subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

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, 0.6× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. remove-double-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. remove-double-neg N/A

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

   5. lift-neg.f64 N/A

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

   6. distribute-neg-in N/A

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

   7. lift-neg.f64 N/A

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

   8. sub-neg N/A

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

   9. lift--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. distribute-frac-neg N/A

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

   13. neg-sub0 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--r+ N/A

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

   17. neg-sub0 N/A

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

   18. lift-neg.f64 N/A

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

   19. div-sub N/A

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

   20. lift-/.f64 N/A

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

   21. lower--.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(n, \frac{-2}{f}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (+ n f) (- n f)) -0.5) (fma n (/ -2.0 f) -1.0) (/ (- n) (- f n))))

C

double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -0.5) {
		tmp = fma(n, (-2.0 / f), -1.0);
	} else {
		tmp = -n / (f - n);
	}
	return tmp;
}

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(n + f) / Float64(n - f)) <= -0.5)
		tmp = fma(n, Float64(-2.0 / f), -1.0);
	else
		tmp = Float64(Float64(-n) / Float64(f - n));
	end
	return tmp
end

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(n + f), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -0.5], N[(n * N[(-2.0 / f), $MachinePrecision] + -1.0), $MachinePrecision], N[((-n) / 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 99.9%

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

   3. Taylor expanded in f around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac2 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2} \cdot n}{\mathsf{neg}\left(f\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \frac{2 \cdot n}{\color{blue}{-1 \cdot f}} + \left(\mathsf{neg}\left(1\right)\right) \]

      9. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2}{-1 \cdot f} \cdot n} + \left(\mathsf{neg}\left(1\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \frac{2}{\color{blue}{\mathsf{neg}\left(f\right)}} \cdot n + \left(\mathsf{neg}\left(1\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{2}{f}\right)\right)} \cdot n + \left(\mathsf{neg}\left(1\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{2 \cdot 1}}{f}\right)\right) \cdot n + \left(\mathsf{neg}\left(1\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{1}{f}}\right)\right) \cdot n + \left(\mathsf{neg}\left(1\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{n \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{f}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto n \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{f}\right)\right) + \color{blue}{-1} \]

      16. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(n, \mathsf{neg}\left(2 \cdot \frac{1}{f}\right), -1\right)} \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(n, \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{f}}\right), -1\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(n, \mathsf{neg}\left(\frac{\color{blue}{2}}{f}\right), -1\right) \]

      19. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(n, \color{blue}{\frac{\mathsf{neg}\left(2\right)}{f}}, -1\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(n, \frac{\color{blue}{-2}}{f}, -1\right) \]

      21. lower-/.f64 97.1

          \[\leadsto \mathsf{fma}\left(n, \color{blue}{\frac{-2}{f}}, -1\right) \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(n, \frac{-2}{f}, -1\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in f around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(n, \frac{-2}{f}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(n + f) / Float64(n - f)) <= -0.5)
		tmp = Float64(Float64(-f) / Float64(f - n));
	else
		tmp = Float64(Float64(-n) / 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 / (f - n);
	else
		tmp = -n / (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[(f - n), $MachinePrecision]), $MachinePrecision], N[((-n) / 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 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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. lower-neg.f64 95.2

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

   5. Applied rewrites 95.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in f around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 96.5%

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

Alternative 4: 97.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(f, n):
	tmp = 0
	if ((n + f) / (n - f)) <= -0.5:
		tmp = -f / (f - n)
	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(Float64(-f) / Float64(f - n));
	else
		tmp = Float64(1.0 - Float64(f / Float64(-n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (((n + f) / (n - f)) <= -0.5)
		tmp = -f / (f - n);
	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[(f - n), $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 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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. lower-neg.f64 95.2

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

   5. Applied rewrites 95.2%

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

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in f around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{1} - \frac{f}{\mathsf{neg}\left(n\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.0%

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

4. Final simplification 96.5%

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

Alternative 5: 97.8% 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 / Float64(-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 99.9%

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

   3. Taylor expanded in f around inf

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

   5. Applied rewrites 95.0%

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

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

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in f around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in n around inf

      \[\leadsto \color{blue}{1} - \frac{f}{\mathsf{neg}\left(n\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.0%

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

4. Final simplification 96.5%

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

Alternative 6: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (+ n f) (- n f)) -1e-309) -1.0 1.0))

C

double code(double f, double n) {
	double tmp;
	if (((n + f) / (n - f)) <= -1e-309) {
		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)) <= (-1d-309)) 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)) <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if ((n + f) / (n - f)) <= -1e-309:
		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)) <= -1e-309)
		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)) <= -1e-309)
		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], -1e-309], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -1.000000000000002e-309

   1. Initial program 99.9%

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

   3. Taylor expanded in f around inf

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

   5. Applied rewrites 95.0%

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

   if -1.000000000000002e-309 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

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

   3. Taylor expanded in f around 0

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

   5. Applied rewrites 97.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{n + f}{n - f} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
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 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. remove-double-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 49.8% 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 99.9%

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

3. Taylor expanded in f around inf

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

5. Applied rewrites 50.1%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))