subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 8

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

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

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 100.0%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. /-lowering-/.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

   11. --lowering--.f64 N/A

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

   12. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{1}{\frac{n - f}{n + f}} \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f + f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (+ n f) (- n f)) -0.5) (/ f (- n f)) (+ 1.0 (/ (+ f f) n))))

C

double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + ((f + f) / n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((n + f) / (n - f)) <= (-0.5d0)) then
        tmp = f / (n - f)
    else
        tmp = 1.0d0 + ((f + f) / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + ((f + f) / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if ((n + f) / (n - f)) <= -0.5:
		tmp = f / (n - f)
	else:
		tmp = 1.0 + ((f + f) / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(n + f) / Float64(n - f)) <= -0.5)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = Float64(1.0 + Float64(Float64(f + f) / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (((n + f) / (n - f)) <= -0.5)
		tmp = f / (n - f);
	else
		tmp = 1.0 + ((f + f) / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(n + f), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -0.5], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(f + f), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot f}}{f - n} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.4

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

   5. Simplified 97.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. remove-double-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. --lowering--.f64 97.4

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

   7. Applied egg-rr 97.4%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{f}{n}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. count-2 N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. +-lowering-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-out N/A

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

      15. remove-double-neg N/A

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

      16. /-lowering-/.f64 N/A

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

      17. +-lowering-+.f64 98.0

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

   5. Simplified 98.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f + f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (+ n f) (- n f)) -0.5) (/ f (- n f)) (+ 1.0 (/ f n))))

C

double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((n + f) / (n - f)) <= (-0.5d0)) then
        tmp = f / (n - f)
    else
        tmp = 1.0d0 + (f / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if ((n + f) / (n - f)) <= -0.5:
		tmp = f / (n - f)
	else:
		tmp = 1.0 + (f / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(n + f) / Float64(n - f)) <= -0.5)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = Float64(1.0 + Float64(f / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (((n + f) / (n - f)) <= -0.5)
		tmp = f / (n - f);
	else
		tmp = 1.0 + (f / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(n + f), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -0.5], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot f}}{f - n} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.4

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

   5. Simplified 97.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. remove-double-neg N/A

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

      9. /-lowering-/.f64 N/A

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

      10. --lowering--.f64 97.4

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

   7. Applied egg-rr 97.4%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 99.9%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in n around inf

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

   7. Simplified 96.6%

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

   8. Taylor expanded in f around 0

      \[\leadsto \color{blue}{1 + \frac{f}{n}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 96.6

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

   10. Simplified 96.6%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (+ n f) (- n f)) -0.5) (- -1.0 (/ n f)) (+ 1.0 (/ f n))))

C

double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((n + f) / (n - f)) <= (-0.5d0)) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = 1.0d0 + (f / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if ((n + f) / (n - f)) <= -0.5:
		tmp = -1.0 - (n / f)
	else:
		tmp = 1.0 + (f / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(n + f) / Float64(n - f)) <= -0.5)
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = Float64(1.0 + Float64(f / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (((n + f) / (n - f)) <= -0.5)
		tmp = -1.0 - (n / f);
	else
		tmp = 1.0 + (f / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(n + f), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -0.5], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot f}}{f - n} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.4

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

   5. Simplified 97.4%

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

   6. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{n}{f} - 1} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 97.4

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

   8. Simplified 97.4%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 99.9%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in n around inf

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

   7. Simplified 96.6%

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

   8. Taylor expanded in f around 0

      \[\leadsto \color{blue}{1 + \frac{f}{n}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 96.6

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

   10. Simplified 96.6%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (+ n f) (- n f)) -0.5) -1.0 (+ 1.0 (/ f n))))

C

double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((n + f) / (n - f)) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + (f / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if ((n + f) / (n - f)) <= -0.5:
		tmp = -1.0
	else:
		tmp = 1.0 + (f / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(n + f) / Float64(n - f)) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(f / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (((n + f) / (n - f)) <= -0.5)
		tmp = -1.0;
	else
		tmp = 1.0 + (f / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(n + f), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 97.3%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 99.9%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in n around inf

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

   7. Simplified 96.6%

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

   8. Taylor expanded in f around 0

      \[\leadsto \color{blue}{1 + \frac{f}{n}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 96.6

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

   10. Simplified 96.6%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (+ n f) (- n f)) -5e-310) -1.0 1.0))

C

double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -5e-310) {
		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 + f) / (n - f)) <= (-5d-310)) 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 + f) / (n - f)) <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if ((n + f) / (n - f)) <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(n + f) / Float64(n - f)) <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (((n + f) / (n - f)) <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(n + f), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -5e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -4.999999999999985e-310

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 97.3%

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

   if -4.999999999999985e-310 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 100.0%

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

   1. frac-2neg N/A

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

   2. remove-double-neg N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. remove-double-neg N/A

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

   9. sub-neg N/A

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

   10. --lowering--.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{n + f}{n - f} \]
6. Add Preprocessing

Alternative 8: 49.5% accurate, 20.0× 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. Add Preprocessing

3. Taylor expanded in f around inf

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 52.4%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))