Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Percentage Accurate: 65.9% → 99.8%

Time: 10.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

C

double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Python

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

C

double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

Python

def code(x, y):
	return 1.0 - (((1.0 - x) * y) / (y + 1.0))

Julia

function code(x, y)
	return Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
end

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + -1\right)}^{2}\\ \mathbf{if}\;y \leq -4500 \lor \neg \left(y \leq 4.3 \cdot 10^{-5}\right):\\ \;\;\;\;x + \frac{1}{y \cdot \left(\frac{\frac{1}{t\_0} - \frac{x}{t\_0}}{y} + \frac{-1}{x + -1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (+ x -1.0) 2.0)))
   (if (or (<= y -4500.0) (not (<= y 4.3e-5)))
     (+
      x
      (/ 1.0 (* y (+ (/ (- (/ 1.0 t_0) (/ x t_0)) y) (/ -1.0 (+ x -1.0))))))
     (- 1.0 (* (- 1.0 x) (/ y (+ y 1.0)))))))

C

double code(double x, double y) {
	double t_0 = pow((x + -1.0), 2.0);
	double tmp;
	if ((y <= -4500.0) || !(y <= 4.3e-5)) {
		tmp = x + (1.0 / (y * ((((1.0 / t_0) - (x / t_0)) / y) + (-1.0 / (x + -1.0)))));
	} else {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + (-1.0d0)) ** 2.0d0
    if ((y <= (-4500.0d0)) .or. (.not. (y <= 4.3d-5))) then
        tmp = x + (1.0d0 / (y * ((((1.0d0 / t_0) - (x / t_0)) / y) + ((-1.0d0) / (x + (-1.0d0))))))
    else
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow((x + -1.0), 2.0);
	double tmp;
	if ((y <= -4500.0) || !(y <= 4.3e-5)) {
		tmp = x + (1.0 / (y * ((((1.0 / t_0) - (x / t_0)) / y) + (-1.0 / (x + -1.0)))));
	} else {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow((x + -1.0), 2.0)
	tmp = 0
	if (y <= -4500.0) or not (y <= 4.3e-5):
		tmp = x + (1.0 / (y * ((((1.0 / t_0) - (x / t_0)) / y) + (-1.0 / (x + -1.0)))))
	else:
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + -1.0) ^ 2.0
	tmp = 0.0
	if ((y <= -4500.0) || !(y <= 4.3e-5))
		tmp = Float64(x + Float64(1.0 / Float64(y * Float64(Float64(Float64(Float64(1.0 / t_0) - Float64(x / t_0)) / y) + Float64(-1.0 / Float64(x + -1.0))))));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + -1.0) ^ 2.0;
	tmp = 0.0;
	if ((y <= -4500.0) || ~((y <= 4.3e-5)))
		tmp = x + (1.0 / (y * ((((1.0 / t_0) - (x / t_0)) / y) + (-1.0 / (x + -1.0)))));
	else
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(x + -1.0), $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[y, -4500.0], N[Not[LessEqual[y, 4.3e-5]], $MachinePrecision]], N[(x + N[(1.0 / N[(y * N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4500 or 4.3000000000000002e-5 < y

   1. Initial program 36.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 58.4%

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

      2. remove-double-neg 58.4%

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

      3. remove-double-neg 58.4%

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

      4. +-commutative 58.4%

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

   3. Simplified 58.4%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 98.1%

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

   7. Simplified 98.1%

      \[\leadsto \color{blue}{x - \frac{\left(-1 + x\right) + \frac{1 - x}{y}}{y}} \]
   8. Step-by-step derivation

      1. clear-num 98.1%

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

      2. inv-pow 98.1%

         \[\leadsto x - \color{blue}{{\left(\frac{y}{\left(-1 + x\right) + \frac{1 - x}{y}}\right)}^{-1}} \]

      3. associate-+l+ 98.1%

         \[\leadsto x - {\left(\frac{y}{\color{blue}{-1 + \left(x + \frac{1 - x}{y}\right)}}\right)}^{-1} \]

   9. Applied egg-rr 98.1%

      \[\leadsto x - \color{blue}{{\left(\frac{y}{-1 + \left(x + \frac{1 - x}{y}\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 98.1%

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

   11. Simplified 98.1%

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

   12. Taylor expanded in y around -inf 100.0%

       \[\leadsto x - \frac{1}{\color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{\frac{x}{{\left(x - 1\right)}^{2}} - \frac{1}{{\left(x - 1\right)}^{2}}}{y} - \frac{1}{x - 1}\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto x - \frac{1}{\color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot \frac{\frac{x}{{\left(x - 1\right)}^{2}} - \frac{1}{{\left(x - 1\right)}^{2}}}{y} - \frac{1}{x - 1}\right)}} \]

       2. neg-mul-1 100.0%

          \[\leadsto x - \frac{1}{\color{blue}{\left(-y\right)} \cdot \left(-1 \cdot \frac{\frac{x}{{\left(x - 1\right)}^{2}} - \frac{1}{{\left(x - 1\right)}^{2}}}{y} - \frac{1}{x - 1}\right)} \]

