Compound Interest

Percentage Accurate: 28.7% → 96.9%

Time: 19.5s

Alternatives: 16

Speedup: 16.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 28.7% 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: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (/ (* n 100.0) (/ i (expm1 (* n (log1p (/ i n))))))
     (if (<= t_0 INFINITY) (* t_0 100.0) (* 100.0 (/ n (+ 1.0 (* i -0.5))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (n * 100.0) / (i / expm1((n * log1p((i / n)))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (n * 100.0) / (i / Math.expm1((n * Math.log1p((i / n)))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (n * 100.0) / (i / math.expm1((n * math.log1p((i / n)))))
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(n * 100.0), $MachinePrecision] / N[(i / N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-/r/ 26.7%

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

      3. sub-neg 26.7%

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

      4. metadata-eval 26.7%

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

      5. associate-*r* 26.7%

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

      6. *-commutative 26.7%

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

      7. clear-num 26.7%

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

      8. un-div-inv 26.7%

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

      9. metadata-eval 26.7%

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

      10. sub-neg 26.7%

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

      11. pow-to-exp 26.1%

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

      12. expm1-def 36.0%

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

      13. add-log-exp 26.1%

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

      14. pow-to-exp 26.7%

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

      15. log-pow 36.0%

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

      16. log1p-udef 96.7%

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

   3. Applied egg-rr 96.7%

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

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

   1. Initial program 99.6%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 88.2%

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

   4. Simplified 88.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{n \cdot 100}{\frac{i}{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 2: 96.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (/ (expm1 (* n (log1p (/ i n)))) (/ i (* n 100.0)))
     (if (<= t_0 INFINITY) (* t_0 100.0) (* 100.0 (/ n (+ 1.0 (* i -0.5))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = expm1((n * log1p((i / n)))) / (i / (n * 100.0));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) / (i / (n * 100.0));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.expm1((n * math.log1p((i / n)))) / (i / (n * 100.0))
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(i / N[(n * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-/r/ 26.7%

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

      3. sub-neg 26.7%

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

      4. metadata-eval 26.7%

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

      5. associate-*r* 26.7%

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

      6. metadata-eval 26.7%

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

      7. sub-neg 26.7%

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

      8. associate-*l/ 26.7%

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

      9. associate-/l* 26.9%

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

   3. Applied egg-rr 96.3%

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

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

   1. Initial program 99.6%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 88.2%

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

   4. Simplified 88.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{i}{n \cdot 100}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{\frac{i}{100}}{n}}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (/ (+ (pow (+ 1.0 (/ i n)) n) -1.0) (/ i n))))
   (if (<= t_0 0.0)
     (/ (expm1 (* n (log1p (/ i n)))) (/ (/ i 100.0) n))
     (if (<= t_0 INFINITY) (* t_0 100.0) (* 100.0 (/ n (+ 1.0 (* i -0.5))))))))

C

double code(double i, double n) {
	double t_0 = (pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = expm1((n * log1p((i / n)))) / ((i / 100.0) / n);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = (Math.pow((1.0 + (i / n)), n) + -1.0) / (i / n);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.expm1((n * Math.log1p((i / n)))) / ((i / 100.0) / n);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * 100.0;
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = (math.pow((1.0 + (i / n)), n) + -1.0) / (i / n)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.expm1((n * math.log1p((i / n)))) / ((i / 100.0) / n)
	elif t_0 <= math.inf:
		tmp = t_0 * 100.0
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

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

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] + -1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(Exp[N[(n * N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(N[(i / 100.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 * 100.0), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 1 (/.f64 i n)) n) 1) (/.f64 i n)) < 0.0

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-/r/ 26.7%

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

      3. sub-neg 26.7%

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

      4. metadata-eval 26.7%

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

      5. associate-*r* 26.7%

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

      6. metadata-eval 26.7%

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

      7. sub-neg 26.7%

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

      8. associate-*l/ 26.7%

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

      9. associate-/l* 26.9%

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

   3. Applied egg-rr 96.3%

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

      1. *-un-lft-identity 96.3%

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

      2. times-frac 96.5%

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

   5. Applied egg-rr 96.5%

      \[\leadsto \frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\color{blue}{\frac{1}{n} \cdot \frac{i}{100}}} \]
   6. Step-by-step derivation

      1. associate-*l/ 96.6%

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

      2. *-lft-identity 96.6%

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

   7. Simplified 96.6%

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

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

   1. Initial program 99.6%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in n around inf 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

      3. expm1-def 88.2%

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

   4. Simplified 88.2%

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

   5. Taylor expanded in i around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(n \cdot \mathsf{log1p}\left(\frac{i}{n}\right)\right)}{\frac{\frac{i}{100}}{n}}\\ \mathbf{elif}\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\frac{{\left(1 + \frac{i}{n}\right)}^{n} + -1}{\frac{i}{n}} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{if}\;n \leq -4.3 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{\frac{n \cdot \left(i \cdot -0.005\right) + n \cdot 0.01}{n \cdot n}}\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-98}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{\frac{i}{\log \left(\frac{i}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* 100.0 (/ n (/ i (expm1 i))))))
   (if (<= n -4.3e-120)
     t_0
     (if (<= n 5.1e-144)
       (/ 1.0 (/ (+ (* n (* i -0.005)) (* n 0.01)) (* n n)))
       (if (<= n 1.25e-98) (* 100.0 (/ (* n n) (/ i (log (/ i n))))) t_0)))))

