Compound Interest

Percentage Accurate: 28.8% → 94.6%

Time: 17.5s

Alternatives: 11

Speedup: 9.5×

• 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: 28.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + \frac{i}{n}\right)}^{n}\\ t_1 := \frac{t\_0 + -1}{\frac{i}{n}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;n \cdot \frac{t\_0 \cdot 100 + -100}{i}\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 (/ i n)) n)) (t_1 (/ (+ t_0 -1.0) (/ i n))))
   (if (<= t_1 0.0)
     (* 100.0 (/ (expm1 (* n (log1p (/ i n)))) (/ i n)))
     (if (<= t_1 INFINITY) (* n (/ (+ (* t_0 100.0) -100.0) i)) (* n 100.0)))))

C

double code(double i, double n) {
	double t_0 = pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (expm1((n * log1p((i / n)))) / (i / n));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = Math.pow((1.0 + (i / n)), n);
	double t_1 = (t_0 + -1.0) / (i / n);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 100.0 * (Math.expm1((n * Math.log1p((i / n)))) / (i / n));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = n * (((t_0 * 100.0) + -100.0) / i);
	} else {
		tmp = n * 100.0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = math.pow((1.0 + (i / n)), n)
	t_1 = (t_0 + -1.0) / (i / n)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 100.0 * (math.expm1((n * math.log1p((i / n)))) / (i / n))
	elif t_1 <= math.inf:
		tmp = n * (((t_0 * 100.0) + -100.0) / i)
	else:
		tmp = n * 100.0
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. metadata-eval 24.2%

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

   4. Applied egg-rr 24.2%

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

      1. metadata-eval 24.2%

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

      2. sub-neg 24.2%

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

      3. exp-to-pow 23.6%

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

      4. log1p-undefine 48.6%

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

      5. *-commutative 48.6%

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

      6. expm1-undefine 99.2%

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

   6. Simplified 99.2%

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

   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 99.8%

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

      1. associate-/r/ 99.7%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-define 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-undefine 100.0%

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

      2. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #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 78.4%

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

      1. *-commutative 78.4%

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

   5. Simplified 78.4%

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

4. Final simplification 95.7%

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

Alternative 2: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-237}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* n (* 100.0 (/ (expm1 i) i)))))
   (if (<= n -6.5e-217)
     t_0
     (if (<= n 2.65e-237) 0.0 (if (<= n 2.0) (* 100.0 (/ i (/ i n))) t_0)))))

C

double code(double i, double n) {
	double t_0 = n * (100.0 * (expm1(i) / i));
	double tmp;
	if (n <= -6.5e-217) {
		tmp = t_0;
	} else if (n <= 2.65e-237) {
		tmp = 0.0;
	} else if (n <= 2.0) {
		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 * (100.0 * (Math.expm1(i) / i));
	double tmp;
	if (n <= -6.5e-217) {
		tmp = t_0;
	} else if (n <= 2.65e-237) {
		tmp = 0.0;
	} else if (n <= 2.0) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = n * (100.0 * (math.expm1(i) / i))
	tmp = 0
	if n <= -6.5e-217:
		tmp = t_0
	elif n <= 2.65e-237:
		tmp = 0.0
	elif n <= 2.0:
		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(expm1(i) / i)))
	tmp = 0.0
	if (n <= -6.5e-217)
		tmp = t_0;
	elseif (n <= 2.65e-237)
		tmp = 0.0;
	elseif (n <= 2.0)
		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[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e-217], t$95$0, If[LessEqual[n, 2.65e-237], 0.0, If[LessEqual[n, 2.0], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.4999999999999996e-217 or 2 < n

   1. Initial program 24.7%

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

      1. associate-/r/ 24.7%

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

      2. associate-*r* 24.8%

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

      3. *-commutative 24.8%

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

      4. associate-*r/ 24.8%

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

      5. sub-neg 24.8%

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

      6. distribute-lft-in 24.8%

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

      7. metadata-eval 24.8%

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

      8. metadata-eval 24.8%

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

      9. metadata-eval 24.8%

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

      10. fma-define 24.8%

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

      11. metadata-eval 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in n around inf 40.8%

