subtraction fraction

Percentage Accurate: 100.0% → 99.9%

Time: 4.3s

Alternatives: 8

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: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{f + n}{n - f}\right)\right) \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (log1p (expm1 (/ (+ f n) (- n f)))))

C

double code(double f, double n) {
	return log1p(expm1(((f + n) / (n - f))));
}

Java

public static double code(double f, double n) {
	return Math.log1p(Math.expm1(((f + n) / (n - f))));
}

Python

def code(f, n):
	return math.log1p(math.expm1(((f + n) / (n - f))))

Julia

function code(f, n)
	return log1p(expm1(Float64(Float64(f + n) / Float64(n - f))))
end

Wolfram

code[f_, n_] := N[Log[1 + N[(Exp[N[(N[(f + n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. distribute-neg-frac2 99.9%

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

   3. sub-neg 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. remove-double-neg 99.9%

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

   7. sub-neg 99.9%

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

3. Simplified 99.9%

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

   1. log1p-expm1-u 99.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{f + n}{n - f}\right)\right)} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{f + n}{n - f}\right)\right)} \]
7. Add Preprocessing

Alternative 2: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{n}{f} + -1\\ \mathbf{if}\;f \leq -7.2 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -1.65 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 3.3 \cdot 10^{+71}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ (* -2.0 (/ n f)) -1.0)))
   (if (<= f -7.2e+55)
     t_0
     (if (<= f -1.2e+39)
       1.0
       (if (<= f -1.65e-18)
         -1.0
         (if (<= f 3.3e+71) (+ 1.0 (* 2.0 (/ f n))) t_0))))))

C

double code(double f, double n) {
	double t_0 = (-2.0 * (n / f)) + -1.0;
	double tmp;
	if (f <= -7.2e+55) {
		tmp = t_0;
	} else if (f <= -1.2e+39) {
		tmp = 1.0;
	} else if (f <= -1.65e-18) {
		tmp = -1.0;
	} else if (f <= 3.3e+71) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) * (n / f)) + (-1.0d0)
    if (f <= (-7.2d+55)) then
        tmp = t_0
    else if (f <= (-1.2d+39)) then
        tmp = 1.0d0
    else if (f <= (-1.65d-18)) then
        tmp = -1.0d0
    else if (f <= 3.3d+71) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = (-2.0 * (n / f)) + -1.0;
	double tmp;
	if (f <= -7.2e+55) {
		tmp = t_0;
	} else if (f <= -1.2e+39) {
		tmp = 1.0;
	} else if (f <= -1.65e-18) {
		tmp = -1.0;
	} else if (f <= 3.3e+71) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = (-2.0 * (n / f)) + -1.0
	tmp = 0
	if f <= -7.2e+55:
		tmp = t_0
	elif f <= -1.2e+39:
		tmp = 1.0
	elif f <= -1.65e-18:
		tmp = -1.0
	elif f <= 3.3e+71:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(Float64(-2.0 * Float64(n / f)) + -1.0)
	tmp = 0.0
	if (f <= -7.2e+55)
		tmp = t_0;
	elseif (f <= -1.2e+39)
		tmp = 1.0;
	elseif (f <= -1.65e-18)
		tmp = -1.0;
	elseif (f <= 3.3e+71)
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = (-2.0 * (n / f)) + -1.0;
	tmp = 0.0;
	if (f <= -7.2e+55)
		tmp = t_0;
	elseif (f <= -1.2e+39)
		tmp = 1.0;
	elseif (f <= -1.65e-18)
		tmp = -1.0;
	elseif (f <= 3.3e+71)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[f, -7.2e+55], t$95$0, If[LessEqual[f, -1.2e+39], 1.0, If[LessEqual[f, -1.65e-18], -1.0, If[LessEqual[f, 3.3e+71], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if f < -7.19999999999999975e55 or 3.2999999999999998e71 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in n around 0 85.6%

