Compound Interest

Percentage Accurate: 28.6% → 93.6%

Time: 17.3s

Alternatives: 15

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;n \cdot \left(\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right) \cdot \frac{100}{i}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -0.0

   1. Initial program 20.3%

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

      1. associate-/r/ 20.0%

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

      2. associate-*r* 20.0%

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

      3. *-commutative 20.0%

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

      4. associate-*r/ 20.0%

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

      5. sub-neg 20.0%

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

      6. distribute-lft-in 20.0%

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

      7. metadata-eval 20.0%

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

      8. metadata-eval 20.0%

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

      9. metadata-eval 20.0%

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

      10. fma-define 20.0%

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

      11. metadata-eval 20.0%

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

   3. Simplified 20.0%

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

      1. fma-undefine 20.0%

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

      2. metadata-eval 20.0%

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

      3. metadata-eval 20.0%

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

      4. distribute-lft-in 20.0%

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

      5. sub-neg 20.0%

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

      6. *-commutative 20.0%

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

      7. add-exp-log 20.0%

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

      8. expm1-define 20.0%

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

      9. log-pow 33.7%

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

      10. log1p-define 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. associate-/l* 98.3%

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

   8. Applied egg-rr 98.3%

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

   if -0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 92.8%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.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/ 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r/ 1.9%

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

      5. sub-neg 1.9%

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

      6. distribute-lft-in 1.9%

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

      7. metadata-eval 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. fma-define 1.9%

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

      11. metadata-eval 1.9%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in i around 0 71.5%

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

      1. *-commutative 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 93.0%

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

Alternative 2: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-248}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-240}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.8e-248)
   (* n (* (/ 100.0 i) (expm1 i)))
   (if (<= n 1.05e-240)
     (/ 0.0 (/ i n))
     (if (<= n 1.05e+32)
       (* 100.0 (/ i (/ i n)))
       (* n (/ (* 100.0 (expm1 i)) i))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.8e-248) {
		tmp = n * ((100.0 / i) * expm1(i));
	} else if (n <= 1.05e-240) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.05e+32) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((100.0 * expm1(i)) / i);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.8e-248) {
		tmp = n * ((100.0 / i) * Math.expm1(i));
	} else if (n <= 1.05e-240) {
		tmp = 0.0 / (i / n);
	} else if (n <= 1.05e+32) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((100.0 * Math.expm1(i)) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.8e-248:
		tmp = n * ((100.0 / i) * math.expm1(i))
	elif n <= 1.05e-240:
		tmp = 0.0 / (i / n)
	elif n <= 1.05e+32:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * ((100.0 * math.expm1(i)) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.8e-248)
		tmp = Float64(n * Float64(Float64(100.0 / i) * expm1(i)));
	elseif (n <= 1.05e-240)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 1.05e+32)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(Float64(100.0 * expm1(i)) / i));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.8e-248], N[(n * N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-240], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e+32], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -4.80000000000000006e-248

   1. Initial program 25.0%

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

      1. associate-/r/ 25.2%

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

      2. associate-*r* 25.3%

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

      3. *-commutative 25.3%

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

      4. associate-*r/ 25.3%

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

      5. sub-neg 25.3%

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

      6. distribute-lft-in 25.3%

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

      7. metadata-eval 25.3%

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

      8. metadata-eval 25.3%

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

      9. metadata-eval 25.3%

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

      10. fma-define 25.3%

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

      11. metadata-eval 25.3%

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

   3. Simplified 25.3%

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

      1. fma-undefine 25.3%

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

      2. metadata-eval 25.3%

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

      3. metadata-eval 25.3%

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

      4. distribute-lft-in 25.3%

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

      5. sub-neg 25.3%

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

      6. *-commutative 25.3%

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

      7. add-exp-log 25.3%

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

      8. expm1-define 25.3%

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

      9. log-pow 24.1%

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

      10. log1p-define 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. associate-/l* 78.0%