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

      1. sub-neg 40.8%

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

      2. metadata-eval 40.8%

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

      3. metadata-eval 40.8%

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

      4. distribute-lft-in 40.8%

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

      5. metadata-eval 40.8%

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

      6. sub-neg 40.8%

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

      7. associate-*r/ 40.8%

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

      8. *-commutative 40.8%

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

      9. expm1-define 89.1%

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

   7. Simplified 89.1%

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

   if -6.4999999999999996e-217 < n < 2.64999999999999977e-237

   1. Initial program 66.7%

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

      1. associate-*r/ 66.7%

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

      2. sub-neg 66.7%

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

      3. distribute-rgt-in 66.7%

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

      4. metadata-eval 66.7%

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

      5. metadata-eval 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in i around 0 81.7%

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

   6. Taylor expanded in i around 0 81.7%

      \[\leadsto \color{blue}{0} \]

   if 2.64999999999999977e-237 < n < 2

   1. Initial program 14.5%

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

   3. Taylor expanded in i around 0 61.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-217}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \mathbf{elif}\;n \leq 2.65 \cdot 10^{-237}:\\ \;\;\;\;0\\ \mathbf{elif}\;n \leq 2:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 \cdot \frac{\mathsf{expm1}\left(i\right)}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.4:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.4)
   0.0
   (*
    100.0
    (+
     n
     (*
      i
      (+
       (* n 0.5)
       (*
        i
        (+ (* 0.041666666666666664 (* i n)) (* n 0.16666666666666666)))))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.4) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.4d0)) then
        tmp = 0.0d0
    else
        tmp = 100.0d0 * (n + (i * ((n * 0.5d0) + (i * ((0.041666666666666664d0 * (i * n)) + (n * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.4) {
		tmp = 0.0;
	} else {
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.4:
		tmp = 0.0
	else:
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.4)
		tmp = 0.0;
	else
		tmp = Float64(100.0 * Float64(n + Float64(i * Float64(Float64(n * 0.5) + Float64(i * Float64(Float64(0.041666666666666664 * Float64(i * n)) + Float64(n * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.4)
		tmp = 0.0;
	else
		tmp = 100.0 * (n + (i * ((n * 0.5) + (i * ((0.041666666666666664 * (i * n)) + (n * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.4], 0.0, N[(100.0 * N[(n + N[(i * N[(N[(n * 0.5), $MachinePrecision] + N[(i * N[(N[(0.041666666666666664 * N[(i * n), $MachinePrecision]), $MachinePrecision] + N[(n * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.3999999999999999

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. sub-neg 60.0%

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

      3. distribute-rgt-in 60.0%

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

      4. metadata-eval 60.0%

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

      5. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in i around 0 37.5%

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

   6. Taylor expanded in i around 0 37.5%

      \[\leadsto \color{blue}{0} \]

   if -1.3999999999999999 < i

   1. Initial program 18.8%

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

   3. Taylor expanded in n around inf 28.3%

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

      1. *-commutative 28.3%

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

      2. associate-/l* 28.3%

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

      3. expm1-define 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in i around 0 72.2%

      \[\leadsto \color{blue}{\left(n + i \cdot \left(0.5 \cdot n + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot n\right)\right)\right)} \cdot 100 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.4:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot \left(n \cdot 0.5 + i \cdot \left(0.041666666666666664 \cdot \left(i \cdot n\right) + n \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.9:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + i \cdot 4.166666666666667\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.9)
   0.0
   (*
    n
    (+
     100.0
     (* i (+ 50.0 (* i (+ 16.666666666666668 (* i 4.166666666666667)))))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.9) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (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 (i <= (-1.9d0)) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * (16.666666666666668d0 + (i * 4.166666666666667d0))))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.9) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.9:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.9)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * Float64(16.666666666666668 + Float64(i * 4.166666666666667)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.9)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * (50.0 + (i * (16.666666666666668 + (i * 4.166666666666667))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.9], 0.0, N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * N[(16.666666666666668 + N[(i * 4.166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.8999999999999999