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

   if -7.19999999999999975e55 < f < -1.2e39

   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 0 100.0%

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

   if -1.2e39 < f < -1.6500000000000001e-18

   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 78.1%

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

   if -1.6500000000000001e-18 < f < 3.2999999999999998e71

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0 75.4%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{f}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -7.2 \cdot 10^{+55}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{elif}\;f \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -1.65 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 3.3 \cdot 10^{+71}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{f + n}{-f}\\ \mathbf{if}\;f \leq -1.02 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq -1.15 \cdot 10^{+41}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.1 \cdot 10^{+71}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ (+ f n) (- f))))
   (if (<= f -1.02e+55)
     t_0
     (if (<= f -1.15e+41)
       1.0
       (if (<= f -1.5e-18)
         -1.0
         (if (<= f 4.1e+71) (+ 1.0 (* 2.0 (/ f n))) t_0))))))

C

double code(double f, double n) {
	double t_0 = (f + n) / -f;
	double tmp;
	if (f <= -1.02e+55) {
		tmp = t_0;
	} else if (f <= -1.15e+41) {
		tmp = 1.0;
	} else if (f <= -1.5e-18) {
		tmp = -1.0;
	} else if (f <= 4.1e+71) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (f + n) / -f
    if (f <= (-1.02d+55)) then
        tmp = t_0
    else if (f <= (-1.15d+41)) then
        tmp = 1.0d0
    else if (f <= (-1.5d-18)) then
        tmp = -1.0d0
    else if (f <= 4.1d+71) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = (f + n) / -f;
	double tmp;
	if (f <= -1.02e+55) {
		tmp = t_0;
	} else if (f <= -1.15e+41) {
		tmp = 1.0;
	} else if (f <= -1.5e-18) {
		tmp = -1.0;
	} else if (f <= 4.1e+71) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = (f + n) / -f
	tmp = 0
	if f <= -1.02e+55:
		tmp = t_0
	elif f <= -1.15e+41:
		tmp = 1.0
	elif f <= -1.5e-18:
		tmp = -1.0
	elif f <= 4.1e+71:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(Float64(f + n) / Float64(-f))
	tmp = 0.0
	if (f <= -1.02e+55)
		tmp = t_0;
	elseif (f <= -1.15e+41)
		tmp = 1.0;
	elseif (f <= -1.5e-18)
		tmp = -1.0;
	elseif (f <= 4.1e+71)
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = (f + n) / -f;
	tmp = 0.0;
	if (f <= -1.02e+55)
		tmp = t_0;
	elseif (f <= -1.15e+41)
		tmp = 1.0;
	elseif (f <= -1.5e-18)
		tmp = -1.0;
	elseif (f <= 4.1e+71)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(N[(f + n), $MachinePrecision] / (-f)), $MachinePrecision]}, If[LessEqual[f, -1.02e+55], t$95$0, If[LessEqual[f, -1.15e+41], 1.0, If[LessEqual[f, -1.5e-18], -1.0, If[LessEqual[f, 4.1e+71], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if f < -1.02000000000000002e55 or 4.1000000000000002e71 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-sqr-sqrt 43.6%

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

      3. times-frac 43.5%

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

   6. Applied egg-rr 43.5%

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

      1. associate-*l/ 43.6%

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

      2. *-lft-identity 43.6%

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

      3. associate-/l/ 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in n around 0 84.7%

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

       1. neg-mul-1 84.7%

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

   11. Simplified 84.7%

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

   if -1.02000000000000002e55 < f < -1.1499999999999999e41

   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 0 100.0%

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

   if -1.1499999999999999e41 < f < -1.49999999999999991e-18

   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 78.1%

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

   if -1.49999999999999991e-18 < f < 4.1000000000000002e71

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0 75.4%

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

Alternative 4: 74.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{f + n}{-f}\\ \mathbf{if}\;f \leq -1.12 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq -1 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 8 \cdot 10^{+71}:\\ \;\;\;\;\frac{f + n}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ (+ f n) (- f))))
   (if (<= f -1.12e+55)
     t_0
     (if (<= f -1e+43)
       1.0
       (if (<= f -2.5e-18) -1.0 (if (<= f 8e+71) (/ (+ f n) n) t_0))))))