C

double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / expm1(i)));
	double tmp;
	if (n <= -4.3e-120) {
		tmp = t_0;
	} else if (n <= 5.1e-144) {
		tmp = 1.0 / (((n * (i * -0.005)) + (n * 0.01)) / (n * n));
	} else if (n <= 1.25e-98) {
		tmp = 100.0 * ((n * n) / (i / log((i / n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double t_0 = 100.0 * (n / (i / Math.expm1(i)));
	double tmp;
	if (n <= -4.3e-120) {
		tmp = t_0;
	} else if (n <= 5.1e-144) {
		tmp = 1.0 / (((n * (i * -0.005)) + (n * 0.01)) / (n * n));
	} else if (n <= 1.25e-98) {
		tmp = 100.0 * ((n * n) / (i / Math.log((i / n))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(i, n):
	t_0 = 100.0 * (n / (i / math.expm1(i)))
	tmp = 0
	if n <= -4.3e-120:
		tmp = t_0
	elif n <= 5.1e-144:
		tmp = 1.0 / (((n * (i * -0.005)) + (n * 0.01)) / (n * n))
	elif n <= 1.25e-98:
		tmp = 100.0 * ((n * n) / (i / math.log((i / n))))
	else:
		tmp = t_0
	return tmp

Julia

function code(i, n)
	t_0 = Float64(100.0 * Float64(n / Float64(i / expm1(i))))
	tmp = 0.0
	if (n <= -4.3e-120)
		tmp = t_0;
	elseif (n <= 5.1e-144)
		tmp = Float64(1.0 / Float64(Float64(Float64(n * Float64(i * -0.005)) + Float64(n * 0.01)) / Float64(n * n)));
	elseif (n <= 1.25e-98)
		tmp = Float64(100.0 * Float64(Float64(n * n) / Float64(i / log(Float64(i / n)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.3e-120], t$95$0, If[LessEqual[n, 5.1e-144], N[(1.0 / N[(N[(N[(n * N[(i * -0.005), $MachinePrecision]), $MachinePrecision] + N[(n * 0.01), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.25e-98], N[(100.0 * N[(N[(n * n), $MachinePrecision] / N[(i / N[Log[N[(i / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.29999999999999982e-120 or 1.25000000000000005e-98 < n

   1. Initial program 20.5%

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

   2. Taylor expanded in n around inf 35.6%

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

      1. *-commutative 35.6%

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

      2. associate-/l* 35.6%

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

      3. expm1-def 91.2%

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

   4. Simplified 91.2%

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

   if -4.29999999999999982e-120 < n < 5.1e-144

   1. Initial program 55.6%

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

   2. Taylor expanded in n around inf 39.9%

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

      1. *-commutative 39.9%

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

      2. associate-/l* 39.9%

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

      3. expm1-def 45.9%

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

   4. Simplified 45.9%

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

      1. associate-*l/ 45.9%

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

      2. clear-num 46.8%

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

   6. Applied egg-rr 46.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   7. Step-by-step derivation

      1. associate-/l/ 31.5%

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

      2. *-commutative 31.5%

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

      3. associate-*r* 31.5%

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

      4. *-commutative 31.5%

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

      5. associate-*r* 31.5%

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

      6. *-commutative 31.5%

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

      7. associate-*l* 31.5%

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

   8. Simplified 31.5%

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

   9. Taylor expanded in i around 0 54.9%

      \[\leadsto \frac{1}{\color{blue}{-0.005 \cdot \frac{i}{n} + 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 54.9%

          \[\leadsto \frac{1}{\color{blue}{\frac{-0.005 \cdot i}{n}} + 0.01 \cdot \frac{1}{n}} \]

       2. un-div-inv 55.0%

          \[\leadsto \frac{1}{\frac{-0.005 \cdot i}{n} + \color{blue}{\frac{0.01}{n}}} \]

       3. frac-add 66.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(-0.005 \cdot i\right) \cdot n + n \cdot 0.01}{n \cdot n}}} \]

       4. *-commutative 66.1%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(i \cdot -0.005\right)} \cdot n + n \cdot 0.01}{n \cdot n}} \]

   11. Applied egg-rr 66.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(i \cdot -0.005\right) \cdot n + n \cdot 0.01}{n \cdot n}}} \]

   if 5.1e-144 < n < 1.25000000000000005e-98

   1. Initial program 46.4%

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

      1. *-commutative 46.4%

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

      2. associate-/r/ 46.4%

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

      3. sub-neg 46.4%

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

      4. metadata-eval 46.4%

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

      5. associate-*r* 46.4%

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

      6. metadata-eval 46.4%

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

      7. sub-neg 46.4%

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

      8. associate-*l/ 46.4%

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

      9. associate-/l* 46.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in n around 0 99.3%

