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

Percentage Accurate: 66.7% → 99.8%

Time: 7.1s

Alternatives: 11

Speedup: 2.1×

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: 66.7% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -250000:\\ \;\;\;\;\left(x + \frac{x + -1}{{y}^{2}}\right) + t_0\\ \mathbf{elif}\;y \leq 245000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -250000.0)
     (+ (+ x (/ (+ x -1.0) (pow y 2.0))) t_0)
     (if (<= y 245000000.0) (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0) (+ x t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -250000.0) {
		tmp = (x + ((x + -1.0) / pow(y, 2.0))) + t_0;
	} else if (y <= 245000000.0) {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -250000.0)
		tmp = Float64(Float64(x + Float64(Float64(x + -1.0) / (y ^ 2.0))) + t_0);
	elseif (y <= 245000000.0)
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -250000.0], N[(N[(x + N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[y, 245000000.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e5

   1. Initial program 36.0%

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

      1. associate-*l/ 56.4%

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

      2. +-commutative 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. associate-+r+ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--l+ 100.0%

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

      5. mul-1-neg 100.0%

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

      6. unsub-neg 100.0%

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

      7. div-sub 100.0%

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

   6. Simplified 100.0%

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

   if -2.5e5 < y < 2.45e8

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. fma-def 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. neg-sub0 100.0%

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

      8. associate--r- 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 2.45e8 < y

   1. Initial program 31.6%

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

      1. associate-*l/ 48.3%

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

      2. +-commutative 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -250000:\\ \;\;\;\;\left(x + \frac{x + -1}{{y}^{2}}\right) + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 245000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -130000000 \lor \neg \left(y \leq 165000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -130000000.0) (not (<= y 165000000.0)))
   (+ x (/ (- 1.0 x) y))
   (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -130000000.0) || !(y <= 165000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -130000000.0) || !(y <= 165000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -130000000.0], N[Not[LessEqual[y, 165000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e8 or 1.65e8 < y

   1. Initial program 33.7%

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

      1. associate-*l/ 52.1%

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

      2. +-commutative 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. div-sub 99.9%

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

   6. Simplified 99.9%

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

   if -1.3e8 < y < 1.65e8

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. fma-def 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. neg-sub0 100.0%

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

      8. associate--r- 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130000000 \lor \neg \left(y \leq 165000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \]