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

   8. Applied egg-rr 78.0%

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

   9. Taylor expanded in n around inf 32.3%

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

       1. expm1-define 83.1%

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

   11. Simplified 83.1%

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

   if -4.80000000000000006e-248 < n < 1.04999999999999997e-240

   1. Initial program 51.1%

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

      1. associate-*r/ 51.1%

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

      2. sub-neg 51.1%

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

      3. distribute-rgt-in 51.1%

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

      4. metadata-eval 51.1%

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

      5. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in i around 0 86.1%

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

   if 1.04999999999999997e-240 < n < 1.05e32

   1. Initial program 13.9%

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

   3. Taylor expanded in i around 0 56.7%

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

   if 1.05e32 < n

   1. Initial program 17.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-*r* 17.6%

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

      3. *-commutative 17.6%

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

      4. associate-*r/ 17.6%

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

      5. sub-neg 17.6%

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

      6. distribute-lft-in 17.6%

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

      7. metadata-eval 17.6%

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

      8. metadata-eval 17.6%

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

      9. metadata-eval 17.6%

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

      10. fma-define 17.6%

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

      11. metadata-eval 17.6%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in n around inf 50.5%

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

      1. sub-neg 50.5%

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

      2. metadata-eval 50.5%

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

      3. metadata-eval 50.5%

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

      4. distribute-lft-in 50.4%

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

      5. metadata-eval 50.4%

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

      6. sub-neg 50.4%

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

      7. expm1-define 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-248}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-240}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{if}\;n \leq -4.4 \cdot 10^{-248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-240}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* (/ 100.0 i) (expm1 i)))))
   (if (<= n -4.4e-248)
     t_0
     (if (<= n 1.06e-240)
       (/ 0.0 (/ i n))
       (if (<= n 2.8) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * ((100.0 / i) * expm1(i));
	double tmp;
	if (n <= -4.4e-248) {
		tmp = t_0;
	} else if (n <= 1.06e-240) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = n * ((100.0 / i) * Math.expm1(i));
	double tmp;
	if (n <= -4.4e-248) {
		tmp = t_0;
	} else if (n <= 1.06e-240) {
		tmp = 0.0 / (i / n);
	} else if (n <= 2.8) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * ((100.0 / i) * math.expm1(i))
	tmp = 0
	if n <= -4.4e-248:
		tmp = t_0
	elif n <= 1.06e-240:
		tmp = 0.0 / (i / n)
	elif n <= 2.8:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(Float64(100.0 / i) * expm1(i)))
	tmp = 0.0
	if (n <= -4.4e-248)
		tmp = t_0;
	elseif (n <= 1.06e-240)
		tmp = Float64(0.0 / Float64(i / n));
	elseif (n <= 2.8)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.4e-248], t$95$0, If[LessEqual[n, 1.06e-240], N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.8], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.39999999999999999e-248 or 2.7999999999999998 < n

   1. Initial program 21.8%

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

      1. associate-/r/ 22.1%

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

      2. associate-*r* 22.1%

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

      3. *-commutative 22.1%

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

      4. associate-*r/ 22.2%

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

      5. sub-neg 22.2%

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

      6. distribute-lft-in 22.1%

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

      7. metadata-eval 22.1%

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

      8. metadata-eval 22.1%

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

      9. metadata-eval 22.1%

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

      10. fma-define 22.2%

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

      11. metadata-eval 22.2%

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

   3. Simplified 22.2%

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

      1. fma-undefine 22.1%

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

      2. metadata-eval 22.1%

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

      3. metadata-eval 22.1%

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

      4. distribute-lft-in 22.2%

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

      5. sub-neg 22.2%

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

      6. *-commutative 22.2%

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

      7. add-exp-log 22.2%

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

      8. expm1-define 22.2%

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

      9. log-pow 20.9%

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

      10. log1p-define 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. associate-/l* 77.3%

