subtraction fraction

Percentage Accurate: 100.0% → 99.9%

Time: 9.0s

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: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   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. unsub-neg N/A

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

   9. remove-double-neg N/A

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

   10. --lowering--.f64 99.9%

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

3. Simplified 99.9%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. flip3-- N/A

      \[\leadsto \frac{\frac{1}{\frac{{n}^{3} - {f}^{3}}{n \cdot n + \left(f \cdot f + n \cdot f\right)}}}{\frac{f - \color{blue}{n}}{f \cdot f - n \cdot n}} \]

   6. clear-num N/A

      \[\leadsto \frac{\frac{n \cdot n + \left(f \cdot f + n \cdot f\right)}{{n}^{3} - {f}^{3}}}{\frac{\color{blue}{f - n}}{f \cdot f - n \cdot n}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{n \cdot n + \left(f \cdot f + n \cdot f\right)}{{n}^{3} - {f}^{3}}\right), \color{blue}{\left(\frac{f - n}{f \cdot f - n \cdot n}\right)}\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{{n}^{3} - {f}^{3}}{n \cdot n + \left(f \cdot f + n \cdot f\right)}}\right), \left(\frac{\color{blue}{f - n}}{f \cdot f - n \cdot n}\right)\right) \]

   9. flip3-- N/A

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

   10. /-lowering-/.f64 N/A

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

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

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

   12. clear-num N/A

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

   13. flip-+ N/A

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

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

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

   15. +-lowering-+.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + n \cdot \frac{-2}{f}\\ \mathbf{if}\;f \leq -9.2 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* n (/ -2.0 f)))))
   (if (<= f -9.2e-97) t_0 (if (<= f 1.85e-18) (+ 1.0 (* 2.0 (/ f n))) t_0))))

C

double code(double f, double n) {
	double t_0 = -1.0 + (n * (-2.0 / f));
	double tmp;
	if (f <= -9.2e-97) {
		tmp = t_0;
	} else if (f <= 1.85e-18) {
		tmp = 1.0 + (2.0 * (f / n));
	} 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 = (-1.0d0) + (n * ((-2.0d0) / f))
    if (f <= (-9.2d-97)) then
        tmp = t_0
    else if (f <= 1.85d-18) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 + (n * (-2.0 / f));
	double tmp;
	if (f <= -9.2e-97) {
		tmp = t_0;
	} else if (f <= 1.85e-18) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 + (n * (-2.0 / f))
	tmp = 0
	if f <= -9.2e-97:
		tmp = t_0
	elif f <= 1.85e-18:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 + Float64(n * Float64(-2.0 / f)))
	tmp = 0.0
	if (f <= -9.2e-97)
		tmp = t_0;
	elseif (f <= 1.85e-18)
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 + (n * (-2.0 / f));
	tmp = 0.0;
	if (f <= -9.2e-97)
		tmp = t_0;
	elseif (f <= 1.85e-18)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 + N[(n * N[(-2.0 / f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -9.2e-97], t$95$0, If[LessEqual[f, 1.85e-18], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if f < -9.19999999999999976e-97 or 1.8500000000000002e-18 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{n \cdot -2}{f}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \color{blue}{\frac{-2}{f}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \frac{\mathsf{neg}\left(2\right)}{f}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \left(\mathsf{neg}\left(\frac{2}{f}\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{f}\right)\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{f}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{f}\right)\right)}\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{f}\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\frac{2}{f}\right)\right)\right)\right) \]

      20. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{f}}\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\frac{-2}{f}\right)\right)\right) \]

      22. /-lowering-/.f64 75.9%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(-2, \color{blue}{f}\right)\right)\right) \]

   7. Simplified 75.9%

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

   if -9.19999999999999976e-97 < f < 1.8500000000000002e-18

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(2 \cdot \frac{1}{n}\right) \cdot f\right)}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(2 \cdot \color{blue}{\left(\frac{1}{n} \cdot f\right)}\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(2 \cdot \frac{1 \cdot f}{\color{blue}{n}}\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(2 \cdot \frac{f}{n}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{f}{n}\right)}\right)\right) \]

