subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 7

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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-/r/ N/A

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

   3. distribute-lft-in N/A

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

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n}\right), \color{blue}{f}, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{1}{\frac{f \cdot f - n \cdot n}{\mathsf{neg}\left(\left(f + n\right)\right)}}\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\frac{f \cdot f - n \cdot n}{f + n}\right)}\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   7. flip-- N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(f - n\right)\right)}\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(f - n\right)\right)\right)\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   11. distribute-neg-in N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \left(\mathsf{neg}\left(f\right)\right)\right)\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \left(n + \left(\mathsf{neg}\left(f\right)\right)\right)\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \left(n - f\right)\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n} \cdot n\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\left(f + n\right)\right)}{f \cdot f - n \cdot n}\right), n\right)\right) \]

4. Applied egg-rr 99.7%

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

   1. associate-/r/ N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \left(\frac{1}{\frac{n - f}{n}}\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \mathsf{/.f64}\left(1, \left(\frac{n - f}{n}\right)\right)\right) \]

   3. div-sub N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \mathsf{/.f64}\left(1, \left(\frac{n}{n} - \frac{f}{n}\right)\right)\right) \]

   4. *-inverses N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \mathsf{/.f64}\left(1, \left(1 - \frac{f}{n}\right)\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(\frac{f}{n}\right)\right)\right)\right) \]

   6. /-lowering-/.f64 99.9%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(n, f\right)\right), f, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(f, n\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{n - f} \cdot f\right), \color{blue}{\left(\frac{1}{1 - \frac{f}{n}}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(f \cdot \frac{1}{n - f}\right), \left(\frac{\color{blue}{1}}{1 - \frac{f}{n}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{f}{n - f}\right), \left(\frac{\color{blue}{1}}{1 - \frac{f}{n}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \left(n - f\right)\right), \left(\frac{\color{blue}{1}}{1 - \frac{f}{n}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \left(\frac{1}{1 - \frac{f}{n}}\right)\right) \]

   6. frac-2neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(1 - \frac{f}{n}\right)\right)}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\left(1 - \frac{f}{n}\right)}\right)}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\left(1 - \frac{f}{n}\right)\right)\right)}\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\frac{f}{n}\right)\right)\right)\right)\right)\right)\right) \]

   10. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{f}{n}\right)\right)\right)\right)}\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{f}{n}\right)\right)}\right)\right)\right)\right)\right) \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \left(-1 + \left(\mathsf{neg}\left(\frac{f}{\mathsf{neg}\left(n\right)}\right)\right)\right)\right)\right) \]

   13. distribute-frac-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \left(-1 + \frac{\mathsf{neg}\left(f\right)}{\color{blue}{\mathsf{neg}\left(n\right)}}\right)\right)\right) \]

   14. frac-2neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \left(-1 + \frac{f}{\color{blue}{n}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{f}{n}\right)}\right)\right)\right) \]

   16. /-lowering-/.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, f\right)\right), \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(f, \color{blue}{n}\right)\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{f}{n - f} + \frac{-1}{-1 + \frac{f}{n}}} \]
9. Add Preprocessing

Alternative 2: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{n - f}\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ n (- n f))))
   (if (<= n -1.35e-32) t_0 (if (<= n 1.8e-29) (/ f (- n f)) t_0))))

C

double code(double f, double n) {
	double t_0 = n / (n - f);
	double tmp;
	if (n <= -1.35e-32) {
		tmp = t_0;
	} else if (n <= 1.8e-29) {
		tmp = f / (n - f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n / (n - f)
    if (n <= (-1.35d-32)) then
        tmp = t_0
    else if (n <= 1.8d-29) then
        tmp = f / (n - f)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = n / (n - f);
	double tmp;
	if (n <= -1.35e-32) {
		tmp = t_0;
	} else if (n <= 1.8e-29) {
		tmp = f / (n - f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = n / (n - f)
	tmp = 0
	if n <= -1.35e-32:
		tmp = t_0
	elif n <= 1.8e-29:
		tmp = f / (n - f)
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(n / Float64(n - f))
	tmp = 0.0
	if (n <= -1.35e-32)
		tmp = t_0;
	elseif (n <= 1.8e-29)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = n / (n - f);
	tmp = 0.0;
	if (n <= -1.35e-32)
		tmp = t_0;
	elseif (n <= 1.8e-29)
		tmp = f / (n - f);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.35e-32], t$95$0, If[LessEqual[n, 1.8e-29], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.3499999999999999e-32 or 1.79999999999999987e-29 < 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