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

      1. associate-/l* 99.6%

         \[\leadsto 100 \cdot \color{blue}{\frac{{n}^{2}}{\frac{i}{\log i + -1 \cdot \log n}}} \]

      2. unpow2 99.6%

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

      3. mul-1-neg 99.6%

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

      4. unsub-neg 99.6%

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

      5. log-div 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.3 \cdot 10^{-120}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{\frac{n \cdot \left(i \cdot -0.005\right) + n \cdot 0.01}{n \cdot n}}\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-98}:\\ \;\;\;\;100 \cdot \frac{n \cdot n}{\frac{i}{\log \left(\frac{i}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \end{array} \]

Alternative 5: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{-13} \lor \neg \left(i \leq 20000000000000\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= i -4.2e-13) (not (<= i 20000000000000.0)))
   (* 100.0 (/ (expm1 i) (/ i n)))
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))))

C

double code(double i, double n) {
	double tmp;
	if ((i <= -4.2e-13) || !(i <= 20000000000000.0)) {
		tmp = 100.0 * (expm1(i) / (i / n));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((i <= -4.2e-13) || !(i <= 20000000000000.0)) {
		tmp = 100.0 * (Math.expm1(i) / (i / n));
	} else {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (i <= -4.2e-13) or not (i <= 20000000000000.0):
		tmp = 100.0 * (math.expm1(i) / (i / n))
	else:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((i <= -4.2e-13) || !(i <= 20000000000000.0))
		tmp = Float64(100.0 * Float64(expm1(i) / Float64(i / n)));
	else
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[i, -4.2e-13], N[Not[LessEqual[i, 20000000000000.0]], $MachinePrecision]], N[(100.0 * N[(N[(Exp[i] - 1), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -4.19999999999999977e-13 or 2e13 < i

   1. Initial program 53.1%

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

   2. Taylor expanded in n around inf 71.7%

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

      1. expm1-def 71.7%

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

   4. Simplified 71.7%

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

   if -4.19999999999999977e-13 < i < 2e13

   1. Initial program 9.4%

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

   2. Taylor expanded in n around inf 8.5%

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

      1. *-commutative 8.5%

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

      2. associate-/l* 8.5%

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

      3. expm1-def 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in i around 0 86.1%

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

      1. *-commutative 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{-13} \lor \neg \left(i \leq 20000000000000\right):\\ \;\;\;\;100 \cdot \frac{\mathsf{expm1}\left(i\right)}{\frac{i}{n}}\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \end{array} \]

Alternative 6: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-132} \lor \neg \left(n \leq 2.2 \cdot 10^{-82}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -1.85e-132) (not (<= n 2.2e-82)))
   (* 100.0 (/ n (/ i (expm1 i))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-132) || !(n <= 2.2e-82)) {
		tmp = 100.0 * (n / (i / expm1(i)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -1.85e-132) || !(n <= 2.2e-82)) {
		tmp = 100.0 * (n / (i / Math.expm1(i)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -1.85e-132) or not (n <= 2.2e-82):
		tmp = 100.0 * (n / (i / math.expm1(i)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -1.85e-132) || !(n <= 2.2e-82))
		tmp = Float64(100.0 * Float64(n / Float64(i / expm1(i))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -1.85e-132], N[Not[LessEqual[n, 2.2e-82]], $MachinePrecision]], N[(100.0 * N[(n / N[(i / N[(Exp[i] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.8500000000000001e-132 or 2.19999999999999986e-82 < n

   1. Initial program 20.6%

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

   2. Taylor expanded in n around inf 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-/l* 35.8%

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

      3. expm1-def 91.7%

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

   4. Simplified 91.7%

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

   if -1.8500000000000001e-132 < n < 2.19999999999999986e-82

   1. Initial program 53.6%

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

      1. *-commutative 53.6%

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

      2. associate-/r/ 52.9%

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

      3. associate-*l* 52.9%

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

      4. sub-neg 52.9%

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

      5. metadata-eval 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in i around 0 62.1%

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

   5. Taylor expanded in i around 0 62.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.85 \cdot 10^{-132} \lor \neg \left(n \leq 2.2 \cdot 10^{-82}\right):\\ \;\;\;\;100 \cdot \frac{n}{\frac{i}{\mathsf{expm1}\left(i\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 63.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-94} \lor \neg \left(n \leq 2.05 \cdot 10^{-81}\right):\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -8.2e-94) (not (<= n 2.05e-81)))
   (* 100.0 (+ n (* n (+ (* i 0.5) (* 0.16666666666666666 (* i i))))))
   0.0))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -8.2e-94) || !(n <= 2.05e-81)) {
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))));
	} 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 ((n <= (-8.2d-94)) .or. (.not. (n <= 2.05d-81))) then
        tmp = 100.0d0 * (n + (n * ((i * 0.5d0) + (0.16666666666666666d0 * (i * i)))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -8.2e-94) || !(n <= 2.05e-81)) {
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -8.2e-94) or not (n <= 2.05e-81):
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -8.2e-94) || !(n <= 2.05e-81))
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(i * 0.5) + Float64(0.16666666666666666 * Float64(i * i))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -8.2e-94) || ~((n <= 2.05e-81)))
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -8.2e-94], N[Not[LessEqual[n, 2.05e-81]], $MachinePrecision]], N[(100.0 * N[(n + N[(n * N[(N[(i * 0.5), $MachinePrecision] + N[(0.16666666666666666 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -8.20000000000000001e-94 or 2.04999999999999992e-81 < n

