subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 6

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (- (+ f n)) (- f n)))

C

double code(double f, double n) {
	return -(f + n) / (f - n);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -(f + n) / (f - n)
end function

Java

public static double code(double f, double n) {
	return -(f + n) / (f - n);
}

Python

def code(f, n):
	return -(f + n) / (f - n)

Julia

function code(f, n)
	return Float64(Float64(-Float64(f + n)) / Float64(f - n))
end

MATLAB

function tmp = code(f, n)
	tmp = -(f + n) / (f - n);
end

Wolfram

code[f_, n_] := N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (- (+ f n)) (- f n)))

C

double code(double f, double n) {
	return -(f + n) / (f - n);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -(f + n) / (f - n)
end function

Java

public static double code(double f, double n) {
	return -(f + n) / (f - n);
}

Python

def code(f, n):
	return -(f + n) / (f - n)

Julia

function code(f, n)
	return Float64(Float64(-Float64(f + n)) / Float64(f - n))
end

MATLAB

function tmp = code(f, n)
	tmp = -(f + n) / (f - n);
end

Wolfram

code[f_, n_] := N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{n - f}\\ t_1 := \frac{f}{n - f}\\ \frac{\frac{f - n}{n - f} \cdot \left(t_0 + t_1\right)}{t_1 - t_0} \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ n (- n f))) (t_1 (/ f (- n f))))
   (/ (* (/ (- f n) (- n f)) (+ t_0 t_1)) (- t_1 t_0))))

C

double code(double f, double n) {
	double t_0 = n / (n - f);
	double t_1 = f / (n - f);
	return (((f - n) / (n - f)) * (t_0 + t_1)) / (t_1 - t_0);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    t_0 = n / (n - f)
    t_1 = f / (n - f)
    code = (((f - n) / (n - f)) * (t_0 + t_1)) / (t_1 - t_0)
end function

Java

public static double code(double f, double n) {
	double t_0 = n / (n - f);
	double t_1 = f / (n - f);
	return (((f - n) / (n - f)) * (t_0 + t_1)) / (t_1 - t_0);
}

Python

def code(f, n):
	t_0 = n / (n - f)
	t_1 = f / (n - f)
	return (((f - n) / (n - f)) * (t_0 + t_1)) / (t_1 - t_0)

Julia

function code(f, n)
	t_0 = Float64(n / Float64(n - f))
	t_1 = Float64(f / Float64(n - f))
	return Float64(Float64(Float64(Float64(f - n) / Float64(n - f)) * Float64(t_0 + t_1)) / Float64(t_1 - t_0))
end

MATLAB

function tmp = code(f, n)
	t_0 = n / (n - f);
	t_1 = f / (n - f);
	tmp = (((f - n) / (n - f)) * (t_0 + t_1)) / (t_1 - t_0);
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(f - n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Step-by-step derivation

   1. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

       \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
4. Step-by-step derivation

   1. clear-num 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]

   2. inv-pow 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]

   2. associate-/r/ 99.7%

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

   3. distribute-lft-in 99.7%

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

   4. flip-+ 99.7%

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

   5. associate-*l/ 99.8%

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

   6. *-un-lft-identity 99.8%

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

   7. associate-*l/ 99.7%

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

   8. *-un-lft-identity 99.7%

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

   9. associate-*l/ 99.9%

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

   10. *-un-lft-identity 99.9%

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

   11. associate-*l/ 99.7%

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

   12. *-un-lft-identity 99.7%

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

   13. associate-*l/ 99.8%

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

   14. *-un-lft-identity 99.8%

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

   15. associate-*l/ 100.0%

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

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\frac{f}{n - f} \cdot \frac{f}{n - f} - \frac{n}{n - f} \cdot \frac{n}{n - f}}{\frac{f}{n - f} - \frac{n}{n - f}}} \]
8. Step-by-step derivation

   1. difference-of-squares 100.0%

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

   2. div-inv 99.9%

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

   3. div-inv 99.7%

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

   4. distribute-rgt-in 99.7%

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

   5. +-commutative 99.7%

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

   6. associate-/r/ 100.0%

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

   7. clear-num 100.0%

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

   8. *-commutative 100.0%

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

   9. sub-div 100.0%

      \[\leadsto \frac{\color{blue}{\frac{f - n}{n - f}} \cdot \frac{n + f}{n - f}}{\frac{f}{n - f} - \frac{n}{n - f}} \]

9. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{\frac{f - n}{n - f} \cdot \frac{n + f}{n - f}}}{\frac{f}{n - f} - \frac{n}{n - f}} \]
10. Step-by-step derivation

    1. clear-num 100.0%

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

    2. associate-/r/ 99.7%

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

    3. distribute-rgt-in 99.7%

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

    4. div-inv 99.9%

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

    5. div-inv 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

    \[\leadsto \frac{\frac{f - n}{n - f} \cdot \left(\frac{n}{n - f} + \frac{f}{n - f}\right)}{\frac{f}{n - f} - \frac{n}{n - f}} \]