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

   8. Applied egg-rr 77.3%

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

   9. Taylor expanded in n around inf 37.5%

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

       1. expm1-define 88.6%

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

   11. Simplified 88.6%

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

   if -4.39999999999999999e-248 < n < 1.06e-240

   1. Initial program 51.1%

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

      1. associate-*r/ 51.1%

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

      2. sub-neg 51.1%

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

      3. distribute-rgt-in 51.1%

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

      4. metadata-eval 51.1%

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

      5. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in i around 0 86.1%

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

   if 1.06e-240 < n < 2.7999999999999998

   1. Initial program 15.1%

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

   3. Taylor expanded in i around 0 52.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.4 \cdot 10^{-248}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{elif}\;n \leq 1.06 \cdot 10^{-240}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \mathbf{elif}\;n \leq 2.8:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{if}\;i \leq -1 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.26 \cdot 10^{+150}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (expm1 i) (/ i n)))))
   (if (<= i -1e-13)
     t_0
     (if (<= i 9.2e-14)
       (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n))))))
       (if (<= i 1.26e+150) t_0 (/ 0.0 (/ i n)))))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (expm1(i) / (i / n));
	double tmp;
	if (i <= -1e-13) {
		tmp = t_0;
	} else if (i <= 9.2e-14) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (i <= 1.26e+150) {
		tmp = t_0;
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (Math.expm1(i) / (i / n));
	double tmp;
	if (i <= -1e-13) {
		tmp = t_0;
	} else if (i <= 9.2e-14) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (i <= 1.26e+150) {
		tmp = t_0;
	} else {
		tmp = 0.0 / (i / n);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (math.expm1(i) / (i / n))
	tmp = 0
	if i <= -1e-13:
		tmp = t_0
	elif i <= 9.2e-14:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	elif i <= 1.26e+150:
		tmp = t_0
	else:
		tmp = 0.0 / (i / n)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(expm1(i) / Float64(i / n)))
	tmp = 0.0
	if (i <= -1e-13)
		tmp = t_0;
	elseif (i <= 9.2e-14)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	elseif (i <= 1.26e+150)
		tmp = t_0;
	else
		tmp = Float64(0.0 / Float64(i / n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1e-13], t$95$0, If[LessEqual[i, 9.2e-14], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.26e+150], t$95$0, N[(0.0 / N[(i / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -1e-13 or 9.19999999999999993e-14 < i < 1.26e150

   1. Initial program 36.9%

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

   3. Taylor expanded in n around inf 70.2%

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

      1. expm1-define 70.6%

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

   5. Simplified 70.6%

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

   if -1e-13 < i < 9.19999999999999993e-14

   1. Initial program 5.6%

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

      1. associate-/r/ 6.1%

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

      2. associate-*r* 6.1%

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

      3. *-commutative 6.1%

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

      4. associate-*r/ 6.1%

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

      5. sub-neg 6.1%

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

      6. distribute-lft-in 6.1%

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

      7. metadata-eval 6.1%

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

      8. metadata-eval 6.1%

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

      9. metadata-eval 6.1%

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

      10. fma-define 6.1%

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

      11. metadata-eval 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in i around 0 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-*r/ 87.9%

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

      3. metadata-eval 87.9%

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

   7. Simplified 87.9%

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

   if 1.26e150 < i

   1. Initial program 59.2%

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

      1. associate-*r/ 59.3%

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

      2. sub-neg 59.3%

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

      3. distribute-rgt-in 59.3%

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

      4. metadata-eval 59.3%

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

      5. metadata-eval 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in i around 0 49.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1 \cdot 10^{-13}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;i \leq 1.26 \cdot 10^{+150}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.15 \cdot 10^{+148}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} + n \cdot \frac{-1}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.15e+148)
   (* n (* (/ 100.0 i) (expm1 i)))
   (* 100.0 (+ (pow (/ i n) (+ n -1.0)) (* n (/ -1.0 i))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.15e+148) {
		tmp = n * ((100.0 / i) * expm1(i));
	} else {
		tmp = 100.0 * (pow((i / n), (n + -1.0)) + (n * (-1.0 / i)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.15e+148) {
		tmp = n * ((100.0 / i) * Math.expm1(i));
	} else {
		tmp = 100.0 * (Math.pow((i / n), (n + -1.0)) + (n * (-1.0 / i)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.15e+148:
		tmp = n * ((100.0 / i) * math.expm1(i))
	else:
		tmp = 100.0 * (math.pow((i / n), (n + -1.0)) + (n * (-1.0 / i)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.15e+148)
		tmp = Float64(n * Float64(Float64(100.0 / i) * expm1(i)));
	else
		tmp = Float64(100.0 * Float64((Float64(i / n) ^ Float64(n + -1.0)) + Float64(n * Float64(-1.0 / i))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.15e+148], N[(n * N[(N[(100.0 / i), $MachinePrecision] * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[Power[N[(i / n), $MachinePrecision], N[(n + -1.0), $MachinePrecision]], $MachinePrecision] + N[(n * N[(-1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.15e148

