subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

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 100.0%

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

   1. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 99.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{f + n}{n - f} \cdot \frac{f + n}{n - f}\right) \cdot \frac{f + n}{n - f}}} \]

   2. pow3 99.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{f + n}{n - f}\right)}^{3}}} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{f + n}{n - f}\right)}^{3}}} \]
6. Step-by-step derivation

   1. rem-cbrt-cube 100.0%

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

   2. clear-num 100.0%

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

   3. associate-/r/ 99.7%

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

   4. +-commutative 99.7%

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

   5. distribute-rgt-in 99.7%

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

   6. un-div-inv 99.9%

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

   7. un-div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 72.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.15 \cdot 10^{+43} \lor \neg \left(f \leq -6.6 \cdot 10^{+24} \lor \neg \left(f \leq -1.8 \cdot 10^{-134}\right) \land f \leq 7.2 \cdot 10^{+78}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.15e+43)
         (not
          (or (<= f -6.6e+24) (and (not (<= f -1.8e-134)) (<= f 7.2e+78)))))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.15e+43) || !((f <= -6.6e+24) || (!(f <= -1.8e-134) && (f <= 7.2e+78)))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-1.15d+43)) .or. (.not. (f <= (-6.6d+24)) .or. (.not. (f <= (-1.8d-134))) .and. (f <= 7.2d+78))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (f / n))
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -1.15e+43) || !((f <= -6.6e+24) || (!(f <= -1.8e-134) && (f <= 7.2e+78)))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -1.15e+43) or not ((f <= -6.6e+24) or (not (f <= -1.8e-134) and (f <= 7.2e+78))):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0 + (2.0 * (f / n))
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -1.15e+43) || !((f <= -6.6e+24) || (!(f <= -1.8e-134) && (f <= 7.2e+78))))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -1.15e+43) || ~(((f <= -6.6e+24) || (~((f <= -1.8e-134)) && (f <= 7.2e+78)))))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0 + (2.0 * (f / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -1.15e+43], N[Not[Or[LessEqual[f, -6.6e+24], And[N[Not[LessEqual[f, -1.8e-134]], $MachinePrecision], LessEqual[f, 7.2e+78]]]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -1.1500000000000001e43 or -6.5999999999999998e24 < f < -1.79999999999999995e-134 or 7.20000000000000039e78 < f

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in n around 0 81.7%

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

   if -1.1500000000000001e43 < f < -6.5999999999999998e24 or -1.79999999999999995e-134 < f < 7.20000000000000039e78

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 80.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.15 \cdot 10^{+43} \lor \neg \left(f \leq -6.6 \cdot 10^{+24} \lor \neg \left(f \leq -1.8 \cdot 10^{-134}\right) \land f \leq 7.2 \cdot 10^{+78}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 3: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.15 \cdot 10^{+43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -2.4 \cdot 10^{+27} \lor \neg \left(f \leq -1.8 \cdot 10^{-134}\right) \land f \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.15e+43)
   -1.0
   (if (or (<= f -2.4e+27) (and (not (<= f -1.8e-134)) (<= f 5.8e+78)))
     (+ 1.0 (* 2.0 (/ f n)))
     -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.15e+43) {
		tmp = -1.0;
	} else if ((f <= -2.4e+27) || (!(f <= -1.8e-134) && (f <= 5.8e+78))) {
		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 <= (-1.15d+43)) then
        tmp = -1.0d0
    else if ((f <= (-2.4d+27)) .or. (.not. (f <= (-1.8d-134))) .and. (f <= 5.8d+78)) 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 <= -1.15e+43) {
		tmp = -1.0;
	} else if ((f <= -2.4e+27) || (!(f <= -1.8e-134) && (f <= 5.8e+78))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.15e+43:
		tmp = -1.0
	elif (f <= -2.4e+27) or (not (f <= -1.8e-134) and (f <= 5.8e+78)):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.15e+43)
		tmp = -1.0;
	elseif ((f <= -2.4e+27) || (!(f <= -1.8e-134) && (f <= 5.8e+78)))
		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 <= -1.15e+43)
		tmp = -1.0;
	elseif ((f <= -2.4e+27) || (~((f <= -1.8e-134)) && (f <= 5.8e+78)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.15e+43], -1.0, If[Or[LessEqual[f, -2.4e+27], And[N[Not[LessEqual[f, -1.8e-134]], $MachinePrecision], LessEqual[f, 5.8e+78]]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.1500000000000001e43 or -2.39999999999999998e27 < f < -1.79999999999999995e-134 or 5.80000000000000034e78 < f

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around inf 80.9%

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

   if -1.1500000000000001e43 < f < -2.39999999999999998e27 or -1.79999999999999995e-134 < f < 5.80000000000000034e78

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 80.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.15 \cdot 10^{+43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -2.4 \cdot 10^{+27} \lor \neg \left(f \leq -1.8 \cdot 10^{-134}\right) \land f \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.3 \cdot 10^{+43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -1 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -1.8 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.3e+43)
   -1.0
   (if (<= f -1e+28)
     1.0
     (if (<= f -1.8e-134) -1.0 (if (<= f 5.8e+78) 1.0 -1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.3e+43) {
		tmp = -1.0;
	} else if (f <= -1e+28) {
		tmp = 1.0;
	} else if (f <= -1.8e-134) {
		tmp = -1.0;
	} else if (f <= 5.8e+78) {
		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 <= (-1.3d+43)) then
        tmp = -1.0d0
    else if (f <= (-1d+28)) then
        tmp = 1.0d0
    else if (f <= (-1.8d-134)) then
        tmp = -1.0d0
    else if (f <= 5.8d+78) 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 <= -1.3e+43) {
		tmp = -1.0;
	} else if (f <= -1e+28) {
		tmp = 1.0;
	} else if (f <= -1.8e-134) {
		tmp = -1.0;
	} else if (f <= 5.8e+78) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.3e+43:
		tmp = -1.0
	elif f <= -1e+28:
		tmp = 1.0
	elif f <= -1.8e-134:
		tmp = -1.0
	elif f <= 5.8e+78:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.3e+43)
		tmp = -1.0;
	elseif (f <= -1e+28)
		tmp = 1.0;
	elseif (f <= -1.8e-134)
		tmp = -1.0;
	elseif (f <= 5.8e+78)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.3e+43)
		tmp = -1.0;
	elseif (f <= -1e+28)
		tmp = 1.0;
	elseif (f <= -1.8e-134)
		tmp = -1.0;
	elseif (f <= 5.8e+78)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.3e+43], -1.0, If[LessEqual[f, -1e+28], 1.0, If[LessEqual[f, -1.8e-134], -1.0, If[LessEqual[f, 5.8e+78], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if f < -1.3000000000000001e43 or -9.99999999999999958e27 < f < -1.79999999999999995e-134 or 5.80000000000000034e78 < f

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around inf 80.9%

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

   if -1.3000000000000001e43 < f < -9.99999999999999958e27 or -1.79999999999999995e-134 < f < 5.80000000000000034e78

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 79.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.3 \cdot 10^{+43}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -1 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -1.8 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 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 100.0%

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

   1. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 6: 48.6% 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. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in f around inf 51.5%

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

5. Final simplification 51.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023319 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))