Compound Interest

Percentage Accurate: 28.9% → 98.1%

Time: 17.6s

Alternatives: 17

Speedup: 38.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.9% 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: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot 2\right)}{i \cdot \left(i + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_0 INFINITY)
       (* t_0 100.0)
       (* 100.0 (/ (* n (* i 2.0)) (* i (+ i 2.0))))))))

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 = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * ((n * (i * 2.0)) / (i * (i + 2.0)));
	}
	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 = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * ((n * (i * 2.0)) / (i * (i + 2.0)));
	}
	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 = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = 100.0 * ((n * (i * 2.0)) / (i * (i + 2.0)))
	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(100.0 * Float64(expm1(Float64(n * log1p(Float64(i / n)))) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * 2.0)) / Float64(i * Float64(i + 2.0))));
	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[(100.0 * N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / N[(i * N[(i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 23.8%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(e^{\log \left(1 + \frac{i}{n}\right) \cdot n} - 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      3. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\left(n \cdot \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \log \left(1 + \frac{i}{n}\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{log1p}\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\left(\frac{i}{n}\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      8. /-lowering-/.f64 98.1%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(i, n\right)\right)\right)\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   4. Applied egg-rr 98.1%

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

   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 99.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(1 + i\right)}, 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(i + 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. +-lowering-+.f64 20.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(i, 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   5. Simplified 20.9%

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

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\left(\left(i + 1\right) - 1\right) \cdot \color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) \]

      2. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1}{\left(i + 1\right) + 1} \cdot \frac{\color{blue}{1}}{\frac{i}{n}}\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1}{\left(i + 1\right) + 1} \cdot \frac{n}{\color{blue}{i}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\color{blue}{\left(\left(i + 1\right) + 1\right) \cdot i}}\right)\right) \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(\left(i + 1\right) + 1\right) \cdot \left(i + \color{blue}{0}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(\left(i + 1\right) + 1\right) \cdot \left(i + \left(1 - \color{blue}{1}\right)\right)}\right)\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(\left(i + 1\right) + 1\right) \cdot \left(\left(i + 1\right) - \color{blue}{1}\right)}\right)\right) \]

      8. difference-of-sqr-1 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(i + 1\right) \cdot \left(i + 1\right) - \color{blue}{1}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot \color{blue}{1}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n\right), \color{blue}{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right)}\right)\right) \]

   7. Applied egg-rr 79.6%

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

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot i\right)}, n\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(i, 2\right), i\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot 2\right), n\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{i}, 2\right), i\right)\right)\right) \]

      2. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, 2\right), n\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{i}, 2\right), i\right)\right)\right) \]

   10. Simplified 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot 2\right)}{i \cdot \left(i + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot 2\right)}{i \cdot \left(i + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (* 100.0 (/ (expm1 i) (/ i n)))
     (if (<= t_0 INFINITY)
       (* t_0 100.0)
       (* 100.0 (/ (* n (* i 2.0)) (* i (+ i 2.0))))))))

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 = 100.0 * (expm1(i) / (i / n));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * ((n * (i * 2.0)) / (i * (i + 2.0)));
	}
	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 = 100.0 * (Math.expm1(i) / (i / n));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * ((n * (i * 2.0)) / (i * (i + 2.0)));
	}
	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 = 100.0 * (math.expm1(i) / (i / n))
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = 100.0 * ((n * (i * 2.0)) / (i * (i + 2.0)))
	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(100.0 * Float64(expm1(i) / Float64(i / n)));
	elseif (t_0 <= Inf)
		tmp = Float64(t_0 * 100.0);
	else
		tmp = Float64(100.0 * Float64(Float64(n * Float64(i * 2.0)) / Float64(i * Float64(i + 2.0))));
	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[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(100.0 * N[(N[(n * N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / N[(i * N[(i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 23.8%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 79.5%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 79.5%

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

   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 99.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(1 + i\right)}, 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(i + 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. +-lowering-+.f64 20.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(i, 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   5. Simplified 20.9%

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

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\left(\left(i + 1\right) - 1\right) \cdot \color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) \]

      2. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1}{\left(i + 1\right) + 1} \cdot \frac{\color{blue}{1}}{\frac{i}{n}}\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1}{\left(i + 1\right) + 1} \cdot \frac{n}{\color{blue}{i}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\color{blue}{\left(\left(i + 1\right) + 1\right) \cdot i}}\right)\right) \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(\left(i + 1\right) + 1\right) \cdot \left(i + \color{blue}{0}\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(\left(i + 1\right) + 1\right) \cdot \left(i + \left(1 - \color{blue}{1}\right)\right)}\right)\right) \]

      7. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(\left(i + 1\right) + 1\right) \cdot \left(\left(i + 1\right) - \color{blue}{1}\right)}\right)\right) \]

      8. difference-of-sqr-1 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(i + 1\right) \cdot \left(i + 1\right) - \color{blue}{1}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\frac{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n}{\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot \color{blue}{1}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right) \cdot n\right), \color{blue}{\left(\left(i + 1\right) \cdot \left(i + 1\right) - 1 \cdot 1\right)}\right)\right) \]

   7. Applied egg-rr 79.6%

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

   8. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot i\right)}, n\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(i, 2\right), i\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot 2\right), n\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{i}, 2\right), i\right)\right)\right) \]

      2. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, 2\right), n\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{i}, 2\right), i\right)\right)\right) \]