       3. associate-*r/ 100.0%

          \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\color{blue}{\frac{-1 \cdot \left(\frac{x}{{\left(x - 1\right)}^{2}} - \frac{1}{{\left(x - 1\right)}^{2}}\right)}{y}} - \frac{1}{x - 1}\right)} \]

       4. mul-1-neg 100.0%

          \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\frac{\color{blue}{-\left(\frac{x}{{\left(x - 1\right)}^{2}} - \frac{1}{{\left(x - 1\right)}^{2}}\right)}}{y} - \frac{1}{x - 1}\right)} \]

       5. sub-neg 100.0%

          \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\frac{-\left(\frac{x}{{\color{blue}{\left(x + \left(-1\right)\right)}}^{2}} - \frac{1}{{\left(x - 1\right)}^{2}}\right)}{y} - \frac{1}{x - 1}\right)} \]

       6. metadata-eval 100.0%

          \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\frac{-\left(\frac{x}{{\left(x + \color{blue}{-1}\right)}^{2}} - \frac{1}{{\left(x - 1\right)}^{2}}\right)}{y} - \frac{1}{x - 1}\right)} \]

       7. sub-neg 100.0%

          \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\frac{-\left(\frac{x}{{\left(x + -1\right)}^{2}} - \frac{1}{{\color{blue}{\left(x + \left(-1\right)\right)}}^{2}}\right)}{y} - \frac{1}{x - 1}\right)} \]

       8. metadata-eval 100.0%

          \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\frac{-\left(\frac{x}{{\left(x + -1\right)}^{2}} - \frac{1}{{\left(x + \color{blue}{-1}\right)}^{2}}\right)}{y} - \frac{1}{x - 1}\right)} \]

       9. sub-neg 100.0%

          \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\frac{-\left(\frac{x}{{\left(x + -1\right)}^{2}} - \frac{1}{{\left(x + -1\right)}^{2}}\right)}{y} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right)} \]

       10. metadata-eval 100.0%

           \[\leadsto x - \frac{1}{\left(-y\right) \cdot \left(\frac{-\left(\frac{x}{{\left(x + -1\right)}^{2}} - \frac{1}{{\left(x + -1\right)}^{2}}\right)}{y} - \frac{1}{x + \color{blue}{-1}}\right)} \]

   14. Simplified 100.0%

       \[\leadsto x - \frac{1}{\color{blue}{\left(-y\right) \cdot \left(\frac{-\left(\frac{x}{{\left(x + -1\right)}^{2}} - \frac{1}{{\left(x + -1\right)}^{2}}\right)}{y} - \frac{1}{x + -1}\right)}} \]

   if -4500 < y < 4.3000000000000002e-5

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4500 \lor \neg \left(y \leq 4.3 \cdot 10^{-5}\right):\\ \;\;\;\;x + \frac{1}{y \cdot \left(\frac{\frac{1}{{\left(x + -1\right)}^{2}} - \frac{x}{{\left(x + -1\right)}^{2}}}{y} + \frac{-1}{x + -1}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.9999 \lor \neg \left(t\_0 \leq 1.0001\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{y}{-1 + \frac{1}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0))))
   (if (or (<= t_0 0.9999) (not (<= t_0 1.0001)))
     (- 1.0 (* (- 1.0 x) (/ y (+ y 1.0))))
     (+ x (/ -1.0 (/ y (+ -1.0 (/ 1.0 y))))))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if ((t_0 <= 0.9999) || !(t_0 <= 1.0001)) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (-1.0 / (y / (-1.0 + (1.0 / y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (1.0d0 - x)) / (y + 1.0d0)
    if ((t_0 <= 0.9999d0) .or. (.not. (t_0 <= 1.0001d0))) then
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (y + 1.0d0)))
    else
        tmp = x + ((-1.0d0) / (y / ((-1.0d0) + (1.0d0 / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if ((t_0 <= 0.9999) || !(t_0 <= 1.0001)) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (-1.0 / (y / (-1.0 + (1.0 / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (y + 1.0)
	tmp = 0
	if (t_0 <= 0.9999) or not (t_0 <= 1.0001):
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)))
	else:
		tmp = x + (-1.0 / (y / (-1.0 + (1.0 / y))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	tmp = 0.0
	if ((t_0 <= 0.9999) || !(t_0 <= 1.0001))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(y / Float64(-1.0 + Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (y + 1.0);
	tmp = 0.0;
	if ((t_0 <= 0.9999) || ~((t_0 <= 1.0001)))
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	else
		tmp = x + (-1.0 / (y / (-1.0 + (1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.9999], N[Not[LessEqual[t$95$0, 1.0001]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(y / N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.99990000000000001 or 1.00009999999999999 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 85.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   if 0.99990000000000001 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.00009999999999999

   1. Initial program 6.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 6.4%

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

      2. remove-double-neg 6.4%

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

      3. remove-double-neg 6.4%

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

      4. +-commutative 6.4%

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

   3. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 99.9%

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{\left(-1 + x\right) + \frac{1 - x}{y}}{y}} \]
   8. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

         \[\leadsto x - \color{blue}{{\left(\frac{y}{\left(-1 + x\right) + \frac{1 - x}{y}}\right)}^{-1}} \]