C

double code(double f, double n) {
	double t_0 = (f + n) / -f;
	double tmp;
	if (f <= -1.12e+55) {
		tmp = t_0;
	} else if (f <= -1e+43) {
		tmp = 1.0;
	} else if (f <= -2.5e-18) {
		tmp = -1.0;
	} else if (f <= 8e+71) {
		tmp = (f + n) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (f + n) / -f
    if (f <= (-1.12d+55)) then
        tmp = t_0
    else if (f <= (-1d+43)) then
        tmp = 1.0d0
    else if (f <= (-2.5d-18)) then
        tmp = -1.0d0
    else if (f <= 8d+71) then
        tmp = (f + n) / n
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = (f + n) / -f;
	double tmp;
	if (f <= -1.12e+55) {
		tmp = t_0;
	} else if (f <= -1e+43) {
		tmp = 1.0;
	} else if (f <= -2.5e-18) {
		tmp = -1.0;
	} else if (f <= 8e+71) {
		tmp = (f + n) / n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = (f + n) / -f
	tmp = 0
	if f <= -1.12e+55:
		tmp = t_0
	elif f <= -1e+43:
		tmp = 1.0
	elif f <= -2.5e-18:
		tmp = -1.0
	elif f <= 8e+71:
		tmp = (f + n) / n
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(Float64(f + n) / Float64(-f))
	tmp = 0.0
	if (f <= -1.12e+55)
		tmp = t_0;
	elseif (f <= -1e+43)
		tmp = 1.0;
	elseif (f <= -2.5e-18)
		tmp = -1.0;
	elseif (f <= 8e+71)
		tmp = Float64(Float64(f + n) / n);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = (f + n) / -f;
	tmp = 0.0;
	if (f <= -1.12e+55)
		tmp = t_0;
	elseif (f <= -1e+43)
		tmp = 1.0;
	elseif (f <= -2.5e-18)
		tmp = -1.0;
	elseif (f <= 8e+71)
		tmp = (f + n) / n;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(N[(f + n), $MachinePrecision] / (-f)), $MachinePrecision]}, If[LessEqual[f, -1.12e+55], t$95$0, If[LessEqual[f, -1e+43], 1.0, If[LessEqual[f, -2.5e-18], -1.0, If[LessEqual[f, 8e+71], N[(N[(f + n), $MachinePrecision] / n), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if f < -1.12000000000000006e55 or 8.0000000000000003e71 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-sqr-sqrt 43.6%

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

      3. times-frac 43.5%

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

   6. Applied egg-rr 43.5%

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

      1. associate-*l/ 43.6%

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

      2. *-lft-identity 43.6%

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

      3. associate-/l/ 43.6%

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

   8. Simplified 43.6%

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

   9. Taylor expanded in n around 0 84.7%

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

       1. neg-mul-1 84.7%

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

   11. Simplified 84.7%

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

   if -1.12000000000000006e55 < f < -1.00000000000000001e43

   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 0 100.0%

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

   if -1.00000000000000001e43 < f < -2.50000000000000018e-18

   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 78.1%

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

   if -2.50000000000000018e-18 < f < 8.0000000000000003e71

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-sqr-sqrt 43.3%

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

      3. times-frac 43.2%

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

   6. Applied egg-rr 43.2%

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

      1. associate-*l/ 43.3%

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

      2. *-lft-identity 43.3%

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

      3. associate-/l/ 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in n around inf 75.1%

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

Alternative 5: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.5 \cdot 10^{+55}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -9.2 \cdot 10^{+42}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -2.45 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 6.5 \cdot 10^{+71}:\\ \;\;\;\;\frac{f + n}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2.5e+55)
   -1.0
   (if (<= f -9.2e+42)
     1.0
     (if (<= f -2.45e-18) -1.0 (if (<= f 6.5e+71) (/ (+ f n) n) -1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2.5e+55) {
		tmp = -1.0;
	} else if (f <= -9.2e+42) {
		tmp = 1.0;
	} else if (f <= -2.45e-18) {
		tmp = -1.0;
	} else if (f <= 6.5e+71) {
		tmp = (f + n) / 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 <= (-2.5d+55)) then
        tmp = -1.0d0
    else if (f <= (-9.2d+42)) then
        tmp = 1.0d0
    else if (f <= (-2.45d-18)) then
        tmp = -1.0d0
    else if (f <= 6.5d+71) then
        tmp = (f + n) / n
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -2.5e+55) {
		tmp = -1.0;
	} else if (f <= -9.2e+42) {
		tmp = 1.0;
	} else if (f <= -2.45e-18) {
		tmp = -1.0;
	} else if (f <= 6.5e+71) {
		tmp = (f + n) / n;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2.5e+55:
		tmp = -1.0
	elif f <= -9.2e+42:
		tmp = 1.0
	elif f <= -2.45e-18:
		tmp = -1.0
	elif f <= 6.5e+71:
		tmp = (f + n) / n
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2.5e+55)
		tmp = -1.0;
	elseif (f <= -9.2e+42)
		tmp = 1.0;
	elseif (f <= -2.45e-18)
		tmp = -1.0;
	elseif (f <= 6.5e+71)
		tmp = Float64(Float64(f + n) / n);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2.5e+55)
		tmp = -1.0;
	elseif (f <= -9.2e+42)
		tmp = 1.0;
	elseif (f <= -2.45e-18)
		tmp = -1.0;
	elseif (f <= 6.5e+71)
		tmp = (f + n) / n;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2.5e+55], -1.0, If[LessEqual[f, -9.2e+42], 1.0, If[LessEqual[f, -2.45e-18], -1.0, If[LessEqual[f, 6.5e+71], N[(N[(f + n), $MachinePrecision] / n), $MachinePrecision], -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if f < -2.50000000000000023e55 or -9.2e42 < f < -2.4500000000000001e-18 or 6.49999999999999954e71 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf 83.8%

