64(Float64(n / i) * -200.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.6)
		tmp = -200.0 / (i / n);
	elseif (i <= 4.6e+218)
		tmp = n * (100.0 + (i * 50.0));
	else
		tmp = (n / i) * -200.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.6], N[(-200.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4.6e+218], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.6000000000000001

   1. Initial program 65.9%

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

      1. associate-/r/ 65.3%

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

      2. sub-neg 65.3%

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

      3. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in n around inf 77.9%

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

      1. associate-/l* 77.9%

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

      2. expm1-def 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in i around 0 34.1%

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

      1. *-commutative 34.1%

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

   10. Simplified 34.1%

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

   11. Taylor expanded in i around inf 34.1%

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

       1. clear-num 34.7%

          \[\leadsto -200 \cdot \color{blue}{\frac{1}{\frac{i}{n}}} \]

       2. un-div-inv 34.7%

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

   13. Applied egg-rr 34.7%

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

   if -1.6000000000000001 < i < 4.6000000000000002e218

   1. Initial program 16.9%

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

      1. associate-*r/ 16.9%

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

      2. sub-neg 16.9%

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

      3. distribute-lft-in 16.9%

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

      4. metadata-eval 16.9%

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

      5. metadata-eval 16.9%

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

      6. metadata-eval 16.9%

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

      7. fma-def 16.9%

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

      8. metadata-eval 16.9%

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

   3. Simplified 16.9%

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

      1. fma-udef 16.9%

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

      2. metadata-eval 16.9%

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

      3. distribute-lft-in 16.9%

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

      4. metadata-eval 16.9%

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

      5. sub-neg 16.9%

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

      6. associate-*r/ 16.9%

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

      7. associate-/r/ 17.2%

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

      8. sub-neg 17.2%

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

      9. metadata-eval 17.2%

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

      10. associate-*r* 17.2%

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

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in i around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. metadata-eval 68.9%

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

   9. Simplified 68.9%

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

   10. Taylor expanded in n around inf 69.2%

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

       1. *-commutative 69.2%

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

   12. Simplified 69.2%

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

   if 4.6000000000000002e218 < i

   1. Initial program 40.2%

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

      1. associate-/r/ 40.9%

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

      2. sub-neg 40.9%

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

      3. metadata-eval 40.9%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in n around inf 22.2%

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

      1. associate-/l* 22.2%

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

      2. expm1-def 22.2%

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

   7. Simplified 22.2%

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

   8. Taylor expanded in i around 0 57.1%

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

      1. *-commutative 57.1%

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

   10. Simplified 57.1%

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

   11. Taylor expanded in i around inf 57.1%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.6:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 4.6 \cdot 10^{+218}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 5.8 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.0) (not (<= i 5.8e+58))) (* (/ n i) -200.0) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 5.8e+58)) {
		tmp = (n / i) * -200.0;
	} 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 <= (-2.0d0)) .or. (.not. (i <= 5.8d+58))) then
        tmp = (n / i) * (-200.0d0)
    else
        tmp = n * 100.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 5.8e+58)) {
		tmp = (n / i) * -200.0;
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.0) or not (i <= 5.8e+58):
		tmp = (n / i) * -200.0
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.0) || !(i <= 5.8e+58))
		tmp = Float64(Float64(n / i) * -200.0);
	else
		tmp = Float64(n * 100.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((i <= -2.0) || ~((i <= 5.8e+58)))
		tmp = (n / i) * -200.0;
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.0], N[Not[LessEqual[i, 5.8e+58]], $MachinePrecision]], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2 or 5.80000000000000004e58 < i

   1. Initial program 55.4%

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

      1. associate-/r/ 55.4%

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

      2. sub-neg 55.4%

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

      3. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in n around inf 59.3%

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

      1. associate-/l* 59.3%

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

      2. expm1-def 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in i around 0 36.2%