   1. Initial program 20.0%

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

   2. Taylor expanded in n around inf 35.7%

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

      1. expm1-def 72.6%

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

   4. Simplified 72.6%

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

   5. Taylor expanded in i around 0 69.9%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + 0.5 \cdot \left(i \cdot n\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 69.9%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)}\right) \]

      2. associate-*r* 69.9%

         \[\leadsto 100 \cdot \left(n + \left(\color{blue}{\left(0.5 \cdot i\right) \cdot n} + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)\right) \]

      3. associate-*r* 69.9%

         \[\leadsto 100 \cdot \left(n + \left(\left(0.5 \cdot i\right) \cdot n + \color{blue}{\left(0.16666666666666666 \cdot {i}^{2}\right) \cdot n}\right)\right) \]

      4. distribute-rgt-out 70.2%

         \[\leadsto 100 \cdot \left(n + \color{blue}{n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot {i}^{2}\right)}\right) \]

      5. *-commutative 70.2%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\color{blue}{i \cdot 0.5} + 0.16666666666666666 \cdot {i}^{2}\right)\right) \]

      6. unpow2 70.2%

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

   7. Simplified 70.2%

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

   if -8.20000000000000001e-94 < n < 2.04999999999999992e-81

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-/r/ 52.2%

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

      3. associate-*l* 52.2%

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

      4. sub-neg 52.2%

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

      5. metadata-eval 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in i around 0 60.6%

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

   5. Taylor expanded in i around 0 60.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-94} \lor \neg \left(n \leq 2.05 \cdot 10^{-81}\right):\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 67.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+132} \lor \neg \left(n \leq 1.5 \cdot 10^{-170}\right):\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{n \cdot \left(i \cdot -0.005\right) + n \cdot 0.01}{n \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (or (<= n -9e+132) (not (<= n 1.5e-170)))
   (* 100.0 (+ n (* n (+ (* i 0.5) (* 0.16666666666666666 (* i i))))))
   (/ 1.0 (/ (+ (* n (* i -0.005)) (* n 0.01)) (* n n)))))

C

double code(double i, double n) {
	double tmp;
	if ((n <= -9e+132) || !(n <= 1.5e-170)) {
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))));
	} else {
		tmp = 1.0 / (((n * (i * -0.005)) + (n * 0.01)) / (n * n));
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-9d+132)) .or. (.not. (n <= 1.5d-170))) then
        tmp = 100.0d0 * (n + (n * ((i * 0.5d0) + (0.16666666666666666d0 * (i * i)))))
    else
        tmp = 1.0d0 / (((n * (i * (-0.005d0))) + (n * 0.01d0)) / (n * n))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if ((n <= -9e+132) || !(n <= 1.5e-170)) {
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))));
	} else {
		tmp = 1.0 / (((n * (i * -0.005)) + (n * 0.01)) / (n * n));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if (n <= -9e+132) or not (n <= 1.5e-170):
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))))
	else:
		tmp = 1.0 / (((n * (i * -0.005)) + (n * 0.01)) / (n * n))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if ((n <= -9e+132) || !(n <= 1.5e-170))
		tmp = Float64(100.0 * Float64(n + Float64(n * Float64(Float64(i * 0.5) + Float64(0.16666666666666666 * Float64(i * i))))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(n * Float64(i * -0.005)) + Float64(n * 0.01)) / Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if ((n <= -9e+132) || ~((n <= 1.5e-170)))
		tmp = 100.0 * (n + (n * ((i * 0.5) + (0.16666666666666666 * (i * i)))));
	else
		tmp = 1.0 / (((n * (i * -0.005)) + (n * 0.01)) / (n * n));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[Or[LessEqual[n, -9e+132], N[Not[LessEqual[n, 1.5e-170]], $MachinePrecision]], N[(100.0 * N[(n + N[(n * N[(N[(i * 0.5), $MachinePrecision] + N[(0.16666666666666666 * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(n * N[(i * -0.005), $MachinePrecision]), $MachinePrecision] + N[(n * 0.01), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.99999999999999944e132 or 1.50000000000000007e-170 < n

   1. Initial program 19.5%

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

   2. Taylor expanded in n around inf 36.4%

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

      1. expm1-def 65.4%

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

   4. Simplified 65.4%

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

   5. Taylor expanded in i around 0 69.3%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + 0.5 \cdot \left(i \cdot n\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 69.3%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)}\right) \]

      2. associate-*r* 69.3%

         \[\leadsto 100 \cdot \left(n + \left(\color{blue}{\left(0.5 \cdot i\right) \cdot n} + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)\right) \]