   1. Initial program 18.6%

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

      1. associate-/r/ 18.7%

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

      2. associate-*r* 18.7%

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

      3. *-commutative 18.7%

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

      4. associate-*r/ 18.7%

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

      5. sub-neg 18.7%

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

      6. distribute-lft-in 18.6%

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

      7. metadata-eval 18.6%

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

      8. metadata-eval 18.6%

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

      9. metadata-eval 18.6%

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

      10. fma-define 18.7%

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

      11. metadata-eval 18.7%

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

   3. Simplified 18.7%

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

      1. fma-undefine 18.6%

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

      2. metadata-eval 18.6%

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

      3. metadata-eval 18.6%

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

      4. distribute-lft-in 18.7%

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

      5. sub-neg 18.7%

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

      6. *-commutative 18.7%

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

      7. add-exp-log 18.7%

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

      8. expm1-define 18.7%

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

      9. log-pow 30.2%

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

      10. log1p-define 82.7%

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

   6. Applied egg-rr 82.7%

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

      1. associate-/l* 82.8%

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

   8. Applied egg-rr 82.8%

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

   9. Taylor expanded in n around inf 32.9%

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

       1. expm1-define 80.3%

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

   11. Simplified 80.3%

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

   if 1.15e148 < i

   1. Initial program 59.2%

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

   3. Taylor expanded in i around inf 59.2%

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

      1. div-sub 59.2%

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

      2. pow1 59.2%

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

      3. pow-div 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. sub-neg 96.2%

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

      2. metadata-eval 96.2%

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

      3. associate-/r/ 91.9%

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

   7. Simplified 91.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.15 \cdot 10^{+148}:\\ \;\;\;\;n \cdot \left(\frac{100}{i} \cdot \mathsf{expm1}\left(i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left({\left(\frac{i}{n}\right)}^{\left(n + -1\right)} + n \cdot \frac{-1}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-35}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.9e-35)
   (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n))))))
   (if (<= n 1.05e+32)
     (* 100.0 (/ i (/ i n)))
     (* n (/ (* 100.0 (* i (+ 1.0 (* i (+ 0.5 (* 0.5 (/ -1.0 n))))))) i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.9e-35) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (n <= 1.05e+32) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((100.0 * (i * (1.0 + (i * (0.5 + (0.5 * (-1.0 / n))))))) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.9d-35)) then
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))
    else if (n <= 1.05d+32) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * ((100.0d0 * (i * (1.0d0 + (i * (0.5d0 + (0.5d0 * ((-1.0d0) / n))))))) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.9e-35) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (n <= 1.05e+32) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((100.0 * (i * (1.0 + (i * (0.5 + (0.5 * (-1.0 / n))))))) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.9e-35:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	elif n <= 1.05e+32:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * ((100.0 * (i * (1.0 + (i * (0.5 + (0.5 * (-1.0 / n))))))) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.9e-35)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	elseif (n <= 1.05e+32)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(Float64(100.0 * Float64(i * Float64(1.0 + Float64(i * Float64(0.5 + Float64(0.5 * Float64(-1.0 / n))))))) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.9e-35)
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	elseif (n <= 1.05e+32)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * ((100.0 * (i * (1.0 + (i * (0.5 + (0.5 * (-1.0 / n))))))) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.9e-35], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e+32], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(100.0 * N[(i * N[(1.0 + N[(i * N[(0.5 + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.9000000000000002e-35

   1. Initial program 22.7%

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

      1. associate-/r/ 23.0%

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

      2. associate-*r* 23.0%

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

      3. *-commutative 23.0%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.0%

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

      7. metadata-eval 23.0%

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

      8. metadata-eval 23.0%

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

      9. metadata-eval 23.0%

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

      10. fma-define 23.1%

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

      11. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in i around 0 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*r/ 60.3%