      3. associate-+l+ 100.0%

         \[\leadsto x - {\left(\frac{y}{\color{blue}{-1 + \left(x + \frac{1 - x}{y}\right)}}\right)}^{-1} \]

   9. Applied egg-rr 100.0%

      \[\leadsto x - \color{blue}{{\left(\frac{y}{-1 + \left(x + \frac{1 - x}{y}\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 0.9999 \lor \neg \left(\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 1.0001\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{y}{-1 + \frac{1}{y}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -7000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-85}:\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -7000.0)
     t_0
     (if (<= y -1.22e-85)
       (* x (/ y (+ y 1.0)))
       (if (<= y 0.92) (- 1.0 y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -7000.0) {
		tmp = t_0;
	} else if (y <= -1.22e-85) {
		tmp = x * (y / (y + 1.0));
	} else if (y <= 0.92) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) / y)
    if (y <= (-7000.0d0)) then
        tmp = t_0
    else if (y <= (-1.22d-85)) then
        tmp = x * (y / (y + 1.0d0))
    else if (y <= 0.92d0) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -7000.0) {
		tmp = t_0;
	} else if (y <= -1.22e-85) {
		tmp = x * (y / (y + 1.0));
	} else if (y <= 0.92) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) / y)
	tmp = 0
	if y <= -7000.0:
		tmp = t_0
	elif y <= -1.22e-85:
		tmp = x * (y / (y + 1.0))
	elif y <= 0.92:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -7000.0)
		tmp = t_0;
	elseif (y <= -1.22e-85)
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	elseif (y <= 0.92)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -7000.0)
		tmp = t_0;
	elseif (y <= -1.22e-85)
		tmp = x * (y / (y + 1.0));
	elseif (y <= 0.92)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7000.0], t$95$0, If[LessEqual[y, -1.22e-85], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.92], N[(1.0 - y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7e3 or 0.92000000000000004 < y

   1. Initial program 35.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 58.0%

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

      2. remove-double-neg 58.0%

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

      3. remove-double-neg 58.0%

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

      4. +-commutative 58.0%

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

   3. Simplified 58.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 98.1%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. neg-sub0 98.1%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 98.1%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 98.1%

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

      6. +-commutative 98.1%

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

      7. sub-neg 98.1%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   if -7e3 < y < -1.22000000000000006e-85

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 64.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   6. Step-by-step derivation

      1. associate-/l* 64.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{1 + y}} \]

      2. +-commutative 64.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{y + 1}} \]

   7. Simplified 64.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{y + 1}} \]

   if -1.22000000000000006e-85 < y < 0.92000000000000004

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r/ 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. add-cube-cbrt 99.5%

         \[\leadsto \left(-\left(1 - x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right) \cdot \sqrt[3]{\frac{y}{1 + y}}\right)} + 1 \]

      9. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right)\right) \cdot \sqrt[3]{\frac{y}{1 + y}}} + 1 \]

      10. fma-define 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right), \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

      11. pow2 99.6%

          \[\leadsto \mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}, \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

   7. Taylor expanded in x around 0 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 75.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      2. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{\color{blue}{y \cdot y}}{{\left(1 + y\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      3. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{y \cdot y}{\color{blue}{\left(1 + y\right) \cdot \left(1 + y\right)}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      4. times-frac 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{\frac{y}{1 + y} \cdot \frac{y}{1 + y}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      5. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   9. Simplified 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   10. Taylor expanded in y around 0 75.2%

       \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   11. Step-by-step derivation

       1. neg-mul-1 75.2%

          \[\leadsto 1 + \color{blue}{\left(-y\right)} \]

       2. unsub-neg 75.2%

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

   12. Simplified 75.2%

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

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -380000:\\ \;\;\;\;x - \frac{1}{\frac{y}{-1 - \left(\frac{x + -1}{y} - x\right)}}\\ \mathbf{elif}\;y \leq 580000000000:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -380000.0)
   (- x (/ 1.0 (/ y (- -1.0 (- (/ (+ x -1.0) y) x)))))
   (if (<= y 580000000000.0)
     (- 1.0 (* (- 1.0 x) (/ y (+ y 1.0))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -380000.0) {
		tmp = x - (1.0 / (y / (-1.0 - (((x + -1.0) / y) - x))));
	} else if (y <= 580000000000.0) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-380000.0d0)) then
        tmp = x - (1.0d0 / (y / ((-1.0d0) - (((x + (-1.0d0)) / y) - x))))
    else if (y <= 580000000000.0d0) then
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (y + 1.0d0)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -380000.0) {
		tmp = x - (1.0 / (y / (-1.0 - (((x + -1.0) / y) - x))));
	} else if (y <= 580000000000.0) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -380000.0:
		tmp = x - (1.0 / (y / (-1.0 - (((x + -1.0) / y) - x))))
	elif y <= 580000000000.0:
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -380000.0)
		tmp = Float64(x - Float64(1.0 / Float64(y / Float64(-1.0 - Float64(Float64(Float64(x + -1.0) / y) - x)))));
	elseif (y <= 580000000000.0)
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -380000.0)
		tmp = x - (1.0 / (y / (-1.0 - (((x + -1.0) / y) - x))));
	elseif (y <= 580000000000.0)
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -380000.0], N[(x - N[(1.0 / N[(y / N[(-1.0 - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 580000000000.0], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e5