   10. Simplified 99.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \left(i \cdot 2\right)}{i \cdot \left(i + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{if}\;n \leq -2.5 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ (* n (expm1 i)) i))))
   (if (<= n -2.5e-57) t_0 (if (<= n 3.7e-23) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = 100.0 * ((n * expm1(i)) / i);
	double tmp;
	if (n <= -2.5e-57) {
		tmp = t_0;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * ((n * Math.expm1(i)) / i);
	double tmp;
	if (n <= -2.5e-57) {
		tmp = t_0;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * ((n * math.expm1(i)) / i)
	tmp = 0
	if n <= -2.5e-57:
		tmp = t_0
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(Float64(n * expm1(i)) / i))
	tmp = 0.0
	if (n <= -2.5e-57)
		tmp = t_0;
	elseif (n <= 3.7e-23)
		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[(100.0 * N[(N[(n * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.5e-57], t$95$0, If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.5000000000000001e-57 or 3.7000000000000003e-23 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 90.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 90.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{n \cdot \left(e^{i} - 1\right)}{i}\right), \color{blue}{100}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), i\right), 100\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), i\right), 100\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), i\right), 100\right) \]

      7. expm1-lowering-expm1.f64 90.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), i\right), 100\right) \]

   7. Applied egg-rr 90.3%

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

   if -2.5000000000000001e-57 < n < 3.7000000000000003e-23

   1. Initial program 31.5%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 69.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{-57}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot \mathsf{expm1}\left(i\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot \mathsf{expm1}\left(i\right)\right) \cdot \frac{100}{i}\\ \mathbf{if}\;n \leq -4.1 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 31000000:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* n (expm1 i)) (/ 100.0 i))))
   (if (<= n -4.1e-57)
     t_0
     (if (<= n 31000000.0) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (n * expm1(i)) * (100.0 / i);
	double tmp;
	if (n <= -4.1e-57) {
		tmp = t_0;
	} else if (n <= 31000000.0) {
		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 * Math.expm1(i)) * (100.0 / i);
	double tmp;
	if (n <= -4.1e-57) {
		tmp = t_0;
	} else if (n <= 31000000.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (n * math.expm1(i)) * (100.0 / i)
	tmp = 0
	if n <= -4.1e-57:
		tmp = t_0
	elif n <= 31000000.0:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(n * expm1(i)) * Float64(100.0 / i))
	tmp = 0.0
	if (n <= -4.1e-57)
		tmp = t_0;
	elseif (n <= 31000000.0)
		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[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.1e-57], t$95$0, If[LessEqual[n, 31000000.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.1000000000000001e-57 or 3.1e7 < n

   1. Initial program 26.0%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 89.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 89.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(n \cdot \left(e^{i} - 1\right)\right), \color{blue}{\left(\frac{100}{i}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right), \left(\frac{\color{blue}{100}}{i}\right)\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right), \left(\frac{100}{i}\right)\right) \]

      6. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), \left(\frac{100}{i}\right)\right) \]

      7. /-lowering-/.f64 89.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right), \mathsf{/.f64}\left(100, \color{blue}{i}\right)\right) \]

   7. Applied egg-rr 89.2%

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

   if -4.1000000000000001e-57 < n < 3.1e7

   1. Initial program 30.0%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 70.6%

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

Alternative 5: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -0.001:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(i \cdot \left(n \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right) + n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -0.001)
   (* 100.0 (/ (expm1 i) (/ i n)))
   (+
    (* i (* i (* n (+ 16.666666666666668 (* i 4.166666666666667)))))
    (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -0.001) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = (i * (i * (n * (16.666666666666668 + (i * 4.166666666666667))))) + (n * (100.0 + (i * 50.0)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -0.001) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = (i * (i * (n * (16.666666666666668 + (i * 4.166666666666667))))) + (n * (100.0 + (i * 50.0)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -0.001:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = (i * (i * (n * (16.666666666666668 + (i * 4.166666666666667))))) + (n * (100.0 + (i * 50.0)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -0.001)
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(i * Float64(n * Float64(16.666666666666668 + Float64(i * 4.166666666666667))))) + Float64(n * Float64(100.0 + Float64(i * 50.0))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -0.001], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(i * N[(n * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1e-3

   1. Initial program 55.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{\left(e^{i} - 1\right)}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(i\right)\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

      2. expm1-lowering-expm1.f64 85.6%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(i\right), \mathsf{/.f64}\left(\color{blue}{i}, n\right)\right)\right) \]

   5. Simplified 85.6%

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

   if -1e-3 < i

   1. Initial program 20.2%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 69.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 69.0%

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

   6. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto 100 \cdot n + \left(\left(50 \cdot n\right) \cdot i + \color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \cdot i}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right) + \color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \cdot i} \]

      3. *-commutative N/A

         \[\leadsto \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right) + i \cdot \color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(n \cdot \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right), \left(\color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right), \left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right), \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)}\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \left(\left(\frac{25}{6} \cdot i\right) \cdot n + \color{blue}{\frac{50}{3}} \cdot n\right)\right)\right)\right) \]

      16. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{25}{6} \cdot i + \frac{50}{3}\right)}\right)\right)\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{25}{6} \cdot i\right), \color{blue}{\frac{50}{3}}\right)\right)\right)\right)\right) \]

