Compound Interest

Percentage Accurate: 29.6% → 74.3%

Time: 24.5s

Alternatives: 13

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: 29.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 74.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i} \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n (/ (* 100.0 (expm1 i)) i)))

C

double code(double i, double n) {
	return n * ((100.0 * expm1(i)) / i);
}

Java

public static double code(double i, double n) {
	return n * ((100.0 * Math.expm1(i)) / i);
}

Python

def code(i, n):
	return n * ((100.0 * math.expm1(i)) / i)

Julia

function code(i, n)
	return Float64(n * Float64(Float64(100.0 * expm1(i)) / i))
end

Wolfram

code[i_, n_] := N[(n * N[(N[(100.0 * N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

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

   1. associate-/r/ 32.8%

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

   2. associate-*r* 32.8%

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

   3. *-commutative 32.8%

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

   4. associate-*r/ 32.8%

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

   5. sub-neg 32.8%

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

   6. distribute-lft-in 32.8%

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

   7. metadata-eval 32.8%

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

   8. metadata-eval 32.8%

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

   9. metadata-eval 32.8%

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

   10. fma-define 32.8%

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

   11. metadata-eval 32.8%

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

3. Simplified 32.8%

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

5. Taylor expanded in n around inf 34.9%

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

   1. sub-neg 34.9%

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

   2. metadata-eval 34.9%

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

   3. metadata-eval 34.9%

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

   4. distribute-lft-in 35.0%

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

   5. metadata-eval 35.0%

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

   6. sub-neg 35.0%

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

   7. expm1-define 71.4%

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

7. Simplified 71.4%

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

8. Final simplification 71.4%

   \[\leadsto n \cdot \frac{100 \cdot \mathsf{expm1}\left(i\right)}{i} \]
9. Add Preprocessing

Alternative 2: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.13:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 0.13)
   (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005))))
   (* 100.0 (/ (expm1 i) (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 0.13) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = 100.0 * (expm1(i) / (i / n));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 0.13) {
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
	} else {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 0.13:
		tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))
	else:
		tmp = 100.0 * (math.expm1(i) / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 0.13)
		tmp = Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))));
	else
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, 0.13], N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.13

   1. Initial program 32.7%

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

      1. associate-/r/ 32.6%

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

      2. associate-*r* 32.6%

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

      3. *-commutative 32.6%

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

      4. associate-*r/ 32.6%

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

      5. sub-neg 32.6%

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

      6. distribute-lft-in 32.6%

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

      7. metadata-eval 32.6%

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

      8. metadata-eval 32.6%

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

      9. metadata-eval 32.6%

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

      10. fma-define 32.6%

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

      11. metadata-eval 32.6%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in n around inf 31.6%

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

      1. sub-neg 31.6%

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

      2. metadata-eval 31.6%

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

      3. metadata-eval 31.6%

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

      4. distribute-lft-in 31.6%

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

      5. metadata-eval 31.6%

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

      6. sub-neg 31.6%

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

      7. expm1-define 65.0%

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

   7. Simplified 65.0%

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

      1. clear-num 65.1%

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

      2. un-div-inv 65.0%

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

      3. *-un-lft-identity 65.0%

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

      4. times-frac 65.0%

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

      5. metadata-eval 65.0%

         \[\leadsto \frac{n}{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}} \]

   9. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 64.9%

          \[\leadsto \frac{n}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   11. Simplified 64.9%

       \[\leadsto \color{blue}{\frac{n}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

   12. Taylor expanded in i around 0 65.8%

       \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]

   if 0.13 < n

   1. Initial program 32.9%

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

   3. Taylor expanded in n around inf 42.9%

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

      1. expm1-define 62.9%

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

   5. Simplified 62.9%

      \[\leadsto 100 \cdot \frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{\frac{i}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.13:\\ \;\;\;\;\frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.5% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-105}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -9.5e-105) (* n 100.0) (* 100.0 (* i (/ n i)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-105) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.5d-105)) then
        tmp = n * 100.0d0
    else
        tmp = 100.0d0 * (i * (n / i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= -9.5e-105) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i * (n / i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= -9.5e-105:
		tmp = n * 100.0
	else:
		tmp = 100.0 * (i * (n / i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -9.5e-105)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= -9.5e-105)
		tmp = n * 100.0;
	else
		tmp = 100.0 * (i * (n / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, -9.5e-105], N[(n * 100.0), $MachinePrecision], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -9.5000000000000002e-105