   1. Initial program 33.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 60.5%

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

      2. remove-double-neg 60.5%

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

      3. remove-double-neg 60.5%

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

      4. +-commutative 60.5%

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

   3. Simplified 60.5%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x - \frac{\left(-1 + x\right) + \frac{1 - x}{y}}{y}} \]
   8. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

         \[\leadsto x - \color{blue}{{\left(\frac{y}{\left(-1 + x\right) + \frac{1 - x}{y}}\right)}^{-1}} \]

      3. associate-+l+ 100.0%

         \[\leadsto x - {\left(\frac{y}{\color{blue}{-1 + \left(x + \frac{1 - x}{y}\right)}}\right)}^{-1} \]

   9. Applied egg-rr 100.0%

      \[\leadsto x - \color{blue}{{\left(\frac{y}{-1 + \left(x + \frac{1 - x}{y}\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

   if -3.8e5 < y < 5.8e11

   1. Initial program 99.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

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

      2. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. +-commutative 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   if 5.8e11 < y

   1. Initial program 35.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 53.1%

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

      2. remove-double-neg 53.1%

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

      3. remove-double-neg 53.1%

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

      4. +-commutative 53.1%

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

   3. Simplified 53.1%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 100.0%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -380000:\\ \;\;\;\;x - \frac{1}{\frac{y}{-1 - \left(\frac{x + -1}{y} - x\right)}}\\ \mathbf{elif}\;y \leq 580000000000:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -410000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{elif}\;y \leq 1060000000000:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -410000.0)
   (+ x (/ (+ (- 1.0 x) (/ (+ x -1.0) y)) y))
   (if (<= y 1060000000000.0)
     (- 1.0 (* (- 1.0 x) (/ y (+ y 1.0))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -410000.0) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	} else if (y <= 1060000000000.0) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-410000.0d0)) then
        tmp = x + (((1.0d0 - x) + ((x + (-1.0d0)) / y)) / y)
    else if (y <= 1060000000000.0d0) then
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (y + 1.0d0)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -410000.0) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	} else if (y <= 1060000000000.0) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -410000.0:
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y)
	elif y <= 1060000000000.0:
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -410000.0)
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + -1.0) / y)) / y));
	elseif (y <= 1060000000000.0)
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -410000.0)
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	elseif (y <= 1060000000000.0)
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -410000.0], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1060000000000.0], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1e5

   1. Initial program 33.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 60.5%

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

      2. remove-double-neg 60.5%

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

      3. remove-double-neg 60.5%

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

      4. +-commutative 60.5%

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

   3. Simplified 60.5%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 100.0%

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

   7. Simplified 100.0%

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

   if -4.1e5 < y < 1.06e12

   1. Initial program 99.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

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

      2. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. +-commutative 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   if 1.06e12 < y

   1. Initial program 35.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 53.1%

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

      2. remove-double-neg 53.1%

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

      3. remove-double-neg 53.1%

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

      4. +-commutative 53.1%

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

   3. Simplified 53.1%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 100.0%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -410000:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{elif}\;y \leq 1060000000000:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260000000000:\\ \;\;\;\;x + \frac{x \cdot \left(-1 + \frac{1}{x}\right)}{y}\\ \mathbf{elif}\;y \leq 11600000000:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -260000000000.0)
   (+ x (/ (* x (+ -1.0 (/ 1.0 x))) y))
   (if (<= y 11600000000.0)
     (- 1.0 (* (- 1.0 x) (/ y (+ y 1.0))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -260000000000.0) {
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y);
	} else if (y <= 11600000000.0) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-260000000000.0d0)) then
        tmp = x + ((x * ((-1.0d0) + (1.0d0 / x))) / y)
    else if (y <= 11600000000.0d0) then
        tmp = 1.0d0 - ((1.0d0 - x) * (y / (y + 1.0d0)))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -260000000000.0) {
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y);
	} else if (y <= 11600000000.0) {
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -260000000000.0:
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y)
	elif y <= 11600000000.0:
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -260000000000.0)
		tmp = Float64(x + Float64(Float64(x * Float64(-1.0 + Float64(1.0 / x))) / y));
	elseif (y <= 11600000000.0)
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) * Float64(y / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -260000000000.0)
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y);
	elseif (y <= 11600000000.0)
		tmp = 1.0 - ((1.0 - x) * (y / (y + 1.0)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -260000000000.0], N[(x + N[(N[(x * N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 11600000000.0], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e11

   1. Initial program 31.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 59.8%

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

      2. remove-double-neg 59.8%

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

      3. remove-double-neg 59.8%

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

      4. +-commutative 59.8%

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

   3. Simplified 59.8%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 100.0%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   8. Taylor expanded in x around inf 100.0%