   8. Simplified 76.0%

      \[\leadsto \color{blue}{n \cdot \left(100 + i \cdot 50\right) + i \cdot \left(i \cdot \left(n \cdot \left(i \cdot 4.166666666666667 + 16.666666666666668\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -0.001:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(i \cdot \left(n \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right) + n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\right)}{i}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.5e+64)
   (/ (* i (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))) i)
   (if (<= n 5.3e-24)
     (* 100.0 (/ i (/ i n)))
     (+
      (* n (+ 100.0 (* i 50.0)))
      (* (* i i) (* n (+ 16.666666666666668 (* i 4.166666666666667))))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i;
	} else if (n <= 5.3e-24) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.5d+64)) then
        tmp = (i * ((n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0)))))) / i
    else if (n <= 5.3d-24) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (n * (100.0d0 + (i * 50.0d0))) + ((i * i) * (n * (16.666666666666668d0 + (i * 4.166666666666667d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i;
	} else if (n <= 5.3e-24) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.5e+64:
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i
	elif n <= 5.3e-24:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.5e+64)
		tmp = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))))) / i);
	elseif (n <= 5.3e-24)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(n * Float64(100.0 + Float64(i * 50.0))) + Float64(Float64(i * i) * Float64(n * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.5e+64)
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i;
	elseif (n <= 5.3e-24)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (n * (100.0 + (i * 50.0))) + ((i * i) * (n * (16.666666666666668 + (i * 4.166666666666667))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.5e+64], N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 5.3e-24], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(i * i), $MachinePrecision] * N[(n * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.5000000000000001e64

   1. Initial program 22.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 88.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 88.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right), i\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(100 \cdot n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)\right), i\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)\right), i\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + 50 \cdot n\right)\right)\right)\right), i\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)\right)\right)\right), i\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(\frac{50}{3} \cdot i + 50\right)\right)\right)\right)\right), i\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), 50\right)\right)\right)\right)\right), i\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(i \cdot \frac{50}{3}\right), 50\right)\right)\right)\right)\right), i\right) \]

      10. *-lowering-*.f64 62.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{50}{3}\right), 50\right)\right)\right)\right)\right), i\right) \]

   8. Simplified 62.1%

      \[\leadsto \frac{\color{blue}{i \cdot \left(100 \cdot n + i \cdot \left(n \cdot \left(i \cdot 16.666666666666668 + 50\right)\right)\right)}}{i} \]

   if -1.5000000000000001e64 < n < 5.29999999999999969e-24

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

   if 5.29999999999999969e-24 < n

   1. Initial program 25.3%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 25.3%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 45.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 45.5%

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

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(50 \cdot n + i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto 100 \cdot n + \left(i \cdot \left(50 \cdot n\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(100 \cdot n + i \cdot \left(50 \cdot n\right)\right) + \color{blue}{i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(100 \cdot n + i \cdot \left(n \cdot 50\right)\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(100 \cdot n + \left(i \cdot n\right) \cdot 50\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(100 \cdot n + \left(\left(100 \cdot \frac{1}{2}\right) \cdot i\right) \cdot n\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(100 \cdot n + \left(100 \cdot \left(\frac{1}{2} \cdot i\right)\right) \cdot n\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto n \cdot \left(100 + 100 \cdot \left(\frac{1}{2} \cdot i\right)\right) + \color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto n \cdot \left(100 \cdot 1 + 100 \cdot \left(\frac{1}{2} \cdot i\right)\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      11. distribute-lft-in N/A

          \[\leadsto n \cdot \left(100 \cdot \left(1 + \frac{1}{2} \cdot i\right)\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(n \cdot 100\right) \cdot \left(1 + \frac{1}{2} \cdot i\right) + \color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(100 \cdot n\right) \cdot \left(1 + \frac{1}{2} \cdot i\right) + i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto 100 \cdot \left(n \cdot \left(1 + \frac{1}{2} \cdot i\right)\right) + \color{blue}{i} \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot \left(n \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\frac{25}{6} \cdot \left(i \cdot n\right) + \frac{50}{3} \cdot n\right)\right)\right)}\right) \]

   10. Simplified 78.5%

       \[\leadsto \color{blue}{n \cdot \left(100 + 50 \cdot i\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(4.166666666666667 \cdot i + 16.666666666666668\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\right)}{i}\\ \mathbf{elif}\;n \leq 5.3 \cdot 10^{-24}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right) + \left(i \cdot i\right) \cdot \left(n \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+64}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\right)}{i}\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.55e+64)
   (/ (* i (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))) i)
   (if (<= n 3.7e-23)
     (* 100.0 (/ i (/ i n)))
     (/ (* i (* n (+ 100.0 (* i 50.0)))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.55e+64) {
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.55d+64)) then
        tmp = (i * ((n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0)))))) / i
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.55e+64) {
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.55e+64:
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.55e+64)
		tmp = Float64(Float64(i * Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))))) / i);
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.55e+64)
		tmp = (i * ((n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668)))))) / i;
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.55e+64], N[(N[(i * N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.55e64

   1. Initial program 22.4%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 88.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 88.3%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right), i\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(100 \cdot n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)\right), i\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right)\right), i\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + 50 \cdot n\right)\right)\right)\right), i\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{50}{3} \cdot i + 50\right)\right)\right)\right)\right), i\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(\frac{50}{3} \cdot i + 50\right)\right)\right)\right)\right), i\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), 50\right)\right)\right)\right)\right), i\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(i \cdot \frac{50}{3}\right), 50\right)\right)\right)\right)\right), i\right) \]

      10. *-lowering-*.f64 62.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{50}{3}\right), 50\right)\right)\right)\right)\right), i\right) \]

