subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 6

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_, n_] := N[(N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision] + N[(f / 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. neg-mul-1 99.9%

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

   2. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

      \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

   5. associate-+l- 99.9%

      \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

   6. neg-sub0 99.9%

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

   7. neg-mul-1 99.9%

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

   8. times-frac 99.9%

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

   9. metadata-eval 99.9%

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

   10. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

   1. add-log-exp 99.9%

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

5. Applied egg-rr 99.9%

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

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

   3. add-log-exp 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. associate-*l/ 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. associate-*l/ 100.0%

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

   8. *-un-lft-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+76} \lor \neg \left(n \leq 2.3 \cdot 10^{-37}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -2e+76) (not (<= n 2.3e-37))) (+ 1.0 (* 2.0 (/ f n))) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -2e+76) || !(n <= 2.3e-37)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2d+76)) .or. (.not. (n <= 2.3d-37))) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -2e+76) || !(n <= 2.3e-37)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -2e+76) or not (n <= 2.3e-37):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -2e+76) || !(n <= 2.3e-37))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -2e+76) || ~((n <= 2.3e-37)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -2e+76], N[Not[LessEqual[n, 2.3e-37]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.0000000000000001e76 or 2.3e-37 < n

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 82.8%

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

   if -2.0000000000000001e76 < n < 2.3e-37

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 81.2%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{+76} \lor \neg \left(n \leq 2.3 \cdot 10^{-37}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+76} \lor \neg \left(n \leq 8 \cdot 10^{-35}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -3.4e+76) (not (<= n 8e-35)))
   (+ 1.0 (* 2.0 (/ f n)))
   (+ (* -2.0 (/ n f)) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -3.4e+76) || !(n <= 8e-35)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = (-2.0 * (n / f)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-3.4d+76)) .or. (.not. (n <= 8d-35))) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -3.4e+76) || !(n <= 8e-35)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = (-2.0 * (n / f)) + -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -3.4e+76) or not (n <= 8e-35):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = (-2.0 * (n / f)) + -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -3.4e+76) || !(n <= 8e-35))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -3.4e+76) || ~((n <= 8e-35)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = (-2.0 * (n / f)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -3.4e+76], N[Not[LessEqual[n, 8e-35]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.3999999999999997e76 or 8.00000000000000006e-35 < n

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 82.8%

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

   if -3.3999999999999997e76 < n < 8.00000000000000006e-35

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in n around 0 82.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+76} \lor \neg \left(n \leq 8 \cdot 10^{-35}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]

Alternative 4: 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. neg-mul-1 99.9%

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

   2. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

      \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

   5. associate-+l- 99.9%

      \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

   6. neg-sub0 99.9%

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

   7. neg-mul-1 99.9%

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

   8. times-frac 99.9%

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

   9. metadata-eval 99.9%

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

   10. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 73.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.5e+76) 1.0 (if (<= n 4.7e-35) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.5e+76) {
		tmp = 1.0;
	} else if (n <= 4.7e-35) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.5d+76)) then
        tmp = 1.0d0
    else if (n <= 4.7d-35) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -1.5e+76) {
		tmp = 1.0;
	} else if (n <= 4.7e-35) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.5e+76:
		tmp = 1.0
	elif n <= 4.7e-35:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.5e+76)
		tmp = 1.0;
	elseif (n <= 4.7e-35)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.5e+76)
		tmp = 1.0;
	elseif (n <= 4.7e-35)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.5e+76], 1.0, If[LessEqual[n, 4.7e-35], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -1.4999999999999999e76 or 4.7e-35 < n

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 81.6%

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

   if -1.4999999999999999e76 < n < 4.7e-35

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

      5. associate-+l- 99.9%

         \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. times-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 81.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 49.5% 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. sub-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. neg-sub0 99.9%

      \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]

   5. associate-+l- 99.9%

      \[\leadsto \frac{-1 \cdot \left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]

   6. neg-sub0 99.9%

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

   7. neg-mul-1 99.9%

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

   8. times-frac 99.9%

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

   9. metadata-eval 99.9%

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

   10. *-lft-identity 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in f around inf 51.9%

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

5. Final simplification 51.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))