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

   if -2.6e11 < y < 1.16e10

   1. Initial program 99.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.4%

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

      2. remove-double-neg 99.4%

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

      3. remove-double-neg 99.4%

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

      4. +-commutative 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   if 1.16e10 < y

   1. Initial program 35.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 53.1%

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

      2. remove-double-neg 53.1%

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

      3. remove-double-neg 53.1%

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

      4. +-commutative 53.1%

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

   3. Simplified 53.1%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 100.0%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260000000000:\\ \;\;\;\;x + \frac{x \cdot \left(-1 + \frac{1}{x}\right)}{y}\\ \mathbf{elif}\;y \leq 11600000000:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -7200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;y \leq 0.0115:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -7200.0)
     t_0
     (if (<= y -2.05e-82)
       (* x (/ y (+ y 1.0)))
       (if (<= y 0.0115) (- 1.0 y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -7200.0) {
		tmp = t_0;
	} else if (y <= -2.05e-82) {
		tmp = x * (y / (y + 1.0));
	} else if (y <= 0.0115) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (1.0d0 / y)
    if (y <= (-7200.0d0)) then
        tmp = t_0
    else if (y <= (-2.05d-82)) then
        tmp = x * (y / (y + 1.0d0))
    else if (y <= 0.0115d0) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -7200.0) {
		tmp = t_0;
	} else if (y <= -2.05e-82) {
		tmp = x * (y / (y + 1.0));
	} else if (y <= 0.0115) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (1.0 / y)
	tmp = 0
	if y <= -7200.0:
		tmp = t_0
	elif y <= -2.05e-82:
		tmp = x * (y / (y + 1.0))
	elif y <= 0.0115:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 / y))
	tmp = 0.0
	if (y <= -7200.0)
		tmp = t_0;
	elseif (y <= -2.05e-82)
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	elseif (y <= 0.0115)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (1.0 / y);
	tmp = 0.0;
	if (y <= -7200.0)
		tmp = t_0;
	elseif (y <= -2.05e-82)
		tmp = x * (y / (y + 1.0));
	elseif (y <= 0.0115)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7200.0], t$95$0, If[LessEqual[y, -2.05e-82], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0115], N[(1.0 - y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7200 or 0.0115 < y

   1. Initial program 35.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 58.0%

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

      2. remove-double-neg 58.0%

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

      3. remove-double-neg 58.0%

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

      4. +-commutative 58.0%

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

   3. Simplified 58.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 98.1%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. neg-sub0 98.1%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 98.1%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 98.1%

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

      6. +-commutative 98.1%

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

      7. sub-neg 98.1%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   8. Taylor expanded in x around 0 97.4%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]

   if -7200 < y < -2.04999999999999998e-82

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 64.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   6. Step-by-step derivation

      1. associate-/l* 64.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{1 + y}} \]

      2. +-commutative 64.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{y + 1}} \]

   7. Simplified 64.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{y + 1}} \]

   if -2.04999999999999998e-82 < y < 0.0115

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r/ 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. add-cube-cbrt 99.5%

         \[\leadsto \left(-\left(1 - x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right) \cdot \sqrt[3]{\frac{y}{1 + y}}\right)} + 1 \]

      9. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right)\right) \cdot \sqrt[3]{\frac{y}{1 + y}}} + 1 \]

      10. fma-define 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right), \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

      11. pow2 99.6%

          \[\leadsto \mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}, \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

   7. Taylor expanded in x around 0 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 75.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      2. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{\color{blue}{y \cdot y}}{{\left(1 + y\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      3. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{y \cdot y}{\color{blue}{\left(1 + y\right) \cdot \left(1 + y\right)}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      4. times-frac 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{\frac{y}{1 + y} \cdot \frac{y}{1 + y}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      5. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   9. Simplified 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   10. Taylor expanded in y around 0 75.2%

       \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   11. Step-by-step derivation

       1. neg-mul-1 75.2%

          \[\leadsto 1 + \color{blue}{\left(-y\right)} \]

       2. unsub-neg 75.2%

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

   12. Simplified 75.2%

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

Alternative 8: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-80}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.065:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y -9.2e-80) (* y x) (if (<= y 0.065) (- 1.0 y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -9.2e-80) {
		tmp = y * x;
	} else if (y <= 0.065) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (1.0d0 / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= (-9.2d-80)) then
        tmp = y * x
    else if (y <= 0.065d0) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= -9.2e-80) {
		tmp = y * x;
	} else if (y <= 0.065) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (1.0 / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= -9.2e-80:
		tmp = y * x
	elif y <= 0.065:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= -9.2e-80)
		tmp = Float64(y * x);
	elseif (y <= 0.065)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (1.0 / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= -9.2e-80)
		tmp = y * x;
	elseif (y <= 0.065)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, -9.2e-80], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.065], N[(1.0 - y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.065000000000000002 < y

   1. Initial program 35.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 58.0%

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

      2. remove-double-neg 58.0%

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

      3. remove-double-neg 58.0%

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

      4. +-commutative 58.0%

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

   3. Simplified 58.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 98.1%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. neg-sub0 98.1%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 98.1%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 98.1%