   8. Simplified 62.1%

      \[\leadsto \frac{\color{blue}{i \cdot \left(100 \cdot n + i \cdot \left(n \cdot \left(i \cdot 16.666666666666668 + 50\right)\right)\right)}}{i} \]

   if -1.55e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

   if 3.7000000000000003e-23 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 96.1%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right)\right), i\right) \]

      12. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right)\right), i\right) \]

   8. Simplified 77.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.55 \cdot 10^{+64}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\right)}{i}\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{-100}{i} \cdot \left(i \cdot \left(i \cdot \left(n \cdot \left(i \cdot -0.16666666666666666 + -0.5\right)\right) - n\right)\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.5e+64)
   (* (/ -100.0 i) (* i (- (* i (* n (+ (* i -0.16666666666666666) -0.5))) n)))
   (if (<= n 3.7e-23)
     (* 100.0 (/ i (/ i n)))
     (/ (* i (* n (+ 100.0 (* i 50.0)))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = (-100.0 / i) * (i * ((i * (n * ((i * -0.16666666666666666) + -0.5))) - n));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.5d+64)) then
        tmp = ((-100.0d0) / i) * (i * ((i * (n * ((i * (-0.16666666666666666d0)) + (-0.5d0)))) - n))
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = (-100.0 / i) * (i * ((i * (n * ((i * -0.16666666666666666) + -0.5))) - n));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.5e+64:
		tmp = (-100.0 / i) * (i * ((i * (n * ((i * -0.16666666666666666) + -0.5))) - n))
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.5e+64)
		tmp = Float64(Float64(-100.0 / i) * Float64(i * Float64(Float64(i * Float64(n * Float64(Float64(i * -0.16666666666666666) + -0.5))) - n)));
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.5e+64)
		tmp = (-100.0 / i) * (i * ((i * (n * ((i * -0.16666666666666666) + -0.5))) - n));
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.5e+64], N[(N[(-100.0 / i), $MachinePrecision] * N[(i * N[(N[(i * N[(n * N[(N[(i * -0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.5000000000000001e64

   1. Initial program 22.4%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 22.3%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 42.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 42.3%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-100}{i}\right), \color{blue}{\left(\frac{1 - e^{i}}{\frac{1}{n}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \left(\frac{\color{blue}{1 - e^{i}}}{\frac{1}{n}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{/.f64}\left(\left(1 - e^{i}\right), \color{blue}{\left(\frac{1}{n}\right)}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(e^{i}\right)\right), \left(\frac{\color{blue}{1}}{n}\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right), \left(\frac{1}{n}\right)\right)\right) \]

      9. /-lowering-/.f64 42.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right), \mathsf{/.f64}\left(1, \color{blue}{n}\right)\right)\right) \]

   9. Applied egg-rr 42.9%

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

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \color{blue}{\left(i \cdot \left(-1 \cdot n + i \cdot \left(\frac{-1}{2} \cdot n + \frac{-1}{6} \cdot \left(i \cdot n\right)\right)\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \color{blue}{\left(-1 \cdot n + i \cdot \left(\frac{-1}{2} \cdot n + \frac{-1}{6} \cdot \left(i \cdot n\right)\right)\right)}\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \left(i \cdot \left(\frac{-1}{2} \cdot n + \frac{-1}{6} \cdot \left(i \cdot n\right)\right) + \color{blue}{-1 \cdot n}\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \left(i \cdot \left(\frac{-1}{2} \cdot n + \frac{-1}{6} \cdot \left(i \cdot n\right)\right) + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right) \]

       4. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \left(i \cdot \left(\frac{-1}{2} \cdot n + \frac{-1}{6} \cdot \left(i \cdot n\right)\right) - \color{blue}{n}\right)\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\left(i \cdot \left(\frac{-1}{2} \cdot n + \frac{-1}{6} \cdot \left(i \cdot n\right)\right)\right), \color{blue}{n}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \left(\frac{-1}{2} \cdot n + \frac{-1}{6} \cdot \left(i \cdot n\right)\right)\right), n\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \left(\frac{-1}{6} \cdot \left(i \cdot n\right) + \frac{-1}{2} \cdot n\right)\right), n\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \left(\left(\frac{-1}{6} \cdot i\right) \cdot n + \frac{-1}{2} \cdot n\right)\right), n\right)\right)\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{-1}{6} \cdot i + \frac{-1}{2}\right)\right)\right), n\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{-1}{6} \cdot i + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), n\right)\right)\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right)\right)\right), n\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right)\right)\right), n\right)\right)\right) \]

       13. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(\frac{-1}{6} \cdot i + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), n\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(\frac{-1}{6} \cdot i + \frac{-1}{2}\right)\right)\right), n\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot i\right), \frac{-1}{2}\right)\right)\right), n\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{6}\right), \frac{-1}{2}\right)\right)\right), n\right)\right)\right) \]

       17. *-lowering-*.f64 60.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, i\right), \mathsf{*.f64}\left(i, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{6}\right), \frac{-1}{2}\right)\right)\right), n\right)\right)\right) \]

   12. Simplified 60.3%

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

   if -1.5000000000000001e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

   if 3.7000000000000003e-23 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 96.1%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right)\right), i\right) \]

      12. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right)\right), i\right) \]