      3. associate-*r* 69.3%

         \[\leadsto 100 \cdot \left(n + \left(\left(0.5 \cdot i\right) \cdot n + \color{blue}{\left(0.16666666666666666 \cdot {i}^{2}\right) \cdot n}\right)\right) \]

      4. distribute-rgt-out 69.6%

         \[\leadsto 100 \cdot \left(n + \color{blue}{n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot {i}^{2}\right)}\right) \]

      5. *-commutative 69.6%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\color{blue}{i \cdot 0.5} + 0.16666666666666666 \cdot {i}^{2}\right)\right) \]

      6. unpow2 69.6%

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

   7. Simplified 69.6%

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

   if -8.99999999999999944e132 < n < 1.50000000000000007e-170

   1. Initial program 40.4%

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

   2. Taylor expanded in n around inf 33.8%

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

      1. *-commutative 33.8%

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

      2. associate-/l* 33.8%

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

      3. expm1-def 66.0%

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

   4. Simplified 66.0%

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

      1. associate-*l/ 66.0%

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

      2. clear-num 66.4%

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

   6. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   7. Step-by-step derivation

      1. associate-/l/ 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*r* 58.0%

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

      4. *-commutative 58.0%

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

      5. associate-*r* 58.0%

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

      6. *-commutative 58.0%

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

      7. associate-*l* 58.1%

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

   8. Simplified 58.1%

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

   9. Taylor expanded in i around 0 60.2%

      \[\leadsto \frac{1}{\color{blue}{-0.005 \cdot \frac{i}{n} + 0.01 \cdot \frac{1}{n}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 60.2%

          \[\leadsto \frac{1}{\color{blue}{\frac{-0.005 \cdot i}{n}} + 0.01 \cdot \frac{1}{n}} \]

       2. un-div-inv 60.3%

          \[\leadsto \frac{1}{\frac{-0.005 \cdot i}{n} + \color{blue}{\frac{0.01}{n}}} \]

       3. frac-add 66.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(-0.005 \cdot i\right) \cdot n + n \cdot 0.01}{n \cdot n}}} \]

       4. *-commutative 66.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(i \cdot -0.005\right)} \cdot n + n \cdot 0.01}{n \cdot n}} \]

   11. Applied egg-rr 66.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(i \cdot -0.005\right) \cdot n + n \cdot 0.01}{n \cdot n}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9 \cdot 10^{+132} \lor \neg \left(n \leq 1.5 \cdot 10^{-170}\right):\\ \;\;\;\;100 \cdot \left(n + n \cdot \left(i \cdot 0.5 + 0.16666666666666666 \cdot \left(i \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{n \cdot \left(i \cdot -0.005\right) + n \cdot 0.01}{n \cdot n}}\\ \end{array} \]

Alternative 9: 62.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0)
   (/ -200.0 (/ i n))
   (if (<= i 1.7e-21) (* n 100.0) (* 16.666666666666668 (* n (* i i))))))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 / (i / n);
	} else if (i <= 1.7e-21) {
		tmp = n * 100.0;
	} else {
		tmp = 16.666666666666668 * (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 <= (-2.0d0)) then
        tmp = (-200.0d0) / (i / n)
    else if (i <= 1.7d-21) then
        tmp = n * 100.0d0
    else
        tmp = 16.666666666666668d0 * (n * (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 / (i / n);
	} else if (i <= 1.7e-21) {
		tmp = n * 100.0;
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = -200.0 / (i / n)
	elif i <= 1.7e-21:
		tmp = n * 100.0
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(-200.0 / Float64(i / n));
	elseif (i <= 1.7e-21)
		tmp = Float64(n * 100.0);
	else
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -2.0)
		tmp = -200.0 / (i / n);
	elseif (i <= 1.7e-21)
		tmp = n * 100.0;
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(-200.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.7e-21], N[(n * 100.0), $MachinePrecision], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 59.9%

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

   2. Taylor expanded in n around inf 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-/l* 78.8%

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

      3. expm1-def 78.8%

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

   4. Simplified 78.8%

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

   5. Taylor expanded in i around 0 33.1%

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

      1. *-commutative 33.1%

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

   7. Simplified 33.1%

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

   8. Taylor expanded in i around inf 33.1%

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

      1. clear-num 34.0%

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

      2. un-div-inv 34.0%

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

   10. Applied egg-rr 34.0%

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

   if -2 < i < 1.7e-21

   1. Initial program 7.5%

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

   2. Taylor expanded in i around 0 87.9%

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

      1. *-commutative 87.9%

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

   4. Simplified 87.9%

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

   if 1.7e-21 < i

   1. Initial program 49.1%

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

   2. Taylor expanded in n around inf 60.9%

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

      1. expm1-def 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in i around 0 39.0%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + 0.5 \cdot \left(i \cdot n\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)}\right) \]

      2. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\color{blue}{\left(0.5 \cdot i\right) \cdot n} + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)\right) \]

      3. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\left(0.5 \cdot i\right) \cdot n + \color{blue}{\left(0.16666666666666666 \cdot {i}^{2}\right) \cdot n}\right)\right) \]