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

      3. metadata-eval 60.3%

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

   7. Simplified 60.3%

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

   if -2.9000000000000002e-35 < n < 1.05e32

   1. Initial program 26.7%

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

   3. Taylor expanded in i around 0 61.3%

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

   if 1.05e32 < n

   1. Initial program 17.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-*r* 17.6%

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

      3. *-commutative 17.6%

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

      4. associate-*r/ 17.6%

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

      5. sub-neg 17.6%

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

      6. distribute-lft-in 17.6%

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

      7. metadata-eval 17.6%

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

      8. metadata-eval 17.6%

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

      9. metadata-eval 17.6%

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

      10. fma-define 17.6%

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

      11. metadata-eval 17.6%

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

   3. Simplified 17.6%

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

      1. fma-undefine 17.6%

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

      2. metadata-eval 17.6%

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

      3. metadata-eval 17.6%

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

      4. distribute-lft-in 17.6%

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

      5. sub-neg 17.6%

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

      6. *-commutative 17.6%

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

      7. add-exp-log 17.6%

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

      8. expm1-define 17.6%

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

      9. log-pow 16.0%

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

      10. log1p-define 74.0%

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

   6. Applied egg-rr 74.0%

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

   7. Taylor expanded in i around 0 68.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-35}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{100 \cdot \left(i \cdot \left(1 + i \cdot \left(0.5 + 0.5 \cdot \frac{-1}{n}\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\\ \mathbf{if}\;n \leq -3.2 \cdot 10^{-35}:\\ \;\;\;\;n \cdot t\_0\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot t\_0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n)))))))
   (if (<= n -3.2e-35)
     (* n t_0)
     (if (<= n 1.05e+32) (* 100.0 (/ i (/ i n))) (* n (/ (* i t_0) i))))))

C

double code(double i, double n) {
	double t_0 = 100.0 + (100.0 * (i * (0.5 - (0.5 / n))));
	double tmp;
	if (n <= -3.2e-35) {
		tmp = n * t_0;
	} else if (n <= 1.05e+32) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * t_0) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n))))
    if (n <= (-3.2d-35)) then
        tmp = n * t_0
    else if (n <= 1.05d+32) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * ((i * t_0) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = 100.0 + (100.0 * (i * (0.5 - (0.5 / n))));
	double tmp;
	if (n <= -3.2e-35) {
		tmp = n * t_0;
	} else if (n <= 1.05e+32) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * ((i * t_0) / i);
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 + (100.0 * (i * (0.5 - (0.5 / n))))
	tmp = 0
	if n <= -3.2e-35:
		tmp = n * t_0
	elif n <= 1.05e+32:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * ((i * t_0) / i)
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n)))))
	tmp = 0.0
	if (n <= -3.2e-35)
		tmp = Float64(n * t_0);
	elseif (n <= 1.05e+32)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(Float64(i * t_0) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 100.0 + (100.0 * (i * (0.5 - (0.5 / n))));
	tmp = 0.0;
	if (n <= -3.2e-35)
		tmp = n * t_0;
	elseif (n <= 1.05e+32)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * ((i * t_0) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3.2e-35], N[(n * t$95$0), $MachinePrecision], If[LessEqual[n, 1.05e+32], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(N[(i * t$95$0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.1999999999999998e-35

   1. Initial program 22.7%

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

      1. associate-/r/ 23.0%

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

      2. associate-*r* 23.0%

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

      3. *-commutative 23.0%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.0%

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

      7. metadata-eval 23.0%

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

      8. metadata-eval 23.0%

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

      9. metadata-eval 23.0%

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

      10. fma-define 23.1%

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

      11. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in i around 0 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*r/ 60.3%

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

      3. metadata-eval 60.3%

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

   7. Simplified 60.3%

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

   if -3.1999999999999998e-35 < n < 1.05e32

   1. Initial program 26.7%

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

   3. Taylor expanded in i around 0 61.3%

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

   if 1.05e32 < n

   1. Initial program 17.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   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. associate-*r* 17.6%

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

      3. *-commutative 17.6%

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

      4. associate-*r/ 17.6%

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

      5. sub-neg 17.6%

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

      6. distribute-lft-in 17.6%

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

      7. metadata-eval 17.6%

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

      8. metadata-eval 17.6%

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

      9. metadata-eval 17.6%

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

      10. fma-define 17.6%

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

      11. metadata-eval 17.6%

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

   3. Simplified 17.6%

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

   5. Taylor expanded in i around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r/ 68.0%

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

      3. metadata-eval 68.0%

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

   7. Simplified 68.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.2 \cdot 10^{-35}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+32}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{i \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+28} \lor \neg \left(n \leq 0.00048\right):\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -2.3e+28) (not (<= n 0.00048)))
   (* (* n 100.0) (+ 1.0 (* i 0.5)))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -2.3e+28) || !(n <= 0.00048)) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.3d+28)) .or. (.not. (n <= 0.00048d0))) then
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -2.3e+28) || !(n <= 0.00048)) {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -2.3e+28) or not (n <= 0.00048):
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -2.3e+28) || !(n <= 0.00048))
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -2.3e+28) || ~((n <= 0.00048)))
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -2.3e+28], N[Not[LessEqual[n, 0.00048]], $MachinePrecision]], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.29999999999999984e28 or 4.80000000000000012e-4 < n

