subtraction fraction

Percentage Accurate: 100.0% → 99.9%

Time: 8.9s

Alternatives: 10

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. 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. flip3-+ N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

   7. clear-num N/A

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

   8. distribute-neg-frac2 N/A

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

   9. flip-- N/A

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

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

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. remove-double-neg N/A

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

   15. sub-neg N/A

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

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

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 73.7% accurate, 0.5× speedup

Math

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

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(Float64(n * 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), $MachinePrecision] / f), $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. 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

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

      12. *-lowering-*.f64 75.9%

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

   5. Simplified 75.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in f around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. count-2 N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-out N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac N/A

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

      10. mul-1-neg N/A

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

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

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-out N/A

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

      17. remove-double-neg N/A

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

      18. count-2 N/A

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

      19. *-commutative N/A

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

      20. associate-*l/ N/A

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

      21. *-commutative N/A

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

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

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

      23. /-lowering-/.f64 85.5%

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

   5. Simplified 85.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -9.2 \cdot 10^{-97}:\\ \;\;\;\;-1 - \frac{n \cdot 2}{f}\\ \mathbf{elif}\;f \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n \cdot 2}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.05 \cdot 10^{-96}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;f \leq 6.4 \cdot 10^{+41}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.05e-96)
   (- -1.0 (/ n f))
   (if (<= f 6.4e+41) (+ 1.0 (* 2.0 (/ f n))) (/ f (- n f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.05e-96) {
		tmp = -1.0 - (n / f);
	} else if (f <= 6.4e+41) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = f / (n - f);
	}
	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) - (n / f)
    else if (f <= 6.4d+41) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = f / (n - f)
    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 - (n / f);
	} else if (f <= 6.4e+41) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = f / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.05e-96:
		tmp = -1.0 - (n / f)
	elif f <= 6.4e+41:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = f / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.05e-96)
		tmp = Float64(-1.0 - Float64(n / f));
	elseif (f <= 6.4e+41)
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = Float64(f / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.05e-96)
		tmp = -1.0 - (n / f);
	elseif (f <= 6.4e+41)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = f / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.05e-96], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 6.4e+41], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -1.05000000000000001e-96

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 73.4%

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

   6. Taylor expanded in f around inf

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

      1. sub-neg N/A

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

      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 73.7%

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

   8. Simplified 73.7%

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

   if -1.05000000000000001e-96 < f < 6.40000000000000019e41

   1. Initial program 99.9%

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

   3. Taylor expanded in f around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. count-2 N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-out N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-neg-frac N/A

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

      10. mul-1-neg N/A

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

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

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-out N/A

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

      17. remove-double-neg N/A

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

      18. count-2 N/A

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

      19. *-commutative N/A

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

      20. associate-*l/ N/A

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

      21. *-commutative N/A

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

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

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

      23. /-lowering-/.f64 82.7%

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

   5. Simplified 82.7%

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

   if 6.40000000000000019e41 < f

   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 83.5%

      \[\leadsto \frac{-\color{blue}{f}}{f - n} \]
   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 83.5%

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

   7. Applied egg-rr 83.5%

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

Alternative 4: 73.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -6.6 \cdot 10^{-97}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;f \leq 1.95 \cdot 10^{-20}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \left(n + f\right)}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -6.6e-97)
   (- -1.0 (/ n f))
   (if (<= f 1.95e-20) (/ n (- n f)) (/ (- 0.0 (+ n f)) f))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -6.6e-97) {
		tmp = -1.0 - (n / f);
	} else if (f <= 1.95e-20) {
		tmp = n / (n - f);
	} else {
		tmp = (0.0 - (n + f)) / f;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-6.6d-97)) then
        tmp = (-1.0d0) - (n / f)
    else if (f <= 1.95d-20) then
        tmp = n / (n - f)
    else
        tmp = (0.0d0 - (n + f)) / f
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -6.6e-97) {
		tmp = -1.0 - (n / f);
	} else if (f <= 1.95e-20) {
		tmp = n / (n - f);
	} else {
		tmp = (0.0 - (n + f)) / f;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -6.6e-97:
		tmp = -1.0 - (n / f)
	elif f <= 1.95e-20:
		tmp = n / (n - f)
	else:
		tmp = (0.0 - (n + f)) / f
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -6.6e-97)
		tmp = Float64(-1.0 - Float64(n / f));
	elseif (f <= 1.95e-20)
		tmp = Float64(n / Float64(n - f));
	else
		tmp = Float64(Float64(0.0 - Float64(n + f)) / f);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -6.6e-97)
		tmp = -1.0 - (n / f);
	elseif (f <= 1.95e-20)
		tmp = n / (n - f);
	else
		tmp = (0.0 - (n + f)) / f;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -6.6e-97], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 1.95e-20], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - N[(n + f), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -6.6000000000000002e-97