      9. /-lowering-/.f64 85.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(f, \color{blue}{n}\right)\right)\right) \]

   7. Simplified 85.5%

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

Alternative 3: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + n \cdot \frac{-2}{f}\\ \mathbf{if}\;f \leq -1.05 \cdot 10^{-96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 2.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* n (/ -2.0 f)))))
   (if (<= f -1.05e-96) t_0 (if (<= f 2.8e-19) (/ n (- n f)) t_0))))

C

double code(double f, double n) {
	double t_0 = -1.0 + (n * (-2.0 / f));
	double tmp;
	if (f <= -1.05e-96) {
		tmp = t_0;
	} else if (f <= 2.8e-19) {
		tmp = n / (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 = (-1.0d0) + (n * ((-2.0d0) / f))
    if (f <= (-1.05d-96)) then
        tmp = t_0
    else if (f <= 2.8d-19) then
        tmp = n / (n - f)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 + (n * (-2.0 / f));
	double tmp;
	if (f <= -1.05e-96) {
		tmp = t_0;
	} else if (f <= 2.8e-19) {
		tmp = n / (n - f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 + (n * (-2.0 / f))
	tmp = 0
	if f <= -1.05e-96:
		tmp = t_0
	elif f <= 2.8e-19:
		tmp = n / (n - f)
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 + Float64(n * Float64(-2.0 / f)))
	tmp = 0.0
	if (f <= -1.05e-96)
		tmp = t_0;
	elseif (f <= 2.8e-19)
		tmp = Float64(n / Float64(n - f));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 + (n * (-2.0 / f));
	tmp = 0.0;
	if (f <= -1.05e-96)
		tmp = t_0;
	elseif (f <= 2.8e-19)
		tmp = n / (n - f);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 + N[(n * N[(-2.0 / f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -1.05e-96], t$95$0, If[LessEqual[f, 2.8e-19], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if f < -1.05000000000000001e-96 or 2.80000000000000003e-19 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{n \cdot -2}{f}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \color{blue}{\frac{-2}{f}}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \frac{\mathsf{neg}\left(2\right)}{f}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \left(\mathsf{neg}\left(\frac{2}{f}\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{f}\right)\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(n \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{f}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{f}\right)\right)}\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{f}\right)\right)\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\mathsf{neg}\left(\frac{2}{f}\right)\right)\right)\right) \]

      20. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{f}}\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \left(\frac{-2}{f}\right)\right)\right) \]

      22. /-lowering-/.f64 75.9%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(n, \mathsf{/.f64}\left(-2, \color{blue}{f}\right)\right)\right) \]

   7. Simplified 75.9%

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

   if -1.05000000000000001e-96 < f < 2.80000000000000003e-19

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   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 85.1%

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

Alternative 4: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{n}{f}\\ \mathbf{if}\;f \leq -4 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))))
   (if (<= f -4e-97) t_0 (if (<= f 6.8e-19) (/ n (- n f)) t_0))))

C

double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double tmp;
	if (f <= -4e-97) {
		tmp = t_0;
	} else if (f <= 6.8e-19) {
		tmp = n / (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 = (-1.0d0) - (n / f)
    if (f <= (-4d-97)) then
        tmp = t_0
    else if (f <= 6.8d-19) then
        tmp = n / (n - f)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double tmp;
	if (f <= -4e-97) {
		tmp = t_0;
	} else if (f <= 6.8e-19) {
		tmp = n / (n - f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 - (n / f)
	tmp = 0
	if f <= -4e-97:
		tmp = t_0
	elif f <= 6.8e-19:
		tmp = n / (n - f)
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 - Float64(n / f))
	tmp = 0.0
	if (f <= -4e-97)
		tmp = t_0;
	elseif (f <= 6.8e-19)
		tmp = Float64(n / Float64(n - f));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 - (n / f);
	tmp = 0.0;
	if (f <= -4e-97)
		tmp = t_0;
	elseif (f <= 6.8e-19)
		tmp = n / (n - f);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -4e-97], t$95$0, If[LessEqual[f, 6.8e-19], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if f < -4.00000000000000014e-97 or 6.8000000000000004e-19 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf