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

      6. +-commutative 98.1%

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

      7. sub-neg 98.1%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   8. Taylor expanded in x around 0 97.4%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]

   if -1 < y < -9.1999999999999993e-80

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 64.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   6. Step-by-step derivation

      1. associate-/l* 64.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{1 + y}} \]

      2. +-commutative 64.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{y + 1}} \]

   7. Simplified 64.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{y + 1}} \]

   8. Taylor expanded in y around 0 61.5%

      \[\leadsto \color{blue}{x \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative 61.5%

         \[\leadsto \color{blue}{y \cdot x} \]

   10. Simplified 61.5%

       \[\leadsto \color{blue}{y \cdot x} \]

   if -9.1999999999999993e-80 < y < 0.065000000000000002

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r/ 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. add-cube-cbrt 99.5%

         \[\leadsto \left(-\left(1 - x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right) \cdot \sqrt[3]{\frac{y}{1 + y}}\right)} + 1 \]

      9. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right)\right) \cdot \sqrt[3]{\frac{y}{1 + y}}} + 1 \]

      10. fma-define 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right), \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

      11. pow2 99.6%

          \[\leadsto \mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}, \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

   7. Taylor expanded in x around 0 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 75.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      2. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{\color{blue}{y \cdot y}}{{\left(1 + y\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      3. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{y \cdot y}{\color{blue}{\left(1 + y\right) \cdot \left(1 + y\right)}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      4. times-frac 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{\frac{y}{1 + y} \cdot \frac{y}{1 + y}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      5. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   9. Simplified 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   10. Taylor expanded in y around 0 75.2%

       \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   11. Step-by-step derivation

       1. neg-mul-1 75.2%

          \[\leadsto 1 + \color{blue}{\left(-y\right)} \]

       2. unsub-neg 75.2%

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

   12. Simplified 75.2%

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

Alternative 9: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{x \cdot \left(-1 + \frac{1}{x}\right)}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ x (/ (* x (+ -1.0 (/ 1.0 x))) y))
   (if (<= y 1.0) (+ 1.0 (* y (+ x -1.0))) (- (+ x (/ 1.0 y)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y);
	} else if (y <= 1.0) {
		tmp = 1.0 + (y * (x + -1.0));
	} else {
		tmp = (x + (1.0 / y)) - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x + ((x * ((-1.0d0) + (1.0d0 / x))) / y)
    else if (y <= 1.0d0) then
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    else
        tmp = (x + (1.0d0 / y)) - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y);
	} else if (y <= 1.0) {
		tmp = 1.0 + (y * (x + -1.0));
	} else {
		tmp = (x + (1.0 / y)) - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y)
	elif y <= 1.0:
		tmp = 1.0 + (y * (x + -1.0))
	else:
		tmp = (x + (1.0 / y)) - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x + Float64(Float64(x * Float64(-1.0 + Float64(1.0 / x))) / y));
	elseif (y <= 1.0)
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	else
		tmp = Float64(Float64(x + Float64(1.0 / y)) - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x + ((x * (-1.0 + (1.0 / x))) / y);
	elseif (y <= 1.0)
		tmp = 1.0 + (y * (x + -1.0));
	else
		tmp = (x + (1.0 / y)) - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x + N[(N[(x * N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 33.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 60.8%

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

      2. remove-double-neg 60.8%

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

      3. remove-double-neg 60.8%

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

      4. +-commutative 60.8%

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 98.4%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. neg-sub0 98.4%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 98.4%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 98.4%

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

      6. +-commutative 98.4%

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

      7. sub-neg 98.4%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 98.4%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   8. Taylor expanded in x around inf 98.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto 1 - \left(1 - x\right) \cdot \color{blue}{y} \]

   if 1 < y

   1. Initial program 38.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 55.0%

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

      2. remove-double-neg 55.0%

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

      3. remove-double-neg 55.0%

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

      4. +-commutative 55.0%

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

   3. Simplified 55.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 97.7%

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{x \cdot \left(-1 + \frac{1}{x}\right)}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 1.0) (+ 1.0 (* y (+ x -1.0))) (- (+ x (/ 1.0 y)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 1.0) {
		tmp = 1.0 + (y * (x + -1.0));
	} else {
		tmp = (x + (1.0 / y)) - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 1.0d0) then
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    else
        tmp = (x + (1.0d0 / y)) - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 1.0) {
		tmp = 1.0 + (y * (x + -1.0));
	} else {
		tmp = (x + (1.0 / y)) - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 1.0:
		tmp = 1.0 + (y * (x + -1.0))
	else:
		tmp = (x + (1.0 / y)) - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 1.0)
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	else
		tmp = Float64(Float64(x + Float64(1.0 / y)) - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 1.0)
		tmp = 1.0 + (y * (x + -1.0));
	else
		tmp = (x + (1.0 / y)) - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 33.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 60.8%

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

      2. remove-double-neg 60.8%

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

      3. remove-double-neg 60.8%

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

      4. +-commutative 60.8%

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 98.4%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. neg-sub0 98.4%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 98.4%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 98.4%