   1. Initial program 25.8%

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

   3. Taylor expanded in i around 0 55.2%

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

      1. *-commutative 55.2%

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

   5. Simplified 55.2%

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

   if -9.5000000000000002e-105 < n

   1. Initial program 37.0%

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

   3. Taylor expanded in i around 0 32.5%

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

      1. +-commutative 32.5%

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

   5. Simplified 32.5%

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

      1. div-inv 32.5%

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

      2. associate--l+ 48.9%

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

      3. metadata-eval 48.9%

         \[\leadsto 100 \cdot \left(\left(i + \color{blue}{0}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

      4. +-rgt-identity 48.9%

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

      5. clear-num 47.8%

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

   7. Applied egg-rr 47.8%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{-105}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n 1.55e+52) (* 100.0 (* i (/ n i))) (* 100.0 (+ n (* i -0.5)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= 1.55e+52) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = 100.0 * (n + (i * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1.55d+52) then
        tmp = 100.0d0 * (i * (n / i))
    else
        tmp = 100.0d0 * (n + (i * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (n <= 1.55e+52) {
		tmp = 100.0 * (i * (n / i));
	} else {
		tmp = 100.0 * (n + (i * -0.5));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if n <= 1.55e+52:
		tmp = 100.0 * (i * (n / i))
	else:
		tmp = 100.0 * (n + (i * -0.5))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (n <= 1.55e+52)
		tmp = Float64(100.0 * Float64(i * Float64(n / i)));
	else
		tmp = Float64(100.0 * Float64(n + Float64(i * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (n <= 1.55e+52)
		tmp = 100.0 * (i * (n / i));
	else
		tmp = 100.0 * (n + (i * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[n, 1.55e+52], N[(100.0 * N[(i * N[(n / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.55e52

   1. Initial program 35.6%

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

   3. Taylor expanded in i around 0 26.9%

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

      1. +-commutative 26.9%

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

   5. Simplified 26.9%

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

      1. div-inv 26.9%

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

      2. associate--l+ 53.2%

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

      3. metadata-eval 53.2%

         \[\leadsto 100 \cdot \left(\left(i + \color{blue}{0}\right) \cdot \frac{1}{\frac{i}{n}}\right) \]

      4. +-rgt-identity 53.2%

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

      5. clear-num 52.3%

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

   7. Applied egg-rr 52.3%

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

   if 1.55e52 < n

   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 47.4%

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

      1. associate-*r/ 47.4%

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

      2. metadata-eval 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in n around 0 20.4%

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

      1. associate-*r/ 20.4%

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

      2. associate-*l/ 20.4%

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

      3. *-commutative 20.4%

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

   8. Simplified 20.4%

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

   9. Taylor expanded in i around 0 52.1%

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

       1. *-commutative 52.1%

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

   11. Simplified 52.1%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.55 \cdot 10^{+52}:\\ \;\;\;\;100 \cdot \left(i \cdot \frac{n}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \left(n + i \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 6 \cdot 10^{-29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 6e-29) (* n 100.0) (* 100.0 (/ i (/ i n)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 6e-29) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 6d-29) then
        tmp = n * 100.0d0
    else
        tmp = 100.0d0 * (i / (i / n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 6e-29) {
		tmp = n * 100.0;
	} else {
		tmp = 100.0 * (i / (i / n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 6e-29:
		tmp = n * 100.0
	else:
		tmp = 100.0 * (i / (i / n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 6e-29)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 6e-29)
		tmp = n * 100.0;
	else
		tmp = 100.0 * (i / (i / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 6e-29], N[(n * 100.0), $MachinePrecision], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 6.0000000000000005e-29

   1. Initial program 27.7%

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

   3. Taylor expanded in i around 0 58.3%

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

      1. *-commutative 58.3%

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

   5. Simplified 58.3%

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

   if 6.0000000000000005e-29 < i

   1. Initial program 46.0%

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

   3. Taylor expanded in i around 0 34.0%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6 \cdot 10^{-29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -8.2 \cdot 10^{-46}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot i}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -8.2e-46) (* 100.0 (/ i (/ i n))) (* 100.0 (/ (* n i) i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -8.2e-46) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * i) / i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-8.2d-46)) then
        tmp = 100.0d0 * (i / (i / n))
    else
        tmp = 100.0d0 * ((n * i) / i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -8.2e-46) {
		tmp = 100.0 * (i / (i / n));
	} else {
		tmp = 100.0 * ((n * i) / i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -8.2e-46:
		tmp = 100.0 * (i / (i / n))
	else:
		tmp = 100.0 * ((n * i) / i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -8.2e-46)
		tmp = Float64(100.0 * Float64(i / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(Float64(n * i) / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -8.2e-46)
		tmp = 100.0 * (i / (i / n));
	else
		tmp = 100.0 * ((n * i) / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -8.2e-46], N[(100.0 * N[(i / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(N[(n * i), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -8.1999999999999998e-46