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

      1. *-commutative 36.2%

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

   10. Simplified 36.2%

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

   11. Taylor expanded in i around inf 36.2%

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

   if -2 < i < 5.80000000000000004e58

   1. Initial program 12.6%

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

      1. associate-/r/ 13.0%

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

      2. associate-*r* 13.0%

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

      3. *-commutative 13.0%

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

      4. associate-*r/ 13.0%

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

      5. sub-neg 13.0%

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

      6. distribute-lft-in 13.0%

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

      7. metadata-eval 13.0%

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

      8. metadata-eval 13.0%

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

      9. metadata-eval 13.0%

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

      10. fma-def 13.0%

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

      11. metadata-eval 13.0%

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

   3. Simplified 13.0%

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

   5. Taylor expanded in i around 0 73.7%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 5.8 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 1.9 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -2.0) (not (<= i 1.9e+53))) (/ -200.0 (/ i n)) (* n 100.0)))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -2.0) || !(i <= 1.9e+53)) {
		tmp = -200.0 / (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 <= (-2.0d0)) .or. (.not. (i <= 1.9d+53))) then
        tmp = (-200.0d0) / (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 <= -2.0) || !(i <= 1.9e+53)) {
		tmp = -200.0 / (i / n);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -2.0) or not (i <= 1.9e+53):
		tmp = -200.0 / (i / n)
	else:
		tmp = n * 100.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -2.0) || !(i <= 1.9e+53))
		tmp = Float64(-200.0 / 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 <= -2.0) || ~((i <= 1.9e+53)))
		tmp = -200.0 / (i / n);
	else
		tmp = n * 100.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -2.0], N[Not[LessEqual[i, 1.9e+53]], $MachinePrecision]], N[(-200.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], N[(n * 100.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2 or 1.89999999999999999e53 < i

   1. Initial program 55.4%

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

      1. associate-/r/ 55.4%

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

      2. sub-neg 55.4%

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

      3. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in n around inf 59.3%

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

      1. associate-/l* 59.3%

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

      2. expm1-def 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in i around 0 36.2%

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

      1. *-commutative 36.2%

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

   10. Simplified 36.2%

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

   11. Taylor expanded in i around inf 36.2%

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

       1. clear-num 37.0%

          \[\leadsto -200 \cdot \color{blue}{\frac{1}{\frac{i}{n}}} \]

       2. un-div-inv 37.0%

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

   13. Applied egg-rr 37.0%

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

   if -2 < i < 1.89999999999999999e53

   1. Initial program 12.6%

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

      1. associate-/r/ 13.0%

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

      2. associate-*r* 13.0%

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

      3. *-commutative 13.0%

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

      4. associate-*r/ 13.0%

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

      5. sub-neg 13.0%

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

      6. distribute-lft-in 13.0%

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

      7. metadata-eval 13.0%

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

      8. metadata-eval 13.0%

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

      9. metadata-eval 13.0%

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

      10. fma-def 13.0%

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

      11. metadata-eval 13.0%

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

   3. Simplified 13.0%

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

   5. Taylor expanded in i around 0 73.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \lor \neg \left(i \leq 1.9 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 6 \cdot 10^{+57}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{-200}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0)
   (* (/ n i) -200.0)
   (if (<= i 6e+57) (* n 100.0) (* n (/ -200.0 i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n / i) * -200.0;
	} else if (i <= 6e+57) {
		tmp = n * 100.0;
	} else {
		tmp = n * (-200.0 / 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 <= (-2.0d0)) then
        tmp = (n / i) * (-200.0d0)
    else if (i <= 6d+57) then
        tmp = n * 100.0d0
    else
        tmp = n * ((-200.0d0) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = (n / i) * -200.0;
	} else if (i <= 6e+57) {
		tmp = n * 100.0;
	} else {
		tmp = n * (-200.0 / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = (n / i) * -200.0
	elif i <= 6e+57:
		tmp = n * 100.0
	else:
		tmp = n * (-200.0 / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(Float64(n / i) * -200.0);
	elseif (i <= 6e+57)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(n * Float64(-200.0 / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = (n / i) * -200.0;
	elseif (i <= 6e+57)
		tmp = n * 100.0;
	else
		tmp = n * (-200.0 / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(N[(n / i), $MachinePrecision] * -200.0), $MachinePrecision], If[LessEqual[i, 6e+57], N[(n * 100.0), $MachinePrecision], N[(n * N[(-200.0 / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 65.9%