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

   if -2.50000000000000023e55 < f < -9.2e42

   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 0 100.0%

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

   if -2.4500000000000001e-18 < f < 6.49999999999999954e71

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-sqr-sqrt 43.3%

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

      3. times-frac 43.2%

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

   6. Applied egg-rr 43.2%

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

      1. associate-*l/ 43.3%

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

      2. *-lft-identity 43.3%

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

      3. associate-/l/ 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in n around inf 75.1%

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

Alternative 6: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.02 \cdot 10^{+55}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -1 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -2.75 \cdot 10^{-18}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 5 \cdot 10^{+67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.02e+55)
   -1.0
   (if (<= f -1e+43)
     1.0
     (if (<= f -2.75e-18) -1.0 (if (<= f 5e+67) 1.0 -1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.02e+55) {
		tmp = -1.0;
	} else if (f <= -1e+43) {
		tmp = 1.0;
	} else if (f <= -2.75e-18) {
		tmp = -1.0;
	} else if (f <= 5e+67) {
		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 <= (-1.02d+55)) then
        tmp = -1.0d0
    else if (f <= (-1d+43)) then
        tmp = 1.0d0
    else if (f <= (-2.75d-18)) then
        tmp = -1.0d0
    else if (f <= 5d+67) 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 <= -1.02e+55) {
		tmp = -1.0;
	} else if (f <= -1e+43) {
		tmp = 1.0;
	} else if (f <= -2.75e-18) {
		tmp = -1.0;
	} else if (f <= 5e+67) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.02e+55:
		tmp = -1.0
	elif f <= -1e+43:
		tmp = 1.0
	elif f <= -2.75e-18:
		tmp = -1.0
	elif f <= 5e+67:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.02e+55)
		tmp = -1.0;
	elseif (f <= -1e+43)
		tmp = 1.0;
	elseif (f <= -2.75e-18)
		tmp = -1.0;
	elseif (f <= 5e+67)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.02e+55)
		tmp = -1.0;
	elseif (f <= -1e+43)
		tmp = 1.0;
	elseif (f <= -2.75e-18)
		tmp = -1.0;
	elseif (f <= 5e+67)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.02e+55], -1.0, If[LessEqual[f, -1e+43], 1.0, If[LessEqual[f, -2.75e-18], -1.0, If[LessEqual[f, 5e+67], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if f < -1.02000000000000002e55 or -1.00000000000000001e43 < f < -2.75e-18 or 4.99999999999999976e67 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf 83.2%

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

   if -1.02000000000000002e55 < f < -1.00000000000000001e43 or -2.75e-18 < f < 4.99999999999999976e67

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0 75.4%

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

Alternative 7: 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 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. distribute-neg-frac2 99.9%

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

   3. sub-neg 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. remove-double-neg 99.9%

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

   7. sub-neg 99.9%

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

3. Simplified 99.9%

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

Alternative 8: 49.0% 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 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. distribute-neg-frac2 99.9%

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

   3. sub-neg 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-neg-in 99.9%

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

   6. remove-double-neg 99.9%

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

   7. sub-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in f around inf 49.0%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))