subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. associate-+l- 99.9%

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

   6. neg-sub0 99.9%

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

   7. neg-mul-1 99.9%

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

   8. times-frac 99.9%

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

   9. metadata-eval 99.9%

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

   10. *-lft-identity 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
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. +-commutative 99.7%

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

   5. associate-*l/ 99.8%

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

   6. *-un-lft-identity 99.8%

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

   7. associate-*l/ 100.0%

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

   8. *-un-lft-identity 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -2.15e+38) 1.0 (if (<= n 3.8e-14) (+ (* -2.0 (/ n f)) -1.0) 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -2.15e+38) {
		tmp = 1.0;
	} else if (n <= 3.8e-14) {
		tmp = (-2.0 * (n / f)) + -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 <= (-2.15d+38)) then
        tmp = 1.0d0
    else if (n <= 3.8d-14) then
        tmp = ((-2.0d0) * (n / f)) + (-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 <= -2.15e+38) {
		tmp = 1.0;
	} else if (n <= 3.8e-14) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -2.15e+38:
		tmp = 1.0
	elif n <= 3.8e-14:
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -2.15e+38)
		tmp = 1.0;
	elseif (n <= 3.8e-14)
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -2.15e+38)
		tmp = 1.0;
	elseif (n <= 3.8e-14)
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -2.15e+38], 1.0, If[LessEqual[n, 3.8e-14], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -2.1499999999999998e38 or 3.8000000000000002e-14 < n

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 76.7%

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

   if -2.1499999999999998e38 < n < 3.8000000000000002e-14

   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. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. neg-sub0 100.0%

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

      7. neg-mul-1 100.0%

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

      8. times-frac 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in n around 0 75.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.15 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. associate-+l- 99.9%

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

   6. neg-sub0 99.9%

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

   7. neg-mul-1 99.9%

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

   8. times-frac 99.9%

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

   9. metadata-eval 99.9%

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

   10. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

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

Alternative 4: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. associate-+l- 99.9%

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

   6. neg-sub0 99.9%

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

   7. neg-mul-1 99.9%

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

   8. times-frac 99.9%

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

   9. metadata-eval 99.9%

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

   10. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-17}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.12 \cdot 10^{-17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -7.5e-17) 1.0 (if (<= n 1.12e-17) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -7.5e-17) {
		tmp = 1.0;
	} else if (n <= 1.12e-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 <= (-7.5d-17)) then
        tmp = 1.0d0
    else if (n <= 1.12d-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 <= -7.5e-17) {
		tmp = 1.0;
	} else if (n <= 1.12e-17) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -7.5e-17:
		tmp = 1.0
	elif n <= 1.12e-17:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -7.5e-17)
		tmp = 1.0;
	elseif (n <= 1.12e-17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -7.5e-17)
		tmp = 1.0;
	elseif (n <= 1.12e-17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -7.5e-17], 1.0, If[LessEqual[n, 1.12e-17], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -7.49999999999999984e-17 or 1.12000000000000005e-17 < n

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 74.5%

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

   if -7.49999999999999984e-17 < n < 1.12000000000000005e-17

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 76.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-17}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.12 \cdot 10^{-17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 50.0% 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 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. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

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

   5. associate-+l- 99.9%

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

   6. neg-sub0 99.9%

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

   7. neg-mul-1 99.9%

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

   8. times-frac 99.9%

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

   9. metadata-eval 99.9%

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

   10. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in f around inf 48.6%

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

5. Final simplification 48.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))