Alternative 2: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-20} \lor \neg \left(n \leq 1.02 \cdot 10^{+18}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -8.8e-20) (not (<= n 1.02e+18))) (+ 1.0 (* 2.0 (/ f n))) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -8.8e-20) || !(n <= 1.02e+18)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8.8d-20)) .or. (.not. (n <= 1.02d+18))) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -8.8e-20) || !(n <= 1.02e+18)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -8.8e-20) or not (n <= 1.02e+18):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -8.8e-20) || !(n <= 1.02e+18))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -8.8e-20) || ~((n <= 1.02e+18)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -8.8e-20], N[Not[LessEqual[n, 1.02e+18]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -8.79999999999999964e-20 or 1.02e18 < n

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

          \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in f around 0 83.2%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{f}{n}} \]

   if -8.79999999999999964e-20 < n < 1.02e18

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. neg-mul-1 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. times-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. *-lft-identity 99.9%

          \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

      12. +-commutative 99.9%

          \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in f around inf 74.4%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-20} \lor \neg \left(n \leq 1.02 \cdot 10^{+18}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{n - f}{f + n}} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ 1.0 (/ (- n f) (+ f n))))

C

double code(double f, double n) {
	return 1.0 / ((n - f) / (f + n));
}

Fortran

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

Java

public static double code(double f, double n) {
	return 1.0 / ((n - f) / (f + n));
}

Python

def code(f, n):
	return 1.0 / ((n - f) / (f + n))

Julia

function code(f, n)
	return Float64(1.0 / Float64(Float64(n - f) / Float64(f + n)))
end

MATLAB

function tmp = code(f, n)
	tmp = 1.0 / ((n - f) / (f + n));
end

Wolfram

code[f_, n_] := N[(1.0 / N[(N[(n - f), $MachinePrecision] / N[(f + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Step-by-step derivation

   1. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

       \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
4. Step-by-step derivation

   1. clear-num 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]

   2. inv-pow 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]

   2. +-commutative 100.0%

      \[\leadsto \frac{1}{\frac{n - f}{\color{blue}{n + f}}} \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{n + f}}} \]

8. Final simplification 100.0%

   \[\leadsto \frac{1}{\frac{n - f}{f + n}} \]

Alternative 4: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{f + n}{n - f} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (+ f n) (- n f)))

C

double code(double f, double n) {
	return (f + n) / (n - f);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = (f + n) / (n - f)
end function

Java

public static double code(double f, double n) {
	return (f + n) / (n - f);
}

Python

def code(f, n):
	return (f + n) / (n - f)

Julia

function code(f, n)
	return Float64(Float64(f + n) / Float64(n - f))
end

MATLAB

function tmp = code(f, n)
	tmp = (f + n) / (n - f);
end

Wolfram

code[f_, n_] := N[(N[(f + n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Step-by-step derivation

   1. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

       \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

4. Final simplification 100.0%

   \[\leadsto \frac{f + n}{n - f} \]

Alternative 5: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.9 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{+17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3.9e-19) 1.0 (if (<= n 4.4e+17) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3.9e-19) {
		tmp = 1.0;
	} else if (n <= 4.4e+17) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.9d-19)) then
        tmp = 1.0d0
    else if (n <= 4.4d+17) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -3.9e-19) {
		tmp = 1.0;
	} else if (n <= 4.4e+17) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3.9e-19:
		tmp = 1.0
	elif n <= 4.4e+17:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3.9e-19)
		tmp = 1.0;
	elseif (n <= 4.4e+17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3.9e-19)
		tmp = 1.0;
	elseif (n <= 4.4e+17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3.9e-19], 1.0, If[LessEqual[n, 4.4e+17], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -3.89999999999999995e-19 or 4.4e17 < n

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

          \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

      12. +-commutative 100.0%

          \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in f around 0 82.7%

      \[\leadsto \color{blue}{1} \]

   if -3.89999999999999995e-19 < n < 4.4e17

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. neg-mul-1 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. times-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. *-lft-identity 99.9%

          \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

      12. +-commutative 99.9%

          \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in f around inf 74.4%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.9 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{+17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 49.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (f n) :precision binary64 -1.0)

C

double code(double f, double n) {
	return -1.0;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -1.0d0
end function

Java

public static double code(double f, double n) {
	return -1.0;
}

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

function tmp = code(f, n)
	tmp = -1.0;
end

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 100.0%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Step-by-step derivation

   1. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

       \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]

   12. +-commutative 100.0%

       \[\leadsto \frac{\color{blue}{f + n}}{n - f} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

4. Taylor expanded in f around inf 50.7%

   \[\leadsto \color{blue}{-1} \]

5. Final simplification 50.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))