   8. Simplified 77.3%

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

Alternative 9: 65.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -2.8e+64)
   (+ (* n 100.0) (* i (* n (+ 50.0 (* i 16.666666666666668)))))
   (if (<= n 3.7e-23)
     (* 100.0 (/ i (/ i n)))
     (/ (* i (* n (+ 100.0 (* i 50.0)))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+64) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.8d+64)) then
        tmp = (n * 100.0d0) + (i * (n * (50.0d0 + (i * 16.666666666666668d0))))
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -2.8e+64) {
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -2.8e+64:
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))))
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -2.8e+64)
		tmp = Float64(Float64(n * 100.0) + Float64(i * Float64(n * Float64(50.0 + Float64(i * 16.666666666666668)))));
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -2.8e+64)
		tmp = (n * 100.0) + (i * (n * (50.0 + (i * 16.666666666666668))));
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -2.8e+64], N[(N[(n * 100.0), $MachinePrecision] + N[(i * N[(n * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.80000000000000024e64

   1. Initial program 22.4%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 22.3%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(e^{i}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. exp-lowering-exp.f64 42.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{\_.f64}\left(1, \mathsf{exp.f64}\left(i\right)\right)\right) \]

   7. Simplified 42.3%

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

   8. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(\color{blue}{i} \cdot \left(\frac{50}{3} \cdot \left(i \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(\left(i \cdot n\right) \cdot \frac{50}{3} + \color{blue}{50} \cdot n\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(i \cdot \left(n \cdot \frac{50}{3}\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \left(i \cdot \left(i \cdot \left(\frac{50}{3} \cdot n\right) + 50 \cdot n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(\frac{50}{3} \cdot n\right) + 50 \cdot n\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(i \cdot \left(n \cdot \frac{50}{3}\right) + 50 \cdot n\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(i \cdot n\right) \cdot \frac{50}{3} + \color{blue}{50} \cdot n\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\frac{50}{3} \cdot \left(i \cdot n\right) + \color{blue}{50} \cdot n\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(\left(\frac{50}{3} \cdot i\right) \cdot n + \color{blue}{50} \cdot n\right)\right)\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{\left(\frac{50}{3} \cdot i + 50\right)}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{50}{3} \cdot i\right), \color{blue}{50}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 57.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(100, n\right), \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{50}{3}, i\right), 50\right)\right)\right)\right) \]

   10. Simplified 57.7%

       \[\leadsto \color{blue}{100 \cdot n + i \cdot \left(n \cdot \left(16.666666666666668 \cdot i + 50\right)\right)} \]

   if -2.80000000000000024e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

   if 3.7000000000000003e-23 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 96.1%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right)\right), i\right) \]

      12. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right)\right), i\right) \]

   8. Simplified 77.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+64}:\\ \;\;\;\;n \cdot 100 + i \cdot \left(n \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;\left(n \cdot -100\right) \cdot \left(i \cdot \left(i \cdot -0.16666666666666666 + -0.5\right) + -1\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.5e+64)
   (* (* n -100.0) (+ (* i (+ (* i -0.16666666666666666) -0.5)) -1.0))
   (if (<= n 3.7e-23)
     (* 100.0 (/ i (/ i n)))
     (/ (* i (* n (+ 100.0 (* i 50.0)))) i))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = (n * -100.0) * ((i * ((i * -0.16666666666666666) + -0.5)) + -1.0);
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.5d+64)) then
        tmp = (n * (-100.0d0)) * ((i * ((i * (-0.16666666666666666d0)) + (-0.5d0))) + (-1.0d0))
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = (n * -100.0) * ((i * ((i * -0.16666666666666666) + -0.5)) + -1.0);
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.5e+64:
		tmp = (n * -100.0) * ((i * ((i * -0.16666666666666666) + -0.5)) + -1.0)
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.5e+64)
		tmp = Float64(Float64(n * -100.0) * Float64(Float64(i * Float64(Float64(i * -0.16666666666666666) + -0.5)) + -1.0));
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.5e+64)
		tmp = (n * -100.0) * ((i * ((i * -0.16666666666666666) + -0.5)) + -1.0);
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = (i * (n * (100.0 + (i * 50.0)))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.5e+64], N[(N[(n * -100.0), $MachinePrecision] * N[(N[(i * N[(N[(i * -0.16666666666666666), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.5000000000000001e64

   1. Initial program 22.4%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. associate-*l/ N/A

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

      14. distribute-neg-frac2 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{\frac{i}{n}}\right)\right) \cdot 100\right), \color{blue}{\left(1 - {\left(1 + \frac{i}{n}\right)}^{n}\right)}\right) \]

   3. Simplified 22.3%

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

   5. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \color{blue}{\left(i \cdot \left(i \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{n} + i \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right) - \frac{1}{2}\right) - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{*.f64}\left(i, \color{blue}{\left(i \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{n} + i \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right) - \frac{1}{2}\right) - 1\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{*.f64}\left(i, \left(i \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{n} + i \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right) - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{*.f64}\left(i, \left(i \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{n} + i \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right) - \frac{1}{2}\right) + -1\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-100, \mathsf{/.f64}\left(i, n\right)\right), \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(i \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{n} + i \cdot \left(\frac{1}{2} \cdot \frac{1}{n} - \left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right)\right)\right) - \frac{1}{2}\right)\right), \color{blue}{-1}\right)\right)\right) \]

   7. Simplified 28.3%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{-100 \cdot \left(n \cdot \left(i \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right) - 1\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-100 \cdot n\right) \cdot \color{blue}{\left(i \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right) - 1\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-100 \cdot n\right), \color{blue}{\left(i \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right) - 1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(\color{blue}{i \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right)} - 1\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(i \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(i \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right) + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\left(i \cdot \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right)\right), \color{blue}{-1}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \left(\frac{-1}{6} \cdot i - \frac{1}{2}\right)\right), -1\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \left(\frac{-1}{6} \cdot i + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \left(\frac{-1}{6} \cdot i + \frac{-1}{2}\right)\right), -1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot i\right), \frac{-1}{2}\right)\right), -1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{6}\right), \frac{-1}{2}\right)\right), -1\right)\right) \]