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. sub-neg 60.0%

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

      3. distribute-rgt-in 60.0%

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

      4. metadata-eval 60.0%

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

      5. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in i around 0 37.5%

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

   6. Taylor expanded in i around 0 37.5%

      \[\leadsto \color{blue}{0} \]

   if -1.8999999999999999 < i

   1. Initial program 18.8%

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

      1. associate-/r/ 19.2%

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

      2. associate-*r* 19.2%

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

      3. *-commutative 19.2%

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

      4. associate-*r/ 19.2%

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

      5. sub-neg 19.2%

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

      6. distribute-lft-in 19.2%

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

      7. metadata-eval 19.2%

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

      8. metadata-eval 19.2%

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

      9. metadata-eval 19.2%

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

      10. fma-define 19.2%

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

      11. metadata-eval 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in n around inf 28.3%

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

      1. sub-neg 28.3%

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

      2. metadata-eval 28.3%

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

      3. metadata-eval 28.3%

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

      4. distribute-lft-in 28.3%

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

      5. metadata-eval 28.3%

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

      6. sub-neg 28.3%

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

      7. associate-*r/ 28.3%

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

      8. *-commutative 28.3%

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

      9. expm1-define 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in i around 0 72.2%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + 4.166666666666667 \cdot i\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto n \cdot \left(100 + i \cdot \left(50 + i \cdot \left(16.666666666666668 + \color{blue}{i \cdot 4.166666666666667}\right)\right)\right) \]

   10. Simplified 72.2%

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

Alternative 5: 60.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -8.6 \cdot 10^{+23}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0152:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -8.6e+23)
   0.0
   (if (<= i 0.0152) (* 100.0 (+ n (* i -0.5))) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -8.6e+23) {
		tmp = 0.0;
	} else if (i <= 0.0152) {
		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 <= (-8.6d+23)) then
        tmp = 0.0d0
    else if (i <= 0.0152d0) 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 <= -8.6e+23) {
		tmp = 0.0;
	} else if (i <= 0.0152) {
		tmp = 100.0 * (n + (i * -0.5));
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -8.6e+23:
		tmp = 0.0
	elif i <= 0.0152:
		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 <= -8.6e+23)
		tmp = 0.0;
	elseif (i <= 0.0152)
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -8.6e+23)
		tmp = 0.0;
	elseif (i <= 0.0152)
		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, -8.6e+23], 0.0, If[LessEqual[i, 0.0152], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -8.5999999999999997e23

   1. Initial program 67.6%

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

      1. associate-*r/ 67.6%

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

      2. sub-neg 67.6%

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

      3. distribute-rgt-in 67.6%

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

      4. metadata-eval 67.6%

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

      5. metadata-eval 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in i around 0 43.6%

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

   6. Taylor expanded in i around 0 43.6%

      \[\leadsto \color{blue}{0} \]

   if -8.5999999999999997e23 < i < 0.0152

   1. Initial program 8.7%

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

   3. Taylor expanded in i around 0 82.7%

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

      1. associate-*r* 82.5%

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

      2. associate-*r/ 82.5%

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

      3. metadata-eval 82.5%

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

   5. Simplified 82.5%

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

   6. Taylor expanded in n around 0 82.7%

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

      1. *-commutative 82.7%

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

   8. Simplified 82.7%

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

   if 0.0152 < i

   1. Initial program 38.5%

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

      1. associate-/r/ 38.7%

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

      2. associate-*r* 38.8%

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

      3. *-commutative 38.8%

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

      4. associate-*r/ 38.8%

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

      5. sub-neg 38.8%

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

      6. distribute-lft-in 38.8%

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

      7. metadata-eval 38.8%

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

      8. metadata-eval 38.8%

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

      9. metadata-eval 38.8%

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

      10. fma-define 38.8%

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

      11. metadata-eval 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in n around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. metadata-eval 63.1%