      4. distribute-rgt-out 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot {i}^{2}\right)}\right) \]

      5. *-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\color{blue}{i \cdot 0.5} + 0.16666666666666666 \cdot {i}^{2}\right)\right) \]

      6. unpow2 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in i around inf 39.1%

      \[\leadsto \color{blue}{16.666666666666668 \cdot \left({i}^{2} \cdot n\right)} \]
   9. Step-by-step derivation

      1. unpow2 39.1%

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

      2. *-commutative 39.1%

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

   10. Simplified 39.1%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 10: 62.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.7e-21)
   (* 100.0 (/ n (+ 1.0 (* i -0.5))))
   (* 16.666666666666668 (* n (* i i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.7e-21) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = 16.666666666666668 * (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 <= 1.7d-21) then
        tmp = 100.0d0 * (n / (1.0d0 + (i * (-0.5d0))))
    else
        tmp = 16.666666666666668d0 * (n * (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.7e-21) {
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.7e-21:
		tmp = 100.0 * (n / (1.0 + (i * -0.5)))
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.7e-21)
		tmp = Float64(100.0 * Float64(n / Float64(1.0 + Float64(i * -0.5))));
	else
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.7e-21)
		tmp = 100.0 * (n / (1.0 + (i * -0.5)));
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.7e-21], N[(100.0 * N[(n / N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.7e-21

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 28.1%

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

      1. *-commutative 28.1%

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

      2. associate-/l* 28.1%

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

      3. expm1-def 85.8%

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

   4. Simplified 85.8%

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

   5. Taylor expanded in i around 0 72.7%

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

      1. *-commutative 72.7%

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

   7. Simplified 72.7%

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

   if 1.7e-21 < i

   1. Initial program 49.1%

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

   2. Taylor expanded in n around inf 60.9%

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

      1. expm1-def 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in i around 0 39.0%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + 0.5 \cdot \left(i \cdot n\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)}\right) \]

      2. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\color{blue}{\left(0.5 \cdot i\right) \cdot n} + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)\right) \]

      3. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\left(0.5 \cdot i\right) \cdot n + \color{blue}{\left(0.16666666666666666 \cdot {i}^{2}\right) \cdot n}\right)\right) \]

      4. distribute-rgt-out 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot {i}^{2}\right)}\right) \]

      5. *-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\color{blue}{i \cdot 0.5} + 0.16666666666666666 \cdot {i}^{2}\right)\right) \]

      6. unpow2 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in i around inf 39.1%

      \[\leadsto \color{blue}{16.666666666666668 \cdot \left({i}^{2} \cdot n\right)} \]
   9. Step-by-step derivation

      1. unpow2 39.1%

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

      2. *-commutative 39.1%

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

   10. Simplified 39.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;100 \cdot \frac{n}{1 + i \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 11: 62.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.7e-21)
   (/ 100.0 (/ (+ 1.0 (* i -0.5)) n))
   (* 16.666666666666668 (* n (* i i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.7e-21) {
		tmp = 100.0 / ((1.0 + (i * -0.5)) / n);
	} else {
		tmp = 16.666666666666668 * (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 <= 1.7d-21) then
        tmp = 100.0d0 / ((1.0d0 + (i * (-0.5d0))) / n)
    else
        tmp = 16.666666666666668d0 * (n * (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.7e-21) {
		tmp = 100.0 / ((1.0 + (i * -0.5)) / n);
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.7e-21:
		tmp = 100.0 / ((1.0 + (i * -0.5)) / n)
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.7e-21)
		tmp = Float64(100.0 / Float64(Float64(1.0 + Float64(i * -0.5)) / n));
	else
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.7e-21)
		tmp = 100.0 / ((1.0 + (i * -0.5)) / n);
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.7e-21], N[(100.0 / N[(N[(1.0 + N[(i * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.7e-21

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 28.1%

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

      1. *-commutative 28.1%

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

      2. associate-/l* 28.1%

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

      3. expm1-def 85.8%

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

   4. Simplified 85.8%

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

      1. *-commutative 85.8%

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

      2. clear-num 85.9%

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

      3. un-div-inv 85.8%

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

   6. Applied egg-rr 85.8%

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

   7. Taylor expanded in i around 0 72.7%

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

      1. *-commutative 72.7%

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

   9. Simplified 72.7%

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

   if 1.7e-21 < i

   1. Initial program 49.1%

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

   2. Taylor expanded in n around inf 60.9%

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

      1. expm1-def 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in i around 0 39.0%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + 0.5 \cdot \left(i \cdot n\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)}\right) \]

      2. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\color{blue}{\left(0.5 \cdot i\right) \cdot n} + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)\right) \]

      3. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\left(0.5 \cdot i\right) \cdot n + \color{blue}{\left(0.16666666666666666 \cdot {i}^{2}\right) \cdot n}\right)\right) \]

      4. distribute-rgt-out 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot {i}^{2}\right)}\right) \]

      5. *-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\color{blue}{i \cdot 0.5} + 0.16666666666666666 \cdot {i}^{2}\right)\right) \]