      12. *-lowering-*.f64 57.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{6}\right), \frac{-1}{2}\right)\right), -1\right)\right) \]

   10. Simplified 57.7%

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

   if -1.5000000000000001e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

   if 3.7000000000000003e-23 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 96.1%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right)\right), i\right) \]

      12. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right)\right), i\right) \]

   8. Simplified 77.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;\left(n \cdot -100\right) \cdot \left(i \cdot \left(i \cdot -0.16666666666666666 + -0.5\right) + -1\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{i \cdot \left(n \cdot \left(100 + i \cdot 50\right)\right)}{i}\\ \mathbf{if}\;n \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (* i (* n (+ 100.0 (* i 50.0)))) i)))
   (if (<= n -2e+64) t_0 (if (<= n 3.7e-23) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -2e+64) {
		tmp = t_0;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (i * (n * (100.0d0 + (i * 50.0d0)))) / i
    if (n <= (-2d+64)) then
        tmp = t_0
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	double tmp;
	if (n <= -2e+64) {
		tmp = t_0;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i
	tmp = 0
	if n <= -2e+64:
		tmp = t_0
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(Float64(i * Float64(n * Float64(100.0 + Float64(i * 50.0)))) / i)
	tmp = 0.0
	if (n <= -2e+64)
		tmp = t_0;
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = (i * (n * (100.0 + (i * 50.0)))) / i;
	tmp = 0.0;
	if (n <= -2e+64)
		tmp = t_0;
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(i * N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]}, If[LessEqual[n, -2e+64], t$95$0, If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.00000000000000004e64 or 3.7000000000000003e-23 < n

   1. Initial program 24.1%

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

   3. Taylor expanded in n around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(100 \cdot \left(n \cdot \left(e^{i} - 1\right)\right)\right), \color{blue}{i}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \left(n \cdot \left(e^{i} - 1\right)\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(e^{i} - 1\right)\right)\right), i\right) \]

      5. expm1-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \left(\mathsf{expm1}\left(i\right)\right)\right)\right), i\right) \]

      6. expm1-lowering-expm1.f64 92.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(100, \mathsf{*.f64}\left(n, \mathsf{expm1.f64}\left(i\right)\right)\right), i\right) \]

   5. Simplified 92.8%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(i \cdot \left(50 \cdot \left(i \cdot n\right) + 100 \cdot n\right)\right)}, i\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot n\right) \cdot i\right)\right), i\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(n \cdot i\right)\right)\right), i\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + 50 \cdot \left(i \cdot n\right)\right)\right), i\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(100 \cdot n + \left(50 \cdot i\right) \cdot n\right)\right), i\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(n \cdot \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \left(100 + 50 \cdot i\right)\right)\right), i\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right)\right), i\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(i \cdot 50\right)\right)\right)\right), i\right) \]

      12. *-lowering-*.f64 68.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(i, 50\right)\right)\right)\right), i\right) \]

   8. Simplified 68.4%

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

   if -2.00000000000000004e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

Alternative 12: 62.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -1.5e+64)
   (* 100.0 (* (* i n) (/ 1.0 i)))
   (if (<= n 3.7e-23) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = 100.0 * ((i * n) * (1.0 / i));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.5d+64)) then
        tmp = 100.0d0 * ((i * n) * (1.0d0 / i))
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -1.5e+64) {
		tmp = 100.0 * ((i * n) * (1.0 / i));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -1.5e+64:
		tmp = 100.0 * ((i * n) * (1.0 / i))
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -1.5e+64)
		tmp = Float64(100.0 * Float64(Float64(i * n) * Float64(1.0 / i)));
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -1.5e+64)
		tmp = 100.0 * ((i * n) * (1.0 / i));
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -1.5e+64], N[(100.0 * N[(N[(i * n), $MachinePrecision] * N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.5000000000000001e64

   1. Initial program 22.4%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(1 + i\right)}, 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(i + 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. +-lowering-+.f64 4.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(i, 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   5. Simplified 4.3%

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

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\left(\left(i + 1\right) - 1\right) \cdot \color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\left(i + \left(1 - 1\right)\right) \cdot \frac{\color{blue}{1}}{\frac{i}{n}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\left(i + 0\right) \cdot \frac{1}{\frac{i}{n}}\right)\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(i \cdot \frac{\color{blue}{1}}{\frac{i}{n}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(i \cdot \frac{n}{\color{blue}{i}}\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(i \cdot \left(n \cdot \color{blue}{\frac{1}{i}}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \left(\left(i \cdot n\right) \cdot \color{blue}{\frac{1}{i}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\left(i \cdot n\right), \color{blue}{\left(\frac{1}{i}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, n\right), \left(\frac{\color{blue}{1}}{i}\right)\right)\right) \]

      10. /-lowering-/.f64 55.6%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{*.f64}\left(\mathsf{*.f64}\left(i, n\right), \mathsf{/.f64}\left(1, \color{blue}{i}\right)\right)\right) \]

   7. Applied egg-rr 55.6%

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

   if -1.5000000000000001e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

   if 3.7000000000000003e-23 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot \color{blue}{i}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \left(i \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{n}}\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 70.7%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{n}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 70.7%

