subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 4

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, 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 100.0%

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

   1. sub-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-neg-in 100.0%

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

   4. neg-mul-1 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. metadata-eval 100.0%

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

   10. /-rgt-identity 100.0%

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

   11. +-commutative 100.0%

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

   12. sub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -6.5 \cdot 10^{+25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-37}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -6.5e+25) -1.0 (if (<= f 4.2e-37) (+ 1.0 (* 2.0 (/ f n))) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -6.5e+25) {
		tmp = -1.0;
	} else if (f <= 4.2e-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 (f <= (-6.5d+25)) then
        tmp = -1.0d0
    else if (f <= 4.2d-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 (f <= -6.5e+25) {
		tmp = -1.0;
	} else if (f <= 4.2e-37) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -6.5e+25:
		tmp = -1.0
	elif f <= 4.2e-37:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -6.5e+25)
		tmp = -1.0;
	elseif (f <= 4.2e-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 (f <= -6.5e+25)
		tmp = -1.0;
	elseif (f <= 4.2e-37)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -6.5e+25], -1.0, If[LessEqual[f, 4.2e-37], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -6.50000000000000005e25 or 4.2000000000000002e-37 < f

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around inf 82.3%

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

   if -6.50000000000000005e25 < f < 4.2000000000000002e-37

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 83.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -6.5 \cdot 10^{+25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-37}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 73.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -5 \cdot 10^{+26}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -5e+26) -1.0 (if (<= f 2.5e-36) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -5e+26) {
		tmp = -1.0;
	} else if (f <= 2.5e-36) {
		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+26)) then
        tmp = -1.0d0
    else if (f <= 2.5d-36) 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+26) {
		tmp = -1.0;
	} else if (f <= 2.5e-36) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -5e+26:
		tmp = -1.0
	elif f <= 2.5e-36:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -5e+26)
		tmp = -1.0;
	elseif (f <= 2.5e-36)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -5e+26)
		tmp = -1.0;
	elseif (f <= 2.5e-36)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -5e+26], -1.0, If[LessEqual[f, 2.5e-36], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -5.0000000000000001e26 or 2.50000000000000002e-36 < f

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around inf 82.3%

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

   if -5.0000000000000001e26 < f < 2.50000000000000002e-36

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 82.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -5 \cdot 10^{+26}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2.5 \cdot 10^{-36}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 50.0% 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 100.0%

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

   1. sub-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-neg-in 100.0%

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

   4. neg-mul-1 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. metadata-eval 100.0%

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

   10. /-rgt-identity 100.0%

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

   11. +-commutative 100.0%

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

   12. sub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in f around inf 52.0%

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

5. Final simplification 52.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023280 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))