   1. Initial program 18.7%

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

   3. Taylor expanded in i around 0 14.1%

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

      1. associate-*r/ 14.1%

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

      2. metadata-eval 14.1%

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

   5. Simplified 14.1%

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

   6. Taylor expanded in n around inf 64.1%

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

      1. associate-*r* 64.1%

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

      2. *-commutative 64.1%

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

      3. *-commutative 64.1%

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

   8. Simplified 64.1%

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

   if -2.29999999999999984e28 < n < 4.80000000000000012e-4

   1. Initial program 28.3%

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

   3. Taylor expanded in i around 0 59.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+28} \lor \neg \left(n \leq 0.00048\right):\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+29} \lor \neg \left(n \leq 1.5\right):\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9.5e+29) (not (<= n 1.5)))
   (* 100.0 (* n (+ 1.0 (* i 0.5))))
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9.5e+29) || !(n <= 1.5)) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-9.5d+29)) .or. (.not. (n <= 1.5d0))) then
        tmp = 100.0d0 * (n * (1.0d0 + (i * 0.5d0)))
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9.5e+29) || !(n <= 1.5)) {
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9.5e+29) or not (n <= 1.5):
		tmp = 100.0 * (n * (1.0 + (i * 0.5)))
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9.5e+29) || !(n <= 1.5))
		tmp = Float64(100.0 * Float64(n * Float64(1.0 + Float64(i * 0.5))));
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -9.5e+29) || ~((n <= 1.5)))
		tmp = 100.0 * (n * (1.0 + (i * 0.5)));
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9.5e+29], N[Not[LessEqual[n, 1.5]], $MachinePrecision]], N[(100.0 * N[(n * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.5000000000000003e29 or 1.5 < n

   1. Initial program 18.7%

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

   3. Taylor expanded in i around 0 14.1%

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

      1. associate-*r/ 14.1%

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

      2. metadata-eval 14.1%

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

   5. Simplified 14.1%

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

   6. Taylor expanded in n around inf 64.1%

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

      1. *-commutative 64.1%

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

   8. Simplified 64.1%

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

   if -9.5000000000000003e29 < n < 1.5

   1. Initial program 28.3%

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

   3. Taylor expanded in i around 0 59.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+29} \lor \neg \left(n \leq 1.5\right):\\ \;\;\;\;100 \cdot \left(n \cdot \left(1 + i \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-35}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.2e-35)
   (* n (+ 100.0 (* 100.0 (* i (- 0.5 (/ 0.5 n))))))
   (if (<= n 1.5) (* 100.0 (/ i (/ i n))) (* (* n 100.0) (+ 1.0 (* i 0.5))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.2e-35) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (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 <= (-9.2d-35)) then
        tmp = n * (100.0d0 + (100.0d0 * (i * (0.5d0 - (0.5d0 / n)))))
    else if (n <= 1.5d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.2e-35) {
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.2e-35:
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))))
	elif n <= 1.5:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.2e-35)
		tmp = Float64(n * Float64(100.0 + Float64(100.0 * Float64(i * Float64(0.5 - Float64(0.5 / n))))));
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.2e-35)
		tmp = n * (100.0 + (100.0 * (i * (0.5 - (0.5 / n)))));
	elseif (n <= 1.5)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.2e-35], N[(n * N[(100.0 + N[(100.0 * N[(i * N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.1999999999999996e-35

   1. Initial program 22.7%

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

      1. associate-/r/ 23.0%

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

      2. associate-*r* 23.0%

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

      3. *-commutative 23.0%

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

      4. associate-*r/ 23.1%

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

      5. sub-neg 23.1%

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

      6. distribute-lft-in 23.0%

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

      7. metadata-eval 23.0%

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

      8. metadata-eval 23.0%

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

      9. metadata-eval 23.0%

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

      10. fma-define 23.1%

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

      11. metadata-eval 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in i around 0 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*r/ 60.3%