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

   6. Taylor expanded in n around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot 1 + \color{blue}{100 \cdot \left(\frac{1}{2} \cdot i\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 + \color{blue}{100} \cdot \left(\frac{1}{2} \cdot i\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(100 \cdot \left(\frac{1}{2} \cdot i\right)\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(\left(100 \cdot \frac{1}{2}\right) \cdot \color{blue}{i}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right) \]

      10. *-lowering-*.f64 70.7%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   8. Simplified 70.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;100 \cdot \left(\left(i \cdot n\right) \cdot \frac{1}{i}\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{100}{\frac{i}{i \cdot n}}\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -4.7e+64)
   (/ 100.0 (/ i (* i n)))
   (if (<= n 3.7e-23) (* 100.0 (/ i (/ i n))) (* n (+ 100.0 (* i 50.0))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -4.7e+64) {
		tmp = 100.0 / (i / (i * n));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.7d+64)) then
        tmp = 100.0d0 / (i / (i * n))
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = n * (100.0d0 + (i * 50.0d0))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -4.7e+64) {
		tmp = 100.0 / (i / (i * n));
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -4.7e+64:
		tmp = 100.0 / (i / (i * n))
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -4.7e+64)
		tmp = Float64(100.0 / Float64(i / Float64(i * n)));
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * 50.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -4.7e+64)
		tmp = 100.0 / (i / (i * n));
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -4.7e+64], N[(100.0 / N[(i / N[(i * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.70000000000000029e64

   1. Initial program 22.4%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(1 + i\right)}, 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(i + 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

      2. +-lowering-+.f64 4.3%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(i, 1\right), 1\right), \mathsf{/.f64}\left(i, n\right)\right)\right) \]

   5. Simplified 4.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \color{blue}{\left(\frac{\frac{i}{n}}{\left(i + 1\right) - 1}\right)}\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{\frac{i}{n}}{i + \color{blue}{\left(1 - 1\right)}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{\frac{i}{n}}{i + 0}\right)\right) \]

      6. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{\frac{i}{n}}{i}\right)\right) \]

      7. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(100, \left(\frac{i}{\color{blue}{i \cdot n}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \color{blue}{\left(i \cdot n\right)}\right)\right) \]

      9. *-lowering-*.f64 55.4%

         \[\leadsto \mathsf{/.f64}\left(100, \mathsf{/.f64}\left(i, \mathsf{*.f64}\left(i, \color{blue}{n}\right)\right)\right) \]

   7. Applied egg-rr 55.4%

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

   if -4.70000000000000029e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

   if 3.7000000000000003e-23 < n

   1. Initial program 25.3%

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

   3. Taylor expanded in i around 0

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot \color{blue}{i}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \left(i \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{n}}\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 70.7%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{n}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 70.7%

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

   6. Taylor expanded in n around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot 1 + \color{blue}{100 \cdot \left(\frac{1}{2} \cdot i\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 + \color{blue}{100} \cdot \left(\frac{1}{2} \cdot i\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(100 \cdot \left(\frac{1}{2} \cdot i\right)\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(\left(100 \cdot \frac{1}{2}\right) \cdot \color{blue}{i}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right) \]

      10. *-lowering-*.f64 70.7%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   8. Simplified 70.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{100}{\frac{i}{i \cdot n}}\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (+ 100.0 (* i 50.0)))))
   (if (<= n -1.5e+64) t_0 (if (<= n 3.7e-23) (* 100.0 (/ i (/ i n))) t_0))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.5e+64) {
		tmp = t_0;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (100.0d0 + (i * 50.0d0))
    if (n <= (-1.5d+64)) then
        tmp = t_0
    else if (n <= 3.7d-23) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double t_0 = n * (100.0 + (i * 50.0));
	double tmp;
	if (n <= -1.5e+64) {
		tmp = t_0;
	} else if (n <= 3.7e-23) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 + (i * 50.0))
	tmp = 0
	if n <= -1.5e+64:
		tmp = t_0
	elif n <= 3.7e-23:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(n * Float64(100.0 + Float64(i * 50.0)))
	tmp = 0.0
	if (n <= -1.5e+64)
		tmp = t_0;
	elseif (n <= 3.7e-23)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = n * (100.0 + (i * 50.0));
	tmp = 0.0;
	if (n <= -1.5e+64)
		tmp = t_0;
	elseif (n <= 3.7e-23)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.5e+64], t$95$0, If[LessEqual[n, 3.7e-23], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.5000000000000001e64 or 3.7000000000000003e-23 < n

   1. Initial program 24.1%

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

   3. Taylor expanded in i around 0

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot \color{blue}{i}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \left(i \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{n}}\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{n}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 64.1%

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

   6. Taylor expanded in n around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \color{blue}{\left(100 \cdot \left(1 + \frac{1}{2} \cdot i\right)\right)}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 \cdot 1 + \color{blue}{100 \cdot \left(\frac{1}{2} \cdot i\right)}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \left(100 + \color{blue}{100} \cdot \left(\frac{1}{2} \cdot i\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \color{blue}{\left(100 \cdot \left(\frac{1}{2} \cdot i\right)\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(\left(100 \cdot \frac{1}{2}\right) \cdot \color{blue}{i}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \left(50 \cdot i\right)\right)\right) \]

      10. *-lowering-*.f64 64.1%

          \[\leadsto \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(100, \mathsf{*.f64}\left(50, \color{blue}{i}\right)\right)\right) \]