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 73.4%

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

   6. Taylor expanded in f around inf

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

      1. sub-neg N/A

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

      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 73.7%

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

   8. Simplified 73.7%

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

   if -6.6000000000000002e-97 < f < 1.95000000000000004e-20

   1. Initial program 100.0%

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

   3. Taylor expanded in f around 0

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

   5. Simplified 85.1%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

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

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

      9. --lowering--.f64 85.1%

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

   7. Applied egg-rr 85.1%

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

   if 1.95000000000000004e-20 < f

   1. Initial program 99.9%

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

   3. Taylor expanded in f around inf

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

   5. Simplified 77.7%

      \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{f}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -6.6 \cdot 10^{-97}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;f \leq 1.95 \cdot 10^{-20}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \left(n + f\right)}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))))
   (if (<= f -1.05e-96) t_0 (if (<= f 1.05e-19) (/ n (- n f)) 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 <= 1.05e-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 <= (-1.05d-96)) then
        tmp = t_0
    else if (f <= 1.05d-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 <= -1.05e-96) {
		tmp = t_0;
	} else if (f <= 1.05e-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 <= -1.05e-96:
		tmp = t_0
	elif f <= 1.05e-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 <= -1.05e-96)
		tmp = t_0;
	elseif (f <= 1.05e-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 <= -1.05e-96)
		tmp = t_0;
	elseif (f <= 1.05e-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, -1.05e-96], t$95$0, If[LessEqual[f, 1.05e-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 1.0499999999999999e-19 < f

   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 75.1%

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

   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 75.4%

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

   8. Simplified 75.4%

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

   if -1.05000000000000001e-96 < f < 1.0499999999999999e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in f around 0

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

   5. Simplified 85.1%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

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

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

      9. --lowering--.f64 85.1%

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

   7. Applied egg-rr 85.1%

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

Alternative 6: 73.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{n}{f}\\ \mathbf{if}\;f \leq -1.05 \cdot 10^{-96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 1.56 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))))
   (if (<= f -1.05e-96) t_0 (if (<= f 1.56e-19) 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 <= 1.56e-19) {
		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 <= 1.56d-19) 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 <= 1.56e-19) {
		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 <= 1.56e-19:
		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 <= 1.56e-19)
		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 <= 1.56e-19)
		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, 1.56e-19], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if f < -1.05000000000000001e-96 or 1.56000000000000003e-19 < f

   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 75.1%

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

   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 75.4%

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

   8. Simplified 75.4%

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

   if -1.05000000000000001e-96 < f < 1.56000000000000003e-19

   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 84.6%

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

Alternative 7: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.05 \cdot 10^{-96}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 10^{-21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.05e-96) -1.0 (if (<= f 1e-21) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.05e-96) {
		tmp = -1.0;
	} else if (f <= 1e-21) {
		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 <= 1d-21) 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 <= 1e-21) {
		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 <= 1e-21:
		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 <= 1e-21)
		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 <= 1e-21)
		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, 1e-21], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.05000000000000001e-96 or 9.99999999999999908e-22 < f

   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 74.6%

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

   if -1.05000000000000001e-96 < f < 9.99999999999999908e-22

   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 84.6%

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

Alternative 8: 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. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

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

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

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. remove-double-neg N/A

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

   10. sub-neg N/A

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

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

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

   12. +-lowering-+.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 9: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. 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. Final simplification 99.9%

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

Alternative 10: 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. Add Preprocessing

3. Taylor expanded in f around inf

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

5. Simplified 48.7%

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

Reproduce

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