      6. unpow2 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in i around inf 39.1%

      \[\leadsto \color{blue}{16.666666666666668 \cdot \left({i}^{2} \cdot n\right)} \]
   9. Step-by-step derivation

      1. unpow2 39.1%

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

      2. *-commutative 39.1%

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

   10. Simplified 39.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{100}{\frac{1 + i \cdot -0.5}{n}}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 12: 62.3% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{n}{i \cdot -0.005 + 0.01}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i 1.7e-21)
   (/ n (+ (* i -0.005) 0.01))
   (* 16.666666666666668 (* n (* i i)))))

C

double code(double i, double n) {
	double tmp;
	if (i <= 1.7e-21) {
		tmp = n / ((i * -0.005) + 0.01);
	} else {
		tmp = 16.666666666666668 * (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 <= 1.7d-21) then
        tmp = n / ((i * (-0.005d0)) + 0.01d0)
    else
        tmp = 16.666666666666668d0 * (n * (i * i))
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= 1.7e-21) {
		tmp = n / ((i * -0.005) + 0.01);
	} else {
		tmp = 16.666666666666668 * (n * (i * i));
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= 1.7e-21:
		tmp = n / ((i * -0.005) + 0.01)
	else:
		tmp = 16.666666666666668 * (n * (i * i))
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= 1.7e-21)
		tmp = Float64(n / Float64(Float64(i * -0.005) + 0.01));
	else
		tmp = Float64(16.666666666666668 * Float64(n * Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= 1.7e-21)
		tmp = n / ((i * -0.005) + 0.01);
	else
		tmp = 16.666666666666668 * (n * (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, 1.7e-21], N[(n / N[(N[(i * -0.005), $MachinePrecision] + 0.01), $MachinePrecision]), $MachinePrecision], N[(16.666666666666668 * N[(n * N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.7e-21

   1. Initial program 22.0%

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

   2. Taylor expanded in n around inf 28.1%

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

      1. *-commutative 28.1%

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

      2. associate-/l* 28.1%

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

      3. expm1-def 85.8%

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

   4. Simplified 85.8%

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

      1. associate-*l/ 85.3%

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

      2. clear-num 85.4%

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

   6. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{i}{\mathsf{expm1}\left(i\right)}}{n \cdot 100}}} \]
   7. Step-by-step derivation

      1. associate-/l/ 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*r* 78.8%

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

      4. *-commutative 78.8%

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

      5. associate-*r* 78.8%

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

      6. *-commutative 78.8%

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

      7. associate-*l* 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in i around 0 72.6%

      \[\leadsto \frac{1}{\color{blue}{-0.005 \cdot \frac{i}{n} + 0.01 \cdot \frac{1}{n}}} \]

   10. Taylor expanded in n around 0 72.6%

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

       1. *-commutative 72.6%

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

   12. Simplified 72.6%

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

   if 1.7e-21 < i

   1. Initial program 49.1%

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

   2. Taylor expanded in n around inf 60.9%

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

      1. expm1-def 59.3%

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

   4. Simplified 59.3%

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

   5. Taylor expanded in i around 0 39.0%

      \[\leadsto 100 \cdot \color{blue}{\left(n + \left(0.16666666666666666 \cdot \left({i}^{2} \cdot n\right) + 0.5 \cdot \left(i \cdot n\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{\left(0.5 \cdot \left(i \cdot n\right) + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)}\right) \]

      2. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\color{blue}{\left(0.5 \cdot i\right) \cdot n} + 0.16666666666666666 \cdot \left({i}^{2} \cdot n\right)\right)\right) \]

      3. associate-*r* 39.0%

         \[\leadsto 100 \cdot \left(n + \left(\left(0.5 \cdot i\right) \cdot n + \color{blue}{\left(0.16666666666666666 \cdot {i}^{2}\right) \cdot n}\right)\right) \]

      4. distribute-rgt-out 39.0%

         \[\leadsto 100 \cdot \left(n + \color{blue}{n \cdot \left(0.5 \cdot i + 0.16666666666666666 \cdot {i}^{2}\right)}\right) \]

      5. *-commutative 39.0%

         \[\leadsto 100 \cdot \left(n + n \cdot \left(\color{blue}{i \cdot 0.5} + 0.16666666666666666 \cdot {i}^{2}\right)\right) \]

      6. unpow2 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in i around inf 39.1%

      \[\leadsto \color{blue}{16.666666666666668 \cdot \left({i}^{2} \cdot n\right)} \]
   9. Step-by-step derivation

      1. unpow2 39.1%

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

      2. *-commutative 39.1%

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

   10. Simplified 39.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{n}{i \cdot -0.005 + 0.01}\\ \mathbf{else}:\\ \;\;\;\;16.666666666666668 \cdot \left(n \cdot \left(i \cdot i\right)\right)\\ \end{array} \]