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

      3. metadata-eval 60.3%

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

   7. Simplified 60.3%

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

   if -9.1999999999999996e-35 < n < 1.5

   1. Initial program 28.0%

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

   3. Taylor expanded in i around 0 59.2%

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

   if 1.5 < n

   1. Initial program 16.1%

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

   3. Taylor expanded in i around 0 18.6%

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

      1. associate-*r/ 18.6%

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

      2. metadata-eval 18.6%

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

   5. Simplified 18.6%

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

   6. Taylor expanded in n around inf 69.1%

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

      1. associate-*r* 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

   8. Simplified 69.1%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-35}:\\ \;\;\;\;n \cdot \left(100 + 100 \cdot \left(i \cdot \left(0.5 - \frac{0.5}{n}\right)\right)\right)\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-34}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.35e-34)
   (* 100.0 (+ n (* (- 0.5 (/ 0.5 n)) (* i n))))
   (if (<= n 1.5) (* 100.0 (/ i (/ i n))) (* (* n 100.0) (+ 1.0 (* i 0.5))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.35e-34) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (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 <= (-1.35d-34)) then
        tmp = 100.0d0 * (n + ((0.5d0 - (0.5d0 / n)) * (i * n)))
    else if (n <= 1.5d0) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * 100.0d0) * (1.0d0 + (i * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.35e-34) {
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	} else if (n <= 1.5) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.35e-34:
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)))
	elif n <= 1.5:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * 100.0) * (1.0 + (i * 0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.35e-34)
		tmp = Float64(100.0 * Float64(n + Float64(Float64(0.5 - Float64(0.5 / n)) * Float64(i * n))));
	elseif (n <= 1.5)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * 100.0) * Float64(1.0 + Float64(i * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.35e-34)
		tmp = 100.0 * (n + ((0.5 - (0.5 / n)) * (i * n)));
	elseif (n <= 1.5)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * 100.0) * (1.0 + (i * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.35e-34], N[(100.0 * N[(n + N[(N[(0.5 - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * 100.0), $MachinePrecision] * N[(1.0 + N[(i * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.35000000000000008e-34

   1. Initial program 22.7%

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

   3. Taylor expanded in i around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-*r/ 60.3%

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

      4. metadata-eval 60.3%

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

   5. Simplified 60.3%

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

   if -1.35000000000000008e-34 < n < 1.5

   1. Initial program 28.0%

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

   3. Taylor expanded in i around 0 59.2%

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

   if 1.5 < n

   1. Initial program 16.1%

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

   3. Taylor expanded in i around 0 18.6%

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

      1. associate-*r/ 18.6%

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

      2. metadata-eval 18.6%

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

   5. Simplified 18.6%

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

   6. Taylor expanded in n around inf 69.1%

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

      1. associate-*r* 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

   8. Simplified 69.1%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-34}:\\ \;\;\;\;100 \cdot \left(n + \left(0.5 - \frac{0.5}{n}\right) \cdot \left(i \cdot n\right)\right)\\ \mathbf{elif}\;n \leq 1.5:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot 100\right) \cdot \left(1 + i \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{+74} \lor \neg \left(n \leq 1.1 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.3e+74) (not (<= n 1.1e+32)))
   (/ (* n (* i 100.0)) i)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e+74) || !(n <= 1.1e+32)) {
		tmp = (n * (i * 100.0)) / i;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.3d+74)) .or. (.not. (n <= 1.1d+32))) then
        tmp = (n * (i * 100.0d0)) / i
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.3e+74) || !(n <= 1.1e+32)) {
		tmp = (n * (i * 100.0)) / i;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.3e+74) or not (n <= 1.1e+32):
		tmp = (n * (i * 100.0)) / i
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.3e+74) || !(n <= 1.1e+32))
		tmp = Float64(Float64(n * Float64(i * 100.0)) / i);
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -1.3e+74) || ~((n <= 1.1e+32)))
		tmp = (n * (i * 100.0)) / i;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.3e+74], N[Not[LessEqual[n, 1.1e+32]], $MachinePrecision]], N[(N[(n * N[(i * 100.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.3e74 or 1.1e32 < n

   1. Initial program 19.1%

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

      1. associate-*r/ 19.1%

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

      2. sub-neg 19.1%

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

      3. distribute-rgt-in 19.1%

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

      4. metadata-eval 19.1%

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

      5. metadata-eval 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in i around 0 4.8%

