subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

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. distribute-frac-neg 100.0%

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

   2. distribute-neg-frac2 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-neg-in 100.0%

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

   6. remove-double-neg 100.0%

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

   7. 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 73.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3e+14)
   1.0
   (if (<= n -2.5e-12)
     -1.0
     (if (<= n -1.85e-13)
       1.0
       (if (<= n 4.3e+45)
         -1.0
         (if (<= n 3e+63) 1.0 (if (<= n 4.2e+65) -1.0 1.0)))))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3e+14) {
		tmp = 1.0;
	} else if (n <= -2.5e-12) {
		tmp = -1.0;
	} else if (n <= -1.85e-13) {
		tmp = 1.0;
	} else if (n <= 4.3e+45) {
		tmp = -1.0;
	} else if (n <= 3e+63) {
		tmp = 1.0;
	} else if (n <= 4.2e+65) {
		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 <= (-3d+14)) then
        tmp = 1.0d0
    else if (n <= (-2.5d-12)) then
        tmp = -1.0d0
    else if (n <= (-1.85d-13)) then
        tmp = 1.0d0
    else if (n <= 4.3d+45) then
        tmp = -1.0d0
    else if (n <= 3d+63) then
        tmp = 1.0d0
    else if (n <= 4.2d+65) 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 <= -3e+14) {
		tmp = 1.0;
	} else if (n <= -2.5e-12) {
		tmp = -1.0;
	} else if (n <= -1.85e-13) {
		tmp = 1.0;
	} else if (n <= 4.3e+45) {
		tmp = -1.0;
	} else if (n <= 3e+63) {
		tmp = 1.0;
	} else if (n <= 4.2e+65) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3e+14:
		tmp = 1.0
	elif n <= -2.5e-12:
		tmp = -1.0
	elif n <= -1.85e-13:
		tmp = 1.0
	elif n <= 4.3e+45:
		tmp = -1.0
	elif n <= 3e+63:
		tmp = 1.0
	elif n <= 4.2e+65:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3e+14)
		tmp = 1.0;
	elseif (n <= -2.5e-12)
		tmp = -1.0;
	elseif (n <= -1.85e-13)
		tmp = 1.0;
	elseif (n <= 4.3e+45)
		tmp = -1.0;
	elseif (n <= 3e+63)
		tmp = 1.0;
	elseif (n <= 4.2e+65)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3e+14)
		tmp = 1.0;
	elseif (n <= -2.5e-12)
		tmp = -1.0;
	elseif (n <= -1.85e-13)
		tmp = 1.0;
	elseif (n <= 4.3e+45)
		tmp = -1.0;
	elseif (n <= 3e+63)
		tmp = 1.0;
	elseif (n <= 4.2e+65)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3e+14], 1.0, If[LessEqual[n, -2.5e-12], -1.0, If[LessEqual[n, -1.85e-13], 1.0, If[LessEqual[n, 4.3e+45], -1.0, If[LessEqual[n, 3e+63], 1.0, If[LessEqual[n, 4.2e+65], -1.0, 1.0]]]]]]

Derivation

1. Split input into 2 regimes

2. if n < -3e14 or -2.49999999999999985e-12 < n < -1.84999999999999994e-13 or 4.3000000000000003e45 < n < 2.99999999999999999e63 or 4.19999999999999983e65 < n

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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. Add Preprocessing

   5. Taylor expanded in f around 0 79.9%

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

   if -3e14 < n < -2.49999999999999985e-12 or -1.84999999999999994e-13 < n < 4.3000000000000003e45 or 2.99999999999999999e63 < n < 4.19999999999999983e65

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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. Add Preprocessing

   5. Taylor expanded in f around inf 76.7%

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

Alternative 3: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{+45}:\\ \;\;\;\;-1 + -2 \cdot \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3.3e+14)
   1.0
   (if (<= n -5e-13)
     -1.0
     (if (<= n -3.5e-13)
       1.0
       (if (<= n 4.3e+45) (+ -1.0 (* -2.0 (/ n f))) 1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3.3e+14) {
		tmp = 1.0;
	} else if (n <= -5e-13) {
		tmp = -1.0;
	} else if (n <= -3.5e-13) {
		tmp = 1.0;
	} else if (n <= 4.3e+45) {
		tmp = -1.0 + (-2.0 * (n / f));
	} 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 <= (-3.3d+14)) then
        tmp = 1.0d0
    else if (n <= (-5d-13)) then
        tmp = -1.0d0
    else if (n <= (-3.5d-13)) then
        tmp = 1.0d0
    else if (n <= 4.3d+45) then
        tmp = (-1.0d0) + ((-2.0d0) * (n / f))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -3.3e+14) {
		tmp = 1.0;
	} else if (n <= -5e-13) {
		tmp = -1.0;
	} else if (n <= -3.5e-13) {
		tmp = 1.0;
	} else if (n <= 4.3e+45) {
		tmp = -1.0 + (-2.0 * (n / f));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3.3e+14:
		tmp = 1.0
	elif n <= -5e-13:
		tmp = -1.0
	elif n <= -3.5e-13:
		tmp = 1.0
	elif n <= 4.3e+45:
		tmp = -1.0 + (-2.0 * (n / f))
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3.3e+14)
		tmp = 1.0;
	elseif (n <= -5e-13)
		tmp = -1.0;
	elseif (n <= -3.5e-13)
		tmp = 1.0;
	elseif (n <= 4.3e+45)
		tmp = Float64(-1.0 + Float64(-2.0 * Float64(n / f)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3.3e+14)
		tmp = 1.0;
	elseif (n <= -5e-13)
		tmp = -1.0;
	elseif (n <= -3.5e-13)
		tmp = 1.0;
	elseif (n <= 4.3e+45)
		tmp = -1.0 + (-2.0 * (n / f));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3.3e+14], 1.0, If[LessEqual[n, -5e-13], -1.0, If[LessEqual[n, -3.5e-13], 1.0, If[LessEqual[n, 4.3e+45], N[(-1.0 + N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.3e14 or -4.9999999999999999e-13 < n < -3.5000000000000002e-13 or 4.3000000000000003e45 < n

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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. Add Preprocessing

   5. Taylor expanded in f around 0 79.2%

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

   if -3.3e14 < n < -4.9999999999999999e-13

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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. Add Preprocessing

   5. Taylor expanded in f around inf 100.0%

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

   if -3.5000000000000002e-13 < n < 4.3000000000000003e45

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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. Add Preprocessing

   5. Taylor expanded in n around 0 76.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -3.5 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.3 \cdot 10^{+45}:\\ \;\;\;\;-1 + -2 \cdot \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.2% 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 100.0%

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

   1. distribute-frac-neg 100.0%

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

   2. distribute-neg-frac2 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-neg-in 100.0%

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

   6. remove-double-neg 100.0%

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

   7. 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. Add Preprocessing

5. Taylor expanded in f around inf 53.5%

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

Reproduce

herbie shell --seed 2024105 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))