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

      2. remove-double-neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n + \left(\mathsf{neg}\left(\color{blue}{f}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - \color{blue}{f}\right)\right) \]

      10. --lowering--.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in f around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{n}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 81.1%

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

   if -1.3499999999999999e-32 < n < 1.79999999999999987e-29

   1. Initial program 99.9%

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

   3. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\color{blue}{f}\right), \mathsf{\_.f64}\left(f, n\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. remove-double-neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(f, \color{blue}{\left(n - f\right)}\right) \]

      10. --lowering--.f64 79.4%

          \[\leadsto \mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   7. Applied egg-rr 79.4%

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

Alternative 3: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -2.3e-32) 1.0 (if (<= n 9.5e-27) (/ f (- n f)) 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -2.3e-32) {
		tmp = 1.0;
	} else if (n <= 9.5e-27) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-2.3d-32)) then
        tmp = 1.0d0
    else if (n <= 9.5d-27) then
        tmp = f / (n - f)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -2.3e-32) {
		tmp = 1.0;
	} else if (n <= 9.5e-27) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -2.3e-32:
		tmp = 1.0
	elif n <= 9.5e-27:
		tmp = f / (n - f)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -2.3e-32)
		tmp = 1.0;
	elseif (n <= 9.5e-27)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -2.3e-32)
		tmp = 1.0;
	elseif (n <= 9.5e-27)
		tmp = f / (n - f);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -2.3e-32], 1.0, If[LessEqual[n, 9.5e-27], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -2.3000000000000001e-32 or 9.50000000000000037e-27 < 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 80.7%

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

   if -2.3000000000000001e-32 < n < 9.50000000000000037e-27

   1. Initial program 99.9%

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

   3. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\color{blue}{f}\right), \mathsf{\_.f64}\left(f, n\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-in N/A

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

      5. sub-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. remove-double-neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(f, \color{blue}{\left(n - f\right)}\right) \]

      10. --lowering--.f64 79.4%

          \[\leadsto \mathsf{/.f64}\left(f, \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   7. Applied egg-rr 79.4%

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

Alternative 4: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-27}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -8.2e-55) 1.0 (if (<= n 4.7e-27) (- -1.0 (/ n f)) 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -8.2e-55) {
		tmp = 1.0;
	} else if (n <= 4.7e-27) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-8.2d-55)) then
        tmp = 1.0d0
    else if (n <= 4.7d-27) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -8.2e-55) {
		tmp = 1.0;
	} else if (n <= 4.7e-27) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -8.2e-55:
		tmp = 1.0
	elif n <= 4.7e-27:
		tmp = -1.0 - (n / f)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -8.2e-55)
		tmp = 1.0;
	elseif (n <= 4.7e-27)
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -8.2e-55)
		tmp = 1.0;
	elseif (n <= 4.7e-27)
		tmp = -1.0 - (n / f);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -8.2e-55], 1.0, If[LessEqual[n, 4.7e-27], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -8.1999999999999996e-55 or 4.70000000000000032e-27 < 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. Simplified 79.5%

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

   if -8.1999999999999996e-55 < n < 4.70000000000000032e-27

   1. Initial program 99.9%

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

   3. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\color{blue}{f}\right), \mathsf{\_.f64}\left(f, n\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 80.8%

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

   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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{n}{f}\right)}\right) \]

      7. /-lowering-/.f64 80.7%

         \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(n, \color{blue}{f}\right)\right) \]

   8. Simplified 80.7%

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

Alternative 5: 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. frac-2neg N/A

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

   2. remove-double-neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n + \left(\mathsf{neg}\left(\color{blue}{f}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - \color{blue}{f}\right)\right) \]

   10. --lowering--.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

4. Applied egg-rr 99.9%

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

Alternative 6: 73.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3e-32) 1.0 (if (<= n 2e-30) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3e-32) {
		tmp = 1.0;
	} else if (n <= 2e-30) {
		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 <= (-3d-32)) then
        tmp = 1.0d0
    else if (n <= 2d-30) 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 <= -3e-32) {
		tmp = 1.0;
	} else if (n <= 2e-30) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3e-32:
		tmp = 1.0
	elif n <= 2e-30:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3e-32)
		tmp = 1.0;
	elseif (n <= 2e-30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3e-32)
		tmp = 1.0;
	elseif (n <= 2e-30)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3e-32], 1.0, If[LessEqual[n, 2e-30], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -3e-32 or 2e-30 < 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 80.7%

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

   if -3e-32 < n < 2e-30

   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. Simplified 78.6%

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

Alternative 7: 50.1% 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. Simplified 46.0%

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

Reproduce

herbie shell --seed 2024193 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))