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

      3. metadata-eval 63.1%

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

      4. distribute-lft-in 63.1%

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

      5. metadata-eval 63.1%

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

      6. sub-neg 63.1%

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

      7. associate-*r/ 63.1%

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

      8. *-commutative 63.1%

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

      9. expm1-define 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in i around 0 29.9%

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

   9. Taylor expanded in i around inf 29.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.6 \cdot 10^{+23}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0152:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5.2 \cdot 10^{+19}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot \left(50 + i \cdot 16.666666666666668\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -5.2e+19)
   0.0
   (* n (+ 100.0 (* i (+ 50.0 (* i 16.666666666666668)))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -5.2e+19) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-5.2d+19)) then
        tmp = 0.0d0
    else
        tmp = n * (100.0d0 + (i * (50.0d0 + (i * 16.666666666666668d0))))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -5.2e+19) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -5.2e+19:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -5.2e+19)
		tmp = 0.0;
	else
		tmp = Float64(n * Float64(100.0 + Float64(i * Float64(50.0 + Float64(i * 16.666666666666668)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -5.2e+19)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * (50.0 + (i * 16.666666666666668))));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -5.2e+19], 0.0, N[(n * N[(100.0 + N[(i * N[(50.0 + N[(i * 16.666666666666668), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -5.2e19

   1. Initial program 67.6%

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

      1. associate-*r/ 67.6%

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

      2. sub-neg 67.6%

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

      3. distribute-rgt-in 67.6%

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

      4. metadata-eval 67.6%

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

      5. metadata-eval 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in i around 0 43.6%

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

   6. Taylor expanded in i around 0 43.6%

      \[\leadsto \color{blue}{0} \]

   if -5.2e19 < i

   1. Initial program 18.7%

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

      1. associate-/r/ 19.1%

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

      2. associate-*r* 19.1%

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

      3. *-commutative 19.1%

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

      4. associate-*r/ 19.1%

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

      5. sub-neg 19.1%

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

      6. distribute-lft-in 19.1%

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

      7. metadata-eval 19.1%

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

      8. metadata-eval 19.1%

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

      9. metadata-eval 19.1%

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

      10. fma-define 19.1%

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

      11. metadata-eval 19.1%

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

   3. Simplified 19.1%

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

   5. Taylor expanded in n around inf 30.3%

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

      1. sub-neg 30.3%

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

      2. metadata-eval 30.3%

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

      3. metadata-eval 30.3%

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

      4. distribute-lft-in 30.3%

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

      5. metadata-eval 30.3%

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

      6. sub-neg 30.3%

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

      7. associate-*r/ 30.3%

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

      8. *-commutative 30.3%

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

      9. expm1-define 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in i around 0 69.3%

      \[\leadsto n \cdot \color{blue}{\left(100 + i \cdot \left(50 + 16.666666666666668 \cdot i\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 69.3%

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

   10. Simplified 69.3%

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

Alternative 7: 60.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{+20}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0152:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.2e+20) 0.0 (if (<= i 0.0152) (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.2e+20) {
		tmp = 0.0;
	} else if (i <= 0.0152) {
		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 <= (-4.2d+20)) then
        tmp = 0.0d0
    else if (i <= 0.0152d0) 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 <= -4.2e+20) {
		tmp = 0.0;
	} else if (i <= 0.0152) {
		tmp = n * 100.0;
	} else {
		tmp = (i * n) * 50.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -4.2e+20:
		tmp = 0.0
	elif i <= 0.0152:
		tmp = n * 100.0
	else:
		tmp = (i * n) * 50.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.2e+20)
		tmp = 0.0;
	elseif (i <= 0.0152)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(Float64(i * n) * 50.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -4.2e+20)
		tmp = 0.0;
	elseif (i <= 0.0152)
		tmp = n * 100.0;
	else
		tmp = (i * n) * 50.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.2e+20], 0.0, If[LessEqual[i, 0.0152], N[(n * 100.0), $MachinePrecision], N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.2e20

