subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

Alternatives: 6

Speedup: 1.1×

Specification

Math

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

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)
use fmin_fmax_functions
    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

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

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)
use fmin_fmax_functions
    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

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

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)
use fmin_fmax_functions
    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 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lower--.f64 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{n + f}{n - f}} \]
4. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{f}, -2, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{f}{n}, 2, 1\right)\\ \end{array} \]

FPCore

(FPCore (f n)
  :precision binary64
  (if (<= (/ (- (+ f n)) (- f n)) -0.5)
  (fma (/ n f) -2.0 -1.0)
  (fma (/ f n) 2.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = fma((n / f), -2.0, -1.0);
	} else {
		tmp = fma((f / n), 2.0, 1.0);
	}
	return tmp;
}

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -0.5)
		tmp = fma(Float64(n / f), -2.0, -1.0);
	else
		tmp = fma(Float64(f / n), 2.0, 1.0);
	end
	return tmp
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(n / f), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision], N[(N[(f / n), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 100.0%

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

   2. Taylor expanded in n around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 50.9%

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

   4. Applied rewrites 50.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 50.9%

         \[\leadsto \mathsf{fma}\left(\frac{n}{f}, \color{blue}{-2}, -1\right) \]

   6. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(\frac{n}{f}, \color{blue}{-2}, -1\right) \]

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

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

   2. Taylor expanded in f around inf

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in f around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.1%

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

   7. Applied rewrites 51.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 51.1%

         \[\leadsto \mathsf{fma}\left(\frac{f}{n}, \color{blue}{2}, 1\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\frac{f}{n}, \color{blue}{2}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{f}{n}, 2, 1\right)\\ \end{array} \]

FPCore

(FPCore (f n)
  :precision binary64
  (if (<= (/ (- (+ f n)) (- f n)) -0.5) -1.0 (fma (/ f n) 2.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = fma((f / n), 2.0, 1.0);
	}
	return tmp;
}

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -0.5)
		tmp = -1.0;
	else
		tmp = fma(Float64(f / n), 2.0, 1.0);
	end
	return tmp
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(N[(f / n), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 100.0%

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

   2. Taylor expanded in f around inf

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

   4. Applied rewrites 49.6%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

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

   2. Taylor expanded in f around inf

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in f around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.1%

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

   7. Applied rewrites 51.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 51.1%

         \[\leadsto \mathsf{fma}\left(\frac{f}{n}, \color{blue}{2}, 1\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\frac{f}{n}, \color{blue}{2}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \]

FPCore

(FPCore (f n)
  :precision binary64
  (if (<= (/ (- (+ f n)) (- f n)) -0.5) -1.0 (/ n (- n f))))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((-(f + n) / (f - n)) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = n / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (-(f + n) / (f - n)) <= -0.5:
		tmp = -1.0
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(n / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((-(f + n) / (f - n)) <= -0.5)
		tmp = -1.0;
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 100.0%

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

   2. Taylor expanded in f around inf

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

   4. Applied rewrites 49.6%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

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

   2. Taylor expanded in f around 0

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift--.f64 N/A

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

      6. frac-2neg-rev N/A

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

      7. lower-/.f64 50.9%

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

   6. Applied rewrites 50.9%

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

Alternative 5: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -1.7 \cdot 10^{-228}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (f n)
  :precision binary64
  (if (<= (/ (- (+ f n)) (- f n)) -1.7e-228) -1.0 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -1.7e-228) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((-(f + n) / (f - n)) <= (-1.7d-228)) 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 + n) / (f - n)) <= -1.7e-228) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (-(f + n) / (f - n)) <= -1.7e-228:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -1.7e-228)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((-(f + n) / (f - n)) <= -1.7e-228)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -1.7e-228], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -1.7e-228

   1. Initial program 100.0%

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

   2. Taylor expanded in f around inf

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

   4. Applied rewrites 49.6%

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

   if -1.7e-228 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

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

   2. Taylor expanded in f around 0

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

   4. Applied rewrites 49.8%

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

Alternative 6: 49.6% accurate, 11.0× speedup

Math

\[-1 \]

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)
use fmin_fmax_functions
    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. Taylor expanded in f around inf

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

4. Applied rewrites 49.6%

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

Reproduce

herbie shell --seed 2025326 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))