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

      6. +-commutative 98.4%

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

      7. sub-neg 98.4%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 98.4%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto 1 - \left(1 - x\right) \cdot \color{blue}{y} \]

   if 1 < y

   1. Initial program 38.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 55.0%

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

      2. remove-double-neg 55.0%

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

      3. remove-double-neg 55.0%

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

      4. +-commutative 55.0%

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

   3. Simplified 55.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 97.7%

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1}{y}\right) - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7500:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-85}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.039:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7500.0)
   x
   (if (<= y -3e-85) (* y x) (if (<= y 0.039) (- 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7500.0) {
		tmp = x;
	} else if (y <= -3e-85) {
		tmp = y * x;
	} else if (y <= 0.039) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7500.0d0)) then
        tmp = x
    else if (y <= (-3d-85)) then
        tmp = y * x
    else if (y <= 0.039d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7500.0) {
		tmp = x;
	} else if (y <= -3e-85) {
		tmp = y * x;
	} else if (y <= 0.039) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7500.0:
		tmp = x
	elif y <= -3e-85:
		tmp = y * x
	elif y <= 0.039:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7500.0)
		tmp = x;
	elseif (y <= -3e-85)
		tmp = Float64(y * x);
	elseif (y <= 0.039)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7500.0)
		tmp = x;
	elseif (y <= -3e-85)
		tmp = y * x;
	elseif (y <= 0.039)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7500.0], x, If[LessEqual[y, -3e-85], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.039], N[(1.0 - y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -7500 or 0.0389999999999999999 < y

   1. Initial program 35.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 57.8%

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

      2. remove-double-neg 57.8%

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

      3. remove-double-neg 57.8%

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

      4. +-commutative 57.8%

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

   3. Simplified 57.8%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 77.9%

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

   if -7500 < y < -3.00000000000000022e-85

   1. Initial program 99.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.2%

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

      2. remove-double-neg 99.2%

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

      3. remove-double-neg 99.2%

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

      4. +-commutative 99.2%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 62.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   6. Step-by-step derivation

      1. associate-/l* 62.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{1 + y}} \]

      2. +-commutative 62.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{y + 1}} \]

   7. Simplified 62.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{y + 1}} \]

   8. Taylor expanded in y around 0 59.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto \color{blue}{y \cdot x} \]

   10. Simplified 59.1%

       \[\leadsto \color{blue}{y \cdot x} \]

   if -3.00000000000000022e-85 < y < 0.0389999999999999999

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r/ 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. add-cube-cbrt 99.5%

         \[\leadsto \left(-\left(1 - x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right) \cdot \sqrt[3]{\frac{y}{1 + y}}\right)} + 1 \]

      9. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right)\right) \cdot \sqrt[3]{\frac{y}{1 + y}}} + 1 \]

      10. fma-define 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right), \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

      11. pow2 99.6%

          \[\leadsto \mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}, \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

   7. Taylor expanded in x around 0 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 75.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{\frac{{y}^{2}}{{\left(1 + y\right)}^{2}}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      2. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{\color{blue}{y \cdot y}}{{\left(1 + y\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      3. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\frac{y \cdot y}{\color{blue}{\left(1 + y\right) \cdot \left(1 + y\right)}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      4. times-frac 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{\frac{y}{1 + y} \cdot \frac{y}{1 + y}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

      5. unpow2 75.5%

         \[\leadsto \mathsf{fma}\left(-\sqrt[3]{\color{blue}{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   9. Simplified 75.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{-\sqrt[3]{{\left(\frac{y}{1 + y}\right)}^{2}}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   10. Taylor expanded in y around 0 75.2%

       \[\leadsto \color{blue}{1 + -1 \cdot y} \]
   11. Step-by-step derivation

       1. neg-mul-1 75.2%

          \[\leadsto 1 + \color{blue}{\left(-y\right)} \]

       2. unsub-neg 75.2%

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

   12. Simplified 75.2%

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

Alternative 12: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 35.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 58.0%

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

      2. remove-double-neg 58.0%

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

      3. remove-double-neg 58.0%

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

      4. +-commutative 58.0%

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

   3. Simplified 58.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around -inf 98.1%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{x - 1}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. neg-sub0 98.1%

         \[\leadsto x + \frac{\color{blue}{0 - \left(x - 1\right)}}{y} \]

      4. associate-+l- 98.1%

         \[\leadsto x + \frac{\color{blue}{\left(0 - x\right) + 1}}{y} \]

      5. neg-sub0 98.1%

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

      6. +-commutative 98.1%

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

      7. sub-neg 98.1%

         \[\leadsto x + \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{x + \frac{1 - x}{y}} \]

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.3%

      \[\leadsto 1 - \left(1 - x\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7500:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-82}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 10000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7500.0) x (if (<= y -1.85e-82) (* y x) (if (<= y 10000.0) 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7500.0) {
		tmp = x;
	} else if (y <= -1.85e-82) {
		tmp = y * x;
	} else if (y <= 10000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7500.0d0)) then
        tmp = x
    else if (y <= (-1.85d-82)) then
        tmp = y * x
    else if (y <= 10000.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7500.0) {
		tmp = x;
	} else if (y <= -1.85e-82) {
		tmp = y * x;
	} else if (y <= 10000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7500.0:
		tmp = x
	elif y <= -1.85e-82:
		tmp = y * x
	elif y <= 10000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7500.0)
		tmp = x;
	elseif (y <= -1.85e-82)
		tmp = Float64(y * x);
	elseif (y <= 10000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7500.0)
		tmp = x;
	elseif (y <= -1.85e-82)
		tmp = y * x;
	elseif (y <= 10000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7500.0], x, If[LessEqual[y, -1.85e-82], N[(y * x), $MachinePrecision], If[LessEqual[y, 10000.0], 1.0, x]]]