   1. Initial program 60.2%

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

   3. Taylor expanded in i around 0 32.4%

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

   if -8.1999999999999998e-46 < i

   1. Initial program 22.2%

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

   3. Taylor expanded in i around 0 17.8%

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

      1. +-commutative 17.8%

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

   5. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. div-inv 17.8%

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

      3. times-frac 18.1%

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

      4. associate--l+ 63.0%

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

      5. metadata-eval 63.0%

         \[\leadsto 100 \cdot \left(\frac{1}{i} \cdot \frac{i + \color{blue}{0}}{\frac{1}{n}}\right) \]

      6. +-rgt-identity 63.0%

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

   7. Applied egg-rr 63.0%

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

      1. frac-times 52.0%

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

      2. div-inv 52.1%

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

      3. associate-*l/ 51.5%

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

      4. clear-num 51.0%

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

      5. associate-*l/ 63.2%

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

   9. Applied egg-rr 63.2%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -8.2 \cdot 10^{-46}:\\ \;\;\;\;100 \cdot \frac{i}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n \cdot i}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.7% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (/ n (+ 0.01 (* i (- (* i 0.0008333333333333334) 0.005)))))

C

double code(double i, double n) {
	return n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n / (0.01d0 + (i * ((i * 0.0008333333333333334d0) - 0.005d0)))
end function

Java

public static double code(double i, double n) {
	return n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
}

Python

def code(i, n):
	return n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)))

Julia

function code(i, n)
	return Float64(n / Float64(0.01 + Float64(i * Float64(Float64(i * 0.0008333333333333334) - 0.005))))
end

MATLAB

function tmp = code(i, n)
	tmp = n / (0.01 + (i * ((i * 0.0008333333333333334) - 0.005)));
end

Wolfram

code[i_, n_] := N[(n / N[(0.01 + N[(i * N[(N[(i * 0.0008333333333333334), $MachinePrecision] - 0.005), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

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

   1. associate-/r/ 32.8%

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

   2. associate-*r* 32.8%

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

   3. *-commutative 32.8%

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

   4. associate-*r/ 32.8%

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

   5. sub-neg 32.8%

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

   6. distribute-lft-in 32.8%

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

   7. metadata-eval 32.8%

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

   8. metadata-eval 32.8%

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

   9. metadata-eval 32.8%

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

   10. fma-define 32.8%

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

   11. metadata-eval 32.8%

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

3. Simplified 32.8%

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

5. Taylor expanded in n around inf 34.9%

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

   1. sub-neg 34.9%

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

   2. metadata-eval 34.9%

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

   3. metadata-eval 34.9%

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

   4. distribute-lft-in 35.0%

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

   5. metadata-eval 35.0%

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

   6. sub-neg 35.0%

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

   7. expm1-define 71.4%

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

7. Simplified 71.4%

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

   1. clear-num 71.4%

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

   2. un-div-inv 71.3%

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

   3. *-un-lft-identity 71.3%

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

   4. times-frac 71.3%

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

   5. metadata-eval 71.3%

      \[\leadsto \frac{n}{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}} \]

9. Applied egg-rr 71.3%

   \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
10. Step-by-step derivation

    1. associate-*r/ 71.2%

       \[\leadsto \frac{n}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

11. Simplified 71.2%

    \[\leadsto \color{blue}{\frac{n}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

12. Taylor expanded in i around 0 60.6%

    \[\leadsto \frac{n}{\color{blue}{0.01 + i \cdot \left(0.0008333333333333334 \cdot i - 0.005\right)}} \]

13. Final simplification 60.6%

    \[\leadsto \frac{n}{0.01 + i \cdot \left(i \cdot 0.0008333333333333334 - 0.005\right)} \]
14. Add Preprocessing

Alternative 8: 51.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(n \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.56e+29) (* n 100.0) (* 50.0 (* n i))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.56e+29) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (n * i);
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= 1.56d+29) then
        tmp = n * 100.0d0
    else
        tmp = 50.0d0 * (n * i)
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.56e+29) {
		tmp = n * 100.0;
	} else {
		tmp = 50.0 * (n * i);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.56e+29:
		tmp = n * 100.0
	else:
		tmp = 50.0 * (n * i)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.56e+29)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(50.0 * Float64(n * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.56e+29)
		tmp = n * 100.0;
	else
		tmp = 50.0 * (n * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.56e+29], N[(n * 100.0), $MachinePrecision], N[(50.0 * N[(n * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.5599999999999999e29