   8. Simplified 64.1%

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

   if -1.5000000000000001e64 < n < 3.7000000000000003e-23

   1. Initial program 31.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 65.8%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.2 \cdot 10^{-19}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.2e-19)
   (* 100.0 (/ i (/ i n)))
   (if (<= i 2.3e-8) (* 100.0 (+ n (* i -0.5))) (* i (* n 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.2e-19) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.3e-8) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = i * (n * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-2.2d-19)) then
        tmp = 100.0d0 * (i / (i / n))
    else if (i <= 2.3d-8) then
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    else
        tmp = i * (n * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.2e-19) {
		tmp = 100.0 * (i / (i / n));
	} else if (i <= 2.3e-8) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = i * (n * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.2e-19:
		tmp = 100.0 * (i / (i / n))
	elif i <= 2.3e-8:
		tmp = 100.0 * (n + (i * -0.5))
	else:
		tmp = i * (n * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.2e-19)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	elseif (i <= 2.3e-8)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	else
		tmp = Float64(i * Float64(n * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.2e-19)
		tmp = 100.0 * (i / (i / n));
	elseif (i <= 2.3e-8)
		tmp = 100.0 * (n + (i * -0.5));
	else
		tmp = i * (n * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.2e-19], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2.3e-8], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.1999999999999998e-19

   1. Initial program 52.9%

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

   3. Taylor expanded in i around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{/.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(i, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 23.8%

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

   if -2.1999999999999998e-19 < i < 2.3000000000000001e-8

   1. Initial program 7.1%

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

   3. Taylor expanded in i around 0

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot \color{blue}{i}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \left(i \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{n}}\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 89.5%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{n}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 89.5%

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

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(\frac{-1}{2} \cdot i\right)}\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(i \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      2. *-lowering-*.f64 89.4%

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   8. Simplified 89.4%

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

   if 2.3000000000000001e-8 < i

   1. Initial program 49.8%

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

   3. Taylor expanded in i around 0

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot \color{blue}{i}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \left(i \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{n}}\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 32.3%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{n}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 32.3%

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

   6. Taylor expanded in n around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-100 \cdot n\right), \color{blue}{\left(\frac{-1}{2} \cdot i - 1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(\color{blue}{\frac{-1}{2} \cdot i} - 1\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(\frac{-1}{2} \cdot i + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(\frac{-1}{2} \cdot i + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{-1}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{2}\right), -1\right)\right) \]

      8. *-lowering-*.f64 32.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), -1\right)\right) \]

   8. Simplified 32.9%

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

   9. Taylor expanded in i around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(50 \cdot n\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{50}\right)\right) \]

       6. *-lowering-*.f64 32.9%

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{50}\right)\right) \]

   11. Simplified 32.9%

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

Alternative 16: 54.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 2.3e-8) (* n 100.0) (* i (* n 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 2.3e-8) {
		tmp = n * 100.0;
	} else {
		tmp = i * (n * 50.0);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 2.3d-8) then
        tmp = n * 100.0d0
    else
        tmp = i * (n * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 2.3e-8) {
		tmp = n * 100.0;
	} else {
		tmp = i * (n * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 2.3e-8:
		tmp = n * 100.0
	else:
		tmp = i * (n * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 2.3e-8)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(i * Float64(n * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 2.3e-8)
		tmp = n * 100.0;
	else
		tmp = i * (n * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 2.3e-8], N[(n * 100.0), $MachinePrecision], N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 2.3000000000000001e-8

   1. Initial program 21.1%

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

   3. Taylor expanded in i around 0

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

      1. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

   5. Simplified 64.1%

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

   if 2.3000000000000001e-8 < i

   1. Initial program 49.8%

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

   3. Taylor expanded in i around 0

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

      1. distribute-lft-out N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{\left(n + i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \color{blue}{\left(i \cdot \left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(\left(n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) \cdot \color{blue}{i}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right) \cdot i\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \left(n \cdot \left(i \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \color{blue}{\left(i \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{n}}\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{n}\right)\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 32.3%

          \[\leadsto \mathsf{*.f64}\left(100, \mathsf{+.f64}\left(n, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{n}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 32.3%

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

   6. Taylor expanded in n around -inf

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-100 \cdot n\right), \color{blue}{\left(\frac{-1}{2} \cdot i - 1\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(\color{blue}{\frac{-1}{2} \cdot i} - 1\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(\frac{-1}{2} \cdot i + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \left(\frac{-1}{2} \cdot i + -1\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot i\right), \color{blue}{-1}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\left(i \cdot \frac{-1}{2}\right), -1\right)\right) \]

      8. *-lowering-*.f64 32.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-100, n\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(i, \frac{-1}{2}\right), -1\right)\right) \]

   8. Simplified 32.9%

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

   9. Taylor expanded in i around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(i, \color{blue}{\left(50 \cdot n\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(i, \left(n \cdot \color{blue}{50}\right)\right) \]

       6. *-lowering-*.f64 32.9%

          \[\leadsto \mathsf{*.f64}\left(i, \mathsf{*.f64}\left(n, \color{blue}{50}\right)\right) \]

   11. Simplified 32.9%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.6% accurate, 38.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in i around 0

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

   1. *-lowering-*.f64 51.0%

      \[\leadsto \mathsf{*.f64}\left(100, \color{blue}{n}\right) \]

5. Simplified 51.0%

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

6. Final simplification 51.0%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer Target 1: 33.9% 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 2024158 
(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))))