Alternative 3: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -6500000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-46}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-115}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-92}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e+132)
   x
   (if (<= y -2.15e+30)
     (/ 1.0 y)
     (if (<= y -6500000000000.0)
       x
       (if (<= y -1.32e-46)
         (- 1.0 y)
         (if (<= y -1.55e-115)
           (* y x)
           (if (<= y 3.65e-127)
             1.0
             (if (<= y 5.8e-92)
               (* y x)
               (if (<= y 0.75)
                 (- 1.0 y)
                 (if (<= y 1.2e+68) (/ 1.0 y) x))))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e+132) {
		tmp = x;
	} else if (y <= -2.15e+30) {
		tmp = 1.0 / y;
	} else if (y <= -6500000000000.0) {
		tmp = x;
	} else if (y <= -1.32e-46) {
		tmp = 1.0 - y;
	} else if (y <= -1.55e-115) {
		tmp = y * x;
	} else if (y <= 3.65e-127) {
		tmp = 1.0;
	} else if (y <= 5.8e-92) {
		tmp = y * x;
	} else if (y <= 0.75) {
		tmp = 1.0 - y;
	} else if (y <= 1.2e+68) {
		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 <= (-4.2d+132)) then
        tmp = x
    else if (y <= (-2.15d+30)) then
        tmp = 1.0d0 / y
    else if (y <= (-6500000000000.0d0)) then
        tmp = x
    else if (y <= (-1.32d-46)) then
        tmp = 1.0d0 - y
    else if (y <= (-1.55d-115)) then
        tmp = y * x
    else if (y <= 3.65d-127) then
        tmp = 1.0d0
    else if (y <= 5.8d-92) then
        tmp = y * x
    else if (y <= 0.75d0) then
        tmp = 1.0d0 - y
    else if (y <= 1.2d+68) 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 <= -4.2e+132) {
		tmp = x;
	} else if (y <= -2.15e+30) {
		tmp = 1.0 / y;
	} else if (y <= -6500000000000.0) {
		tmp = x;
	} else if (y <= -1.32e-46) {
		tmp = 1.0 - y;
	} else if (y <= -1.55e-115) {
		tmp = y * x;
	} else if (y <= 3.65e-127) {
		tmp = 1.0;
	} else if (y <= 5.8e-92) {
		tmp = y * x;
	} else if (y <= 0.75) {
		tmp = 1.0 - y;
	} else if (y <= 1.2e+68) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e+132:
		tmp = x
	elif y <= -2.15e+30:
		tmp = 1.0 / y
	elif y <= -6500000000000.0:
		tmp = x
	elif y <= -1.32e-46:
		tmp = 1.0 - y
	elif y <= -1.55e-115:
		tmp = y * x
	elif y <= 3.65e-127:
		tmp = 1.0
	elif y <= 5.8e-92:
		tmp = y * x
	elif y <= 0.75:
		tmp = 1.0 - y
	elif y <= 1.2e+68:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e+132)
		tmp = x;
	elseif (y <= -2.15e+30)
		tmp = Float64(1.0 / y);
	elseif (y <= -6500000000000.0)
		tmp = x;
	elseif (y <= -1.32e-46)
		tmp = Float64(1.0 - y);
	elseif (y <= -1.55e-115)
		tmp = Float64(y * x);
	elseif (y <= 3.65e-127)
		tmp = 1.0;
	elseif (y <= 5.8e-92)
		tmp = Float64(y * x);
	elseif (y <= 0.75)
		tmp = Float64(1.0 - y);
	elseif (y <= 1.2e+68)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e+132)
		tmp = x;
	elseif (y <= -2.15e+30)
		tmp = 1.0 / y;
	elseif (y <= -6500000000000.0)
		tmp = x;
	elseif (y <= -1.32e-46)
		tmp = 1.0 - y;
	elseif (y <= -1.55e-115)
		tmp = y * x;
	elseif (y <= 3.65e-127)
		tmp = 1.0;
	elseif (y <= 5.8e-92)
		tmp = y * x;
	elseif (y <= 0.75)
		tmp = 1.0 - y;
	elseif (y <= 1.2e+68)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e+132], x, If[LessEqual[y, -2.15e+30], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -6500000000000.0], x, If[LessEqual[y, -1.32e-46], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, -1.55e-115], N[(y * x), $MachinePrecision], If[LessEqual[y, 3.65e-127], 1.0, If[LessEqual[y, 5.8e-92], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.75], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 1.2e+68], N[(1.0 / y), $MachinePrecision], x]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -4.19999999999999987e132 or -2.15e30 < y < -6.5e12 or 1.20000000000000004e68 < y

   1. Initial program 35.1%

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

      1. associate-*l/ 59.1%

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

      2. +-commutative 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in y around inf 84.4%

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

   if -4.19999999999999987e132 < y < -2.15e30 or 0.75 < y < 1.20000000000000004e68

   1. Initial program 31.7%

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

      1. associate-*l/ 34.2%

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

      2. +-commutative 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in x around 0 3.4%

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

   5. Taylor expanded in y around inf 67.1%

      \[\leadsto \color{blue}{\frac{1}{y} - \frac{1}{{y}^{2}}} \]

   6. Taylor expanded in y around inf 67.1%

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

   if -6.5e12 < y < -1.32000000000000001e-46 or 5.79999999999999969e-92 < y < 0.75

   1. Initial program 97.8%

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

      1. associate-*l/ 97.8%

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

      2. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in y around 0 94.4%

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

   5. Taylor expanded in x around 0 74.5%

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

   if -1.32000000000000001e-46 < y < -1.55000000000000003e-115 or 3.65000000000000017e-127 < y < 5.79999999999999969e-92

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 70.5%

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

      1. *-commutative 70.5%

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

   7. Simplified 70.5%

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

   if -1.55000000000000003e-115 < y < 3.65000000000000017e-127

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 84.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -6500000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-46}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-115}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-92}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-89}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.94:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 3.65e-127)
       (- 1.0 y)
       (if (<= y 2.2e-89) (* y x) (if (<= y 0.94) (- 1.0 y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 3.65e-127) {
		tmp = 1.0 - y;
	} else if (y