   1. Initial program 28.8%

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

   3. Taylor expanded in i around 0 56.9%

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

      1. *-commutative 56.9%

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

   5. Simplified 56.9%

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

   if 1.5599999999999999e29 < i

   1. Initial program 45.0%

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

      1. associate-/r/ 45.3%

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

      2. associate-*r* 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-*r/ 45.3%

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

      5. sub-neg 45.3%

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

      6. distribute-lft-in 45.3%

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

      7. metadata-eval 45.3%

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

      8. metadata-eval 45.3%

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

      9. metadata-eval 45.3%

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

      10. fma-define 45.3%

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

      11. metadata-eval 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in n around inf 47.5%

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

      1. sub-neg 47.5%

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

      2. metadata-eval 47.5%

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

      3. metadata-eval 47.5%

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

      4. distribute-lft-in 47.5%

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

      5. metadata-eval 47.5%

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

      6. sub-neg 47.5%

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

      7. expm1-define 47.5%

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

   7. Simplified 47.5%

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

   8. Taylor expanded in i around 0 34.7%

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

   9. Taylor expanded in i around inf 34.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;50 \cdot \left(n \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.56e+29) (* n 100.0) (* i (* n 50.0))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.56e+29) {
		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 <= 1.56d+29) 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 <= 1.56e+29) {
		tmp = n * 100.0;
	} else {
		tmp = i * (n * 50.0);
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.56e+29:
		tmp = n * 100.0
	else:
		tmp = i * (n * 50.0)
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.56e+29)
		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 <= 1.56e+29)
		tmp = n * 100.0;
	else
		tmp = i * (n * 50.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.56e+29], N[(n * 100.0), $MachinePrecision], N[(i * N[(n * 50.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.5599999999999999e29

   1. Initial program 28.8%

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

   3. Taylor expanded in i around 0 56.9%

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

      1. *-commutative 56.9%

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

   5. Simplified 56.9%

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

   if 1.5599999999999999e29 < i

   1. Initial program 45.0%

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

      1. associate-/r/ 45.3%

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

      2. associate-*r* 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-*r/ 45.3%

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

      5. sub-neg 45.3%

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

      6. distribute-lft-in 45.3%

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

      7. metadata-eval 45.3%

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

      8. metadata-eval 45.3%

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

      9. metadata-eval 45.3%

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

      10. fma-define 45.3%

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

      11. metadata-eval 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in n around inf 47.5%

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

      1. sub-neg 47.5%

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

      2. metadata-eval 47.5%

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

      3. metadata-eval 47.5%

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

      4. distribute-lft-in 47.5%

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

      5. metadata-eval 47.5%

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

      6. sub-neg 47.5%

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

      7. expm1-define 47.5%

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

   7. Simplified 47.5%

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

   8. Taylor expanded in i around 0 34.7%

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

   9. Taylor expanded in i around inf 34.7%

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

       1. *-commutative 34.7%

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

       2. associate-*r* 34.7%

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

   11. Simplified 34.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(n \cdot 50\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ n \cdot \left(100 + i \cdot 50\right) \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* n (+ 100.0 (* i 50.0))))

C

double code(double i, double n) {
	return n * (100.0 + (i * 50.0));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n * (100.0d0 + (i * 50.0d0))
end function

Java

public static double code(double i, double n) {
	return n * (100.0 + (i * 50.0));
}

Python

def code(i, n):
	return n * (100.0 + (i * 50.0))

Julia

function code(i, n)
	return Float64(n * Float64(100.0 + Float64(i * 50.0)))
end

MATLAB

function tmp = code(i, n)
	tmp = n * (100.0 + (i * 50.0));
end

Wolfram

code[i_, n_] := N[(n * N[(100.0 + N[(i * 50.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.7%

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

   1. associate-/r/ 32.8%

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

   2. associate-*r* 32.8%

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

   3. *-commutative 32.8%

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

   4. associate-*r/ 32.8%

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

   5. sub-neg 32.8%

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

   6. distribute-lft-in 32.8%

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

   7. metadata-eval 32.8%

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

   8. metadata-eval 32.8%

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

   9. metadata-eval 32.8%

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

   10. fma-define 32.8%

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

   11. metadata-eval 32.8%

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

3. Simplified 32.8%

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

5. Taylor expanded in n around inf 34.9%

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

   1. sub-neg 34.9%

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

   2. metadata-eval 34.9%

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

   3. metadata-eval 34.9%

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

   4. distribute-lft-in 35.0%

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

   5. metadata-eval 35.0%

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

   6. sub-neg 35.0%

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

   7. expm1-define 71.4%

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

7. Simplified 71.4%

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

8. Taylor expanded in i around 0 52.2%

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

9. Final simplification 52.2%

   \[\leadsto n \cdot \left(100 + i \cdot 50\right) \]
10. Add Preprocessing

Alternative 11: 61.0% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \frac{n}{0.01 + i \cdot -0.005} \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (/ n (+ 0.01 (* i -0.005))))

