subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 5

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{f}{n - f}\\ t_1 := \frac{n}{n - f}\\ \frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1} \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ f (- n f))) (t_1 (/ n (- n f))))
   (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))))

C

double code(double f, double n) {
	double t_0 = f / (n - f);
	double t_1 = n / (n - f);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    t_0 = f / (n - f)
    t_1 = n / (n - f)
    code = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
end function

Java

public static double code(double f, double n) {
	double t_0 = f / (n - f);
	double t_1 = n / (n - f);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Python

def code(f, n):
	t_0 = f / (n - f)
	t_1 = n / (n - f)
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)

Julia

function code(f, n)
	t_0 = Float64(f / Float64(n - f))
	t_1 = Float64(n / Float64(n - f))
	return Float64(Float64(Float64(t_0 * t_0) - Float64(t_1 * t_1)) / Float64(t_0 - t_1))
end

MATLAB

function tmp = code(f, n)
	t_0 = f / (n - f);
	t_1 = n / (n - f);
	tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

   1. neg-mul-1 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. div-sub 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/l* 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-/l* 99.9%

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

   13. *-commutative 99.9%

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

   14. neg-mul-1 99.9%

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

   15. div-sub 99.9%

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

   16. unsub-neg 99.9%

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

   17. remove-double-neg 99.9%

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

   18. +-commutative 99.9%

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

   19. sub-neg 99.9%

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

   20. metadata-eval 99.9%

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

   21. /-rgt-identity 99.9%

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

3. Simplified 99.9%

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

   1. log1p-expm1-u 99.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{f + n}{n - f}\right)\right)} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{f + n}{n - f}\right)\right)} \]
6. Step-by-step derivation

   1. log1p-expm1-u 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. associate-*l/ 99.7%

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

   4. distribute-rgt-in 99.7%

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

   5. flip-+ 99.7%

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

   6. un-div-inv 99.8%

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

   7. un-div-inv 99.7%

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

   8. un-div-inv 99.9%

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

   9. un-div-inv 99.7%

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

   10. un-div-inv 99.8%

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

   11. un-div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -4.6 \cdot 10^{-7} \lor \neg \left(f \leq 4.9 \cdot 10^{+58}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -4.6e-7) (not (<= f 4.9e+58))) (+ (* -2.0 (/ n f)) -1.0) 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -4.6e-7) || !(f <= 4.9e+58)) {
		tmp = (-2.0 * (n / f)) + -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 <= (-4.6d-7)) .or. (.not. (f <= 4.9d+58))) then
        tmp = ((-2.0d0) * (n / f)) + (-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 <= -4.6e-7) || !(f <= 4.9e+58)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -4.6e-7) or not (f <= 4.9e+58):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -4.6e-7) || !(f <= 4.9e+58))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -4.6e-7) || ~((f <= 4.9e+58)))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -4.6e-7], N[Not[LessEqual[f, 4.9e+58]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if f < -4.5999999999999999e-7 or 4.90000000000000018e58 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in n around 0 78.9%

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

   if -4.5999999999999999e-7 < f < 4.90000000000000018e58

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 75.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -4.6 \cdot 10^{-7} \lor \neg \left(f \leq 4.9 \cdot 10^{+58}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-mul-1 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. div-sub 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/l* 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-/l* 99.9%

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

   13. *-commutative 99.9%

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

   14. neg-mul-1 99.9%

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

   15. div-sub 99.9%

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

   16. unsub-neg 99.9%

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

   17. remove-double-neg 99.9%

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

   18. +-commutative 99.9%

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

   19. sub-neg 99.9%

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

   20. metadata-eval 99.9%

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

   21. /-rgt-identity 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 72.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -5 \cdot 10^{+33}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -5e+33) -1.0 (if (<= f 1.4e+59) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -5e+33) {
		tmp = -1.0;
	} else if (f <= 1.4e+59) {
		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 <= (-5d+33)) then
        tmp = -1.0d0
    else if (f <= 1.4d+59) 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 <= -5e+33) {
		tmp = -1.0;
	} else if (f <= 1.4e+59) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -5e+33:
		tmp = -1.0
	elif f <= 1.4e+59:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -5e+33)
		tmp = -1.0;
	elseif (f <= 1.4e+59)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -5e+33)
		tmp = -1.0;
	elseif (f <= 1.4e+59)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -5e+33], -1.0, If[LessEqual[f, 1.4e+59], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -4.99999999999999973e33 or 1.3999999999999999e59 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 80.1%

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

   if -4.99999999999999973e33 < f < 1.3999999999999999e59

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 74.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -5 \cdot 10^{+33}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 49.3% 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. neg-mul-1 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. div-sub 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/l* 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-/l* 99.9%

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

   13. *-commutative 99.9%

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

   14. neg-mul-1 99.9%

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

   15. div-sub 99.9%

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

   16. unsub-neg 99.9%

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

   17. remove-double-neg 99.9%

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

   18. +-commutative 99.9%

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

   19. sub-neg 99.9%

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

   20. metadata-eval 99.9%

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

   21. /-rgt-identity 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in f around inf 45.5%

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

5. Final simplification 45.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))