Derivation

1. Split input into 3 regimes

2. if y < -7500 or 1e4 < y

   1. Initial program 34.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 57.3%

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

      2. remove-double-neg 57.3%

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

      3. remove-double-neg 57.3%

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

      4. +-commutative 57.3%

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

   3. Simplified 57.3%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 79.1%

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

   if -7500 < y < -1.85e-82

   1. Initial program 99.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.2%

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

      2. remove-double-neg 99.2%

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

      3. remove-double-neg 99.2%

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

      4. +-commutative 99.2%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 62.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + y}} \]
   6. Step-by-step derivation

      1. associate-/l* 62.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{1 + y}} \]

      2. +-commutative 62.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{y + 1}} \]

   7. Simplified 62.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{y + 1}} \]

   8. Taylor expanded in y around 0 59.1%

      \[\leadsto \color{blue}{x \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto \color{blue}{y \cdot x} \]

   10. Simplified 59.1%

       \[\leadsto \color{blue}{y \cdot x} \]

   if -1.85e-82 < y < 1e4

   1. Initial program 99.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-*r/ 99.8%

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

      7. distribute-lft-neg-in 99.8%

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

      8. add-cube-cbrt 99.3%

         \[\leadsto \left(-\left(1 - x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right) \cdot \sqrt[3]{\frac{y}{1 + y}}\right)} + 1 \]

      9. associate-*r* 99.3%

         \[\leadsto \color{blue}{\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right)\right) \cdot \sqrt[3]{\frac{y}{1 + y}}} + 1 \]

      10. fma-define 99.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right), \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

      11. pow2 99.3%

          \[\leadsto \mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}, \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

   7. Taylor expanded in y around 0 73.8%

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

Alternative 14: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 10000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 10000.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 10000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 10000.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 10000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 10000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 10000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 10000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 10000.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1e4 < y

   1. Initial program 35.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 57.5%

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

      2. remove-double-neg 57.5%

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

      3. remove-double-neg 57.5%

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

      4. +-commutative 57.5%

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

   3. Simplified 57.5%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 78.5%

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

   if -1 < y < 1e4

   1. Initial program 99.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-*r/ 99.8%

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

      7. distribute-lft-neg-in 99.8%

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

      8. add-cube-cbrt 99.1%

         \[\leadsto \left(-\left(1 - x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right) \cdot \sqrt[3]{\frac{y}{1 + y}}\right)} + 1 \]

      9. associate-*r* 99.1%

         \[\leadsto \color{blue}{\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right)\right) \cdot \sqrt[3]{\frac{y}{1 + y}}} + 1 \]

      10. fma-define 99.2%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right), \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

      11. pow2 99.2%

          \[\leadsto \mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}, \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

   7. Taylor expanded in y around 0 67.8%

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

Alternative 15: 38.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 68.4%

   \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
2. Step-by-step derivation

   1. associate-/l* 79.3%

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

   2. remove-double-neg 79.3%

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

   3. remove-double-neg 79.3%

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

   4. +-commutative 79.3%

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

3. Simplified 79.3%

   \[\leadsto \color{blue}{1 - \left(1 - x\right) \cdot \frac{y}{1 + y}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sub-neg 79.3%

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

   2. associate-*r/ 68.4%

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

   3. +-commutative 68.4%

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

   4. +-commutative 68.4%

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

   5. +-commutative 68.4%

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

   6. associate-*r/ 79.3%

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

   7. distribute-lft-neg-in 79.3%

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

   8. add-cube-cbrt 78.9%

      \[\leadsto \left(-\left(1 - x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right) \cdot \sqrt[3]{\frac{y}{1 + y}}\right)} + 1 \]

   9. associate-*r* 78.9%

      \[\leadsto \color{blue}{\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right)\right) \cdot \sqrt[3]{\frac{y}{1 + y}}} + 1 \]

   10. fma-define 79.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \left(\sqrt[3]{\frac{y}{1 + y}} \cdot \sqrt[3]{\frac{y}{1 + y}}\right), \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

   11. pow2 79.0%

       \[\leadsto \mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}}, \sqrt[3]{\frac{y}{1 + y}}, 1\right) \]

6. Applied egg-rr 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-\left(1 - x\right)\right) \cdot {\left(\sqrt[3]{\frac{y}{1 + y}}\right)}^{2}, \sqrt[3]{\frac{y}{1 + y}}, 1\right)} \]

7. Taylor expanded in y around 0 36.7%

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

Developer Target 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024181 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ 1 y) (- (/ x y) x)) (if (< y 679931050341891/100000) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x)))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))