   1. Initial program 67.6%

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

      1. associate-*r/ 67.6%

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

      2. sub-neg 67.6%

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

      3. distribute-rgt-in 67.6%

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

      4. metadata-eval 67.6%

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

      5. metadata-eval 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in i around 0 43.6%

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

   6. Taylor expanded in i around 0 43.6%

      \[\leadsto \color{blue}{0} \]

   if -4.2e20 < i < 0.0152

   1. Initial program 8.7%

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

   3. Taylor expanded in i around 0 82.4%

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

      1. *-commutative 82.4%

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

   5. Simplified 82.4%

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

   if 0.0152 < i

   1. Initial program 38.5%

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

      1. associate-/r/ 38.7%

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

      2. associate-*r* 38.8%

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

      3. *-commutative 38.8%

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

      4. associate-*r/ 38.8%

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

      5. sub-neg 38.8%

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

      6. distribute-lft-in 38.8%

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

      7. metadata-eval 38.8%

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

      8. metadata-eval 38.8%

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

      9. metadata-eval 38.8%

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

      10. fma-define 38.8%

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

      11. metadata-eval 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in n around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. metadata-eval 63.1%

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

      3. metadata-eval 63.1%

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

      4. distribute-lft-in 63.1%

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

      5. metadata-eval 63.1%

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

      6. sub-neg 63.1%

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

      7. associate-*r/ 63.1%

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

      8. *-commutative 63.1%

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

      9. expm1-define 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in i around 0 29.9%

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

   9. Taylor expanded in i around inf 29.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{+20}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 0.0152:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0) 0.0 (+ (* n 100.0) (* (* i n) 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) + ((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 <= (-2.0d0)) then
        tmp = 0.0d0
    else
        tmp = (n * 100.0d0) + ((i * n) * 50.0d0)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = 0.0;
	} else {
		tmp = (n * 100.0) + ((i * n) * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = 0.0
	else:
		tmp = (n * 100.0) + ((i * n) * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(n * 100.0) + Float64(Float64(i * n) * 50.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = 0.0;
	else
		tmp = (n * 100.0) + ((i * n) * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], 0.0, N[(N[(n * 100.0), $MachinePrecision] + N[(N[(i * n), $MachinePrecision] * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. sub-neg 60.0%

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

      3. distribute-rgt-in 60.0%

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

      4. metadata-eval 60.0%

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

      5. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in i around 0 37.5%

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

   6. Taylor expanded in i around 0 37.5%

      \[\leadsto \color{blue}{0} \]

   if -2 < i

   1. Initial program 18.8%

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

      1. associate-/r/ 19.2%

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

      2. associate-*r* 19.2%

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

      3. *-commutative 19.2%

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

      4. associate-*r/ 19.2%

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

      5. sub-neg 19.2%

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

      6. distribute-lft-in 19.2%

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

      7. metadata-eval 19.2%

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

      8. metadata-eval 19.2%

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

      9. metadata-eval 19.2%

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

      10. fma-define 19.2%

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

      11. metadata-eval 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in n around inf 28.3%

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

      1. sub-neg 28.3%

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

      2. metadata-eval 28.3%

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

      3. metadata-eval 28.3%

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

      4. distribute-lft-in 28.3%

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

      5. metadata-eval 28.3%

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

      6. sub-neg 28.3%

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

      7. associate-*r/ 28.3%

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

      8. *-commutative 28.3%

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

      9. expm1-define 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in i around 0 67.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot 100 + \left(i \cdot n\right) \cdot 50\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.4 \cdot 10^{+25}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 2.8 \cdot 10^{+19}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.4e+25) 0.0 (if (<= i 2.8e+19) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.4e+25) {
		tmp = 0.0;
	} else if (i <= 2.8e+19) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-1.4d+25)) then
        tmp = 0.0d0
    else if (i <= 2.8d+19) then
        tmp = n * 100.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -1.4e+25) {
		tmp = 0.0;
	} else if (i <= 2.8e+19) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -1.4e+25:
		tmp = 0.0
	elif i <= 2.8e+19:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.4e+25)
		tmp = 0.0;
	elseif (i <= 2.8e+19)
		tmp = Float64(n * 100.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -1.4e+25)
		tmp = 0.0;
	elseif (i <= 2.8e+19)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.4e+25], 0.0, If[LessEqual[i, 2.8e+19], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -1.4000000000000001e25 or 2.8e19 < i

