subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_, n_] := N[(N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision] - N[(f / N[(f - n), $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. 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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. sub-neg N/A

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

   16. lift--.f64 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{n}{n - f} - \frac{f}{f - n}} \]
7. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \color{blue}{-1 \cdot \frac{n - -1 \cdot n}{f} - 1} \]
   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   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. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{n}, f, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

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

Alternative 3: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double f, double n) {
	double tmp;
	if (((f + n) / (n - f)) <= -0.5) {
		tmp = -1.0 - ((n + n) / f);
	} 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 (((f + n) / (n - f)) <= (-0.5d0)) then
        tmp = (-1.0d0) - ((n + n) / f)
    else
        tmp = -n / (f - n)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(f + n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -0.5], N[(-1.0 - N[(N[(n + n), $MachinePrecision] / f), $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 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   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

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 97.2%

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

Alternative 4: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double f, double n) {
	double tmp;
	if (((f + n) / (n - f)) <= -0.5) {
		tmp = (f + n) / -f;
	} 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 (((f + n) / (n - f)) <= (-0.5d0)) then
        tmp = (f + n) / -f
    else
        tmp = -n / (f - n)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in f around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.3

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

   7. Applied rewrites 96.3%

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

   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

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 96.3%

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

Alternative 5: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(f, n):
	tmp = 0
	if ((f + n) / (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(f + n) / 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 (((f + n) / (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[(f + n), $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 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. lower-neg.f64 96.3

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

   5. Applied rewrites 96.3%

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

   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

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 96.3%

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

Alternative 6: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double f, double n) {
	double tmp;
	if (((f + n) / (n - f)) <= -0.5) {
		tmp = -f / (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 (((f + n) / (n - f)) <= (-0.5d0)) then
        tmp = -f / (f - n)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. lower-neg.f64 96.3

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

   5. Applied rewrites 96.3%

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

   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

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

   5. Applied rewrites 96.2%

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

4. Final simplification 96.2%

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

Alternative 7: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(f + n) / 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 (((f + n) / (n - f)) <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(f + n), $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 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. Applied rewrites 96.1%

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

   if -1.000000000000002e-309 < (/.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. Applied rewrites 96.2%

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

4. Final simplification 96.2%

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

Alternative 8: 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 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

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. remove-double-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 9: 50.2% 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 48.1%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))