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

      1. *-commutative 4.8%

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

   7. Simplified 4.8%

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

      1. clear-num 4.7%

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

      2. associate-/r/ 4.8%

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

   9. Applied egg-rr 4.8%

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

       1. *-un-lft-identity 4.8%

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

       2. times-frac 5.3%

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

       3. clear-num 5.3%

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

       4. /-rgt-identity 5.3%

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

       5. +-commutative 5.3%

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

       6. associate-+l+ 53.2%

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

       7. metadata-eval 53.2%

          \[\leadsto n \cdot \frac{i \cdot 100 + \color{blue}{0}}{i} \]

   11. Applied egg-rr 53.2%

       \[\leadsto \color{blue}{n \cdot \frac{i \cdot 100 + 0}{i}} \]
   12. Step-by-step derivation

       1. associate-*r/ 62.5%

          \[\leadsto \color{blue}{\frac{n \cdot \left(i \cdot 100 + 0\right)}{i}} \]

       2. +-rgt-identity 62.5%

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

       3. *-commutative 62.5%

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

   13. Simplified 62.5%

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

   if -1.3e74 < n < 1.1e32

   1. Initial program 26.6%

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

   3. Taylor expanded in i around 0 61.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.3 \cdot 10^{+74} \lor \neg \left(n \leq 1.1 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{n \cdot \left(i \cdot 100\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-35} \lor \neg \left(n \leq 1.65 \cdot 10^{-62}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -5e-35) (not (<= n 1.65e-62)))
   (* n 100.0)
   (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -5e-35) || !(n <= 1.65e-62)) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-5d-35)) .or. (.not. (n <= 1.65d-62))) then
        tmp = n * 100.0d0
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -5e-35) || !(n <= 1.65e-62)) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -5e-35) or not (n <= 1.65e-62):
		tmp = n * 100.0
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -5e-35) || !(n <= 1.65e-62))
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -5e-35) || ~((n <= 1.65e-62)))
		tmp = n * 100.0;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -5e-35], N[Not[LessEqual[n, 1.65e-62]], $MachinePrecision]], N[(n * 100.0), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -4.99999999999999964e-35 or 1.65000000000000002e-62 < n

   1. Initial program 20.9%

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

      1. associate-/r/ 21.3%

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

      2. associate-*r* 21.3%

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

      3. *-commutative 21.3%

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

      4. associate-*r/ 21.3%

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

      5. sub-neg 21.3%

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

      6. distribute-lft-in 21.3%

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

      7. metadata-eval 21.3%

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

      8. metadata-eval 21.3%

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

      9. metadata-eval 21.3%

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

      10. fma-define 21.3%

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

      11. metadata-eval 21.3%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in i around 0 54.5%

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

      1. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   if -4.99999999999999964e-35 < n < 1.65000000000000002e-62

   1. Initial program 27.2%

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

   3. Taylor expanded in i around 0 63.4%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-35} \lor \neg \left(n \leq 1.65 \cdot 10^{-62}\right):\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.3% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.9%

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

   1. associate-/r/ 23.0%

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

   2. associate-*r* 23.0%

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

   3. *-commutative 23.0%

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

   4. associate-*r/ 23.0%

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

   5. sub-neg 23.0%

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

   6. distribute-lft-in 23.0%

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

   7. metadata-eval 23.0%

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

   8. metadata-eval 23.0%

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

   9. metadata-eval 23.0%

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

   10. fma-define 23.0%

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

   11. metadata-eval 23.0%

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

3. Simplified 23.0%

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

5. Taylor expanded in i around 0 48.8%

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

   1. *-commutative 48.8%

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

7. Simplified 48.8%

   \[\leadsto \color{blue}{n \cdot 100} \]
8. Add Preprocessing

Alternative 15: 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 22.9%

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

3. Taylor expanded in i around 0 9.9%

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

   1. associate-*r/ 9.9%

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

   2. metadata-eval 9.9%

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

5. Simplified 9.9%

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

6. Taylor expanded in n around 0 2.9%

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

   1. *-commutative 2.9%

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

8. Simplified 2.9%

   \[\leadsto \color{blue}{i \cdot -50} \]
9. Add Preprocessing

Developer Target 1: 34.8% 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 2024150 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))