Alternative 13: 59.2% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -350:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -350.0) 0.0 (if (<= i 1.7e-21) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -350.0) {
		tmp = 0.0;
	} else if (i <= 1.7e-21) {
		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 <= (-350.0d0)) then
        tmp = 0.0d0
    else if (i <= 1.7d-21) 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 <= -350.0) {
		tmp = 0.0;
	} else if (i <= 1.7e-21) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -350.0:
		tmp = 0.0
	elif i <= 1.7e-21:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -350.0)
		tmp = 0.0;
	elseif (i <= 1.7e-21)
		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 <= -350.0)
		tmp = 0.0;
	elseif (i <= 1.7e-21)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -350.0], 0.0, If[LessEqual[i, 1.7e-21], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -350 or 1.7e-21 < i

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-/r/ 54.6%

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

      3. associate-*l* 54.5%

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

      4. sub-neg 54.5%

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

      5. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in i around 0 29.2%

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

   5. Taylor expanded in i around 0 29.2%

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

   if -350 < i < 1.7e-21

   1. Initial program 7.4%

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

   2. Taylor expanded in i around 0 87.4%

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

      1. *-commutative 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -350:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 58.9% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0) (* -200.0 (/ n i)) (if (<= i 1.7e-21) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 * (n / i);
	} else if (i <= 1.7e-21) {
		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 <= (-2.0d0)) then
        tmp = (-200.0d0) * (n / i)
    else if (i <= 1.7d-21) 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 <= -2.0) {
		tmp = -200.0 * (n / i);
	} else if (i <= 1.7e-21) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = -200.0 * (n / i)
	elif i <= 1.7e-21:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(-200.0 * Float64(n / i));
	elseif (i <= 1.7e-21)
		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 <= -2.0)
		tmp = -200.0 * (n / i);
	elseif (i <= 1.7e-21)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(-200.0 * N[(n / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.7e-21], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 59.9%

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

   2. Taylor expanded in n around inf 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-/l* 78.8%

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

      3. expm1-def 78.8%

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

   4. Simplified 78.8%

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

   5. Taylor expanded in i around 0 33.1%

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

      1. *-commutative 33.1%

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

   7. Simplified 33.1%

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

   8. Taylor expanded in i around inf 33.1%

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

   if -2 < i < 1.7e-21

   1. Initial program 7.5%

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

   2. Taylor expanded in i around 0 87.9%

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

      1. *-commutative 87.9%

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

   4. Simplified 87.9%

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

   if 1.7e-21 < i

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.2%

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

      3. associate-*l* 49.2%

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

      4. sub-neg 49.2%

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

      5. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in i around 0 25.9%

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

   5. Taylor expanded in i around 0 25.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;-200 \cdot \frac{n}{i}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 59.1% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -2.0) (/ -200.0 (/ i n)) (if (<= i 1.7e-21) (* n 100.0) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -2.0) {
		tmp = -200.0 / (i / n);
	} else if (i <= 1.7e-21) {
		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 <= (-2.0d0)) then
        tmp = (-200.0d0) / (i / n)
    else if (i <= 1.7d-21) 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 <= -2.0) {
		tmp = -200.0 / (i / n);
	} else if (i <= 1.7e-21) {
		tmp = n * 100.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -2.0:
		tmp = -200.0 / (i / n)
	elif i <= 1.7e-21:
		tmp = n * 100.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -2.0)
		tmp = Float64(-200.0 / Float64(i / n));
	elseif (i <= 1.7e-21)
		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 <= -2.0)
		tmp = -200.0 / (i / n);
	elseif (i <= 1.7e-21)
		tmp = n * 100.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -2.0], N[(-200.0 / N[(i / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.7e-21], N[(n * 100.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if i < -2

   1. Initial program 59.9%

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

   2. Taylor expanded in n around inf 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-/l* 78.8%

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

      3. expm1-def 78.8%

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

   4. Simplified 78.8%

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

   5. Taylor expanded in i around 0 33.1%

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

      1. *-commutative 33.1%

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

   7. Simplified 33.1%

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

   8. Taylor expanded in i around inf 33.1%

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

      1. clear-num 34.0%

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

      2. un-div-inv 34.0%

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

   10. Applied egg-rr 34.0%

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

   if -2 < i < 1.7e-21

   1. Initial program 7.5%

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

   2. Taylor expanded in i around 0 87.9%

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

      1. *-commutative 87.9%

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

   4. Simplified 87.9%

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

   if 1.7e-21 < i

   1. Initial program 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-/r/ 49.2%

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

      3. associate-*l* 49.2%

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

      4. sub-neg 49.2%

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

      5. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in i around 0 25.9%

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

   5. Taylor expanded in i around 0 25.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2:\\ \;\;\;\;\frac{-200}{\frac{i}{n}}\\ \mathbf{elif}\;i \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;n \cdot 100\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 17.8% 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 28.2%

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

   1. *-commutative 28.2%

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

   2. associate-/r/ 28.3%

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

   3. associate-*l* 28.3%

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

   4. sub-neg 28.3%

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

   5. metadata-eval 28.3%

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

3. Simplified 28.3%

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

4. Taylor expanded in i around 0 17.2%

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

5. Taylor expanded in i around 0 17.2%

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

6. Final simplification 17.2%

   \[\leadsto 0 \]

Developer target: 34.2% 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 2023283 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

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

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