   1. Initial program 52.2%

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

      1. associate-*r/ 52.2%

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

      2. sub-neg 52.2%

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

      3. distribute-rgt-in 52.2%

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

      4. metadata-eval 52.2%

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

      5. metadata-eval 52.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in i around 0 30.7%

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

   6. Taylor expanded in i around 0 30.7%

      \[\leadsto \color{blue}{0} \]

   if -1.4000000000000001e25 < i < 2.8e19

   1. Initial program 8.9%

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

   3. Taylor expanded in i around 0 77.8%

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

      1. *-commutative 77.8%

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

   5. Simplified 77.8%

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

Alternative 10: 60.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;n \cdot \left(100 + i \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0) 0.0 (* n (+ 100.0 (* i 50.0)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = 0.0;
	} 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 (i <= (-2.0d0)) then
        tmp = 0.0d0
    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 (i <= -2.0) {
		tmp = 0.0;
	} else {
		tmp = n * (100.0 + (i * 50.0));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = 0.0
	else:
		tmp = n * (100.0 + (i * 50.0))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = 0.0;
	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 (i <= -2.0)
		tmp = 0.0;
	else
		tmp = n * (100.0 + (i * 50.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], 0.0, N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -2

   1. Initial program 60.0%

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

      1. associate-*r/ 60.0%

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

      2. sub-neg 60.0%

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

      3. distribute-rgt-in 60.0%

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

      4. metadata-eval 60.0%

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

      5. metadata-eval 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in i around 0 37.5%

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

   6. Taylor expanded in i around 0 37.5%

      \[\leadsto \color{blue}{0} \]

   if -2 < i

   1. Initial program 18.8%

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

      1. associate-/r/ 19.2%

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

      2. associate-*r* 19.2%

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

      3. *-commutative 19.2%

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

      4. associate-*r/ 19.2%

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

      5. sub-neg 19.2%

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

      6. distribute-lft-in 19.2%

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

      7. metadata-eval 19.2%

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

      8. metadata-eval 19.2%

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

      9. metadata-eval 19.2%

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

      10. fma-define 19.2%

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

      11. metadata-eval 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in n around inf 28.3%

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

      1. sub-neg 28.3%

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

      2. metadata-eval 28.3%

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

      3. metadata-eval 28.3%

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

      4. distribute-lft-in 28.3%

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

      5. metadata-eval 28.3%

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

      6. sub-neg 28.3%

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

      7. associate-*r/ 28.3%

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

      8. *-commutative 28.3%

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

      9. expm1-define 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in i around 0 66.9%

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

      1. *-commutative 66.9%

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

   10. Simplified 66.9%

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

Alternative 11: 17.9% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 26.5%

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

   1. associate-*r/ 26.5%

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

   2. sub-neg 26.5%

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

   3. distribute-rgt-in 26.5%

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

   4. metadata-eval 26.5%

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

   5. metadata-eval 26.5%

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

3. Simplified 26.5%

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

5. Taylor expanded in i around 0 17.0%

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

6. Taylor expanded in i around 0 17.3%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Developer target: 34.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

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

Reproduce

herbie shell --seed 2024100 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (* 100.0 (/ (- (exp (* n (if (== (+ 1.0 (/ i n)) 1.0) (/ i n) (/ (* (/ i n) (log (+ 1.0 (/ i n)))) (- (+ (/ i n) 1.0) 1.0))))) 1.0) (/ i n)))

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