C

double code(double i, double n) {
	return n / (0.01 + (i * -0.005));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = n / (0.01d0 + (i * (-0.005d0)))
end function

Java

public static double code(double i, double n) {
	return n / (0.01 + (i * -0.005));
}

Python

def code(i, n):
	return n / (0.01 + (i * -0.005))

Julia

function code(i, n)
	return Float64(n / Float64(0.01 + Float64(i * -0.005)))
end

MATLAB

function tmp = code(i, n)
	tmp = n / (0.01 + (i * -0.005));
end

Wolfram

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

Derivation

1. Initial program 32.7%

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

   1. associate-/r/ 32.8%

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

   2. associate-*r* 32.8%

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

   3. *-commutative 32.8%

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

   4. associate-*r/ 32.8%

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

   5. sub-neg 32.8%

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

   6. distribute-lft-in 32.8%

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

   7. metadata-eval 32.8%

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

   8. metadata-eval 32.8%

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

   9. metadata-eval 32.8%

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

   10. fma-define 32.8%

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

   11. metadata-eval 32.8%

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

3. Simplified 32.8%

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

5. Taylor expanded in n around inf 34.9%

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

   1. sub-neg 34.9%

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

   2. metadata-eval 34.9%

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

   3. metadata-eval 34.9%

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

   4. distribute-lft-in 35.0%

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

   5. metadata-eval 35.0%

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

   6. sub-neg 35.0%

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

   7. expm1-define 71.4%

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

7. Simplified 71.4%

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

   1. clear-num 71.4%

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

   2. un-div-inv 71.3%

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

   3. *-un-lft-identity 71.3%

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

   4. times-frac 71.3%

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

   5. metadata-eval 71.3%

      \[\leadsto \frac{n}{\color{blue}{0.01} \cdot \frac{i}{\mathsf{expm1}\left(i\right)}} \]

9. Applied egg-rr 71.3%

   \[\leadsto \color{blue}{\frac{n}{0.01 \cdot \frac{i}{\mathsf{expm1}\left(i\right)}}} \]
10. Step-by-step derivation

    1. associate-*r/ 71.2%

       \[\leadsto \frac{n}{\color{blue}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

11. Simplified 71.2%

    \[\leadsto \color{blue}{\frac{n}{\frac{0.01 \cdot i}{\mathsf{expm1}\left(i\right)}}} \]

12. Taylor expanded in i around 0 59.5%

    \[\leadsto \frac{n}{\color{blue}{0.01 + -0.005 \cdot i}} \]
13. Step-by-step derivation

    1. *-commutative 59.5%

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

14. Simplified 59.5%

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

15. Final simplification 59.5%

    \[\leadsto \frac{n}{0.01 + i \cdot -0.005} \]
16. Add Preprocessing

Alternative 12: 2.7% accurate, 38.0× speedup

Math

\[\begin{array}{l} \\ i \cdot -50 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 (* i -50.0))

C

double code(double i, double n) {
	return i * -50.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = i * (-50.0d0)
end function

Java

public static double code(double i, double n) {
	return i * -50.0;
}

Python

def code(i, n):
	return i * -50.0

Julia

function code(i, n)
	return Float64(i * -50.0)
end

MATLAB

function tmp = code(i, n)
	tmp = i * -50.0;
end

Wolfram

code[i_, n_] := N[(i * -50.0), $MachinePrecision]

Derivation

1. Initial program 32.7%

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

3. Taylor expanded in i around 0 41.4%

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

   1. associate-*r/ 41.4%

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

   2. metadata-eval 41.4%

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

5. Simplified 41.4%

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

6. Taylor expanded in n around 0 2.7%

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

   1. *-commutative 2.7%

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

8. Simplified 2.7%

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

9. Final simplification 2.7%

   \[\leadsto i \cdot -50 \]
10. Add Preprocessing

Alternative 13: 46.6% 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 32.7%

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

3. Taylor expanded in i around 0 44.4%

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

   1. *-commutative 44.4%

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

5. Simplified 44.4%

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

6. Final simplification 44.4%

   \[\leadsto n \cdot 100 \]
7. Add Preprocessing

Developer target: 32.2% 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 2024066 
(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))))