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

      1. associate-/r/ 65.3%

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

      2. sub-neg 65.3%

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

      3. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in n around inf 77.9%

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

      1. associate-/l* 77.9%

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

      2. expm1-def 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in i around 0 34.1%

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

      1. *-commutative 34.1%

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

   10. Simplified 34.1%

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

   11. Taylor expanded in i around inf 34.1%

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

   if -2 < i < 5.9999999999999999e57

   1. Initial program 12.6%

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

      1. associate-/r/ 13.0%

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

      2. associate-*r* 13.0%

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

      3. *-commutative 13.0%

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

      4. associate-*r/ 13.0%

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

      5. sub-neg 13.0%

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

      6. distribute-lft-in 13.0%

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

      7. metadata-eval 13.0%

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

      8. metadata-eval 13.0%

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

      9. metadata-eval 13.0%

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

      10. fma-def 13.0%

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

      11. metadata-eval 13.0%

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

   3. Simplified 13.0%

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

   5. Taylor expanded in i around 0 73.7%

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

   if 5.9999999999999999e57 < i

   1. Initial program 41.9%

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

      1. associate-/r/ 42.5%

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

      2. sub-neg 42.5%

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

      3. metadata-eval 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in n around inf 35.1%

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

      1. associate-/l* 35.1%

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

      2. expm1-def 35.1%

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

   7. Simplified 35.1%

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

   8. Taylor expanded in i around 0 39.0%

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

      1. *-commutative 39.0%

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

   10. Simplified 39.0%

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

       1. associate-*r/ 39.0%

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

       2. *-commutative 39.0%

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

       3. clear-num 40.0%

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

       4. +-commutative 40.0%

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

       5. fma-def 40.0%

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

       6. *-commutative 40.0%

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

   12. Applied egg-rr 40.0%

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

   13. Taylor expanded in i around inf 39.0%

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

       1. associate-*r/ 39.0%

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

       2. associate-/l* 40.0%

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

       3. associate-/r/ 39.0%

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

   15. Simplified 39.0%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{n}{i} \cdot -200\\ \mathbf{elif}\;i \leq 6 \cdot 10^{+57}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{-200}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 29.5%

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

   1. associate-/r/ 29.7%

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

   2. sub-neg 29.7%

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

   3. metadata-eval 29.7%

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

3. Simplified 29.7%

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

5. Taylor expanded in i around 0 50.0%

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

   1. associate-*r/ 50.0%

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

   2. metadata-eval 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in n around 0 2.9%

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

   1. *-commutative 2.9%

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

10. Simplified 2.9%

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

11. Final simplification 2.9%

    \[\leadsto i \cdot -50 \]
12. Add Preprocessing

Alternative 17: 49.8% 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 29.5%

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

   1. associate-/r/ 29.7%

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

   2. associate-*r* 29.7%

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

   3. *-commutative 29.7%

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

   4. associate-*r/ 29.7%

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

   5. sub-neg 29.7%

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

   6. distribute-lft-in 29.7%

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

   7. metadata-eval 29.7%

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

   8. metadata-eval 29.7%

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

   9. metadata-eval 29.7%

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

   10. fma-def 29.7%

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

   11. metadata-eval 29.7%

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

3. Simplified 29.7%

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

5. Taylor expanded in i around 0 47.1%

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

6. Final simplification 47.1%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 34.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024019 
(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))))