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

   7. Simplified 75.1%

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

   8. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{n}{f} - 1} \]
   9. 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 75.4%

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

   10. Simplified 75.4%

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

   if -4.00000000000000014e-97 < f < 6.8000000000000004e-19

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   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 85.1%

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

Alternative 5: 72.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{n}{f}\\ \mathbf{if}\;f \leq -1.05 \cdot 10^{-96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))))
   (if (<= f -1.05e-96) t_0 (if (<= f 2.2e-17) 1.0 t_0))))

C

double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double tmp;
	if (f <= -1.05e-96) {
		tmp = t_0;
	} else if (f <= 2.2e-17) {
		tmp = 1.0;
	} 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 = (-1.0d0) - (n / f)
    if (f <= (-1.05d-96)) then
        tmp = t_0
    else if (f <= 2.2d-17) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double tmp;
	if (f <= -1.05e-96) {
		tmp = t_0;
	} else if (f <= 2.2e-17) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 - (n / f)
	tmp = 0
	if f <= -1.05e-96:
		tmp = t_0
	elif f <= 2.2e-17:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 - Float64(n / f))
	tmp = 0.0
	if (f <= -1.05e-96)
		tmp = t_0;
	elseif (f <= 2.2e-17)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 - (n / f);
	tmp = 0.0;
	if (f <= -1.05e-96)
		tmp = t_0;
	elseif (f <= 2.2e-17)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -1.05e-96], t$95$0, If[LessEqual[f, 2.2e-17], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if f < -1.05000000000000001e-96 or 2.2e-17 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf

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

   7. Simplified 75.1%

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

   8. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{n}{f} - 1} \]
   9. 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 75.4%

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

   10. Simplified 75.4%

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

   if -1.05000000000000001e-96 < f < 2.2e-17

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0

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

   7. Simplified 84.6%

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

Alternative 6: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.05 \cdot 10^{-96}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.5 \cdot 10^{-17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.05e-96) -1.0 (if (<= f 4.5e-17) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.05e-96) {
		tmp = -1.0;
	} else if (f <= 4.5e-17) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-1.05d-96)) then
        tmp = -1.0d0
    else if (f <= 4.5d-17) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -1.05e-96) {
		tmp = -1.0;
	} else if (f <= 4.5e-17) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.05e-96:
		tmp = -1.0
	elif f <= 4.5e-17:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.05e-96)
		tmp = -1.0;
	elseif (f <= 4.5e-17)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.05e-96)
		tmp = -1.0;
	elseif (f <= 4.5e-17)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.05e-96], -1.0, If[LessEqual[f, 4.5e-17], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.05000000000000001e-96 or 4.49999999999999978e-17 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf

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

   7. Simplified 74.6%

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

   if -1.05000000000000001e-96 < f < 4.49999999999999978e-17

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      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. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. --lowering--.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0

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

   7. Simplified 84.6%

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

Alternative 7: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   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. unsub-neg N/A

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

   9. remove-double-neg N/A

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

   10. --lowering--.f64 99.9%

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

3. Simplified 99.9%

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

   1. clear-num N/A

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

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

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

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

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

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

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

   5. +-lowering-+.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 8: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   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. unsub-neg N/A

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

   9. remove-double-neg N/A

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

   10. --lowering--.f64 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 9: 50.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

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

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   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. unsub-neg N/A

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

   9. remove-double-neg N/A

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

   10. --lowering--.f64 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in f around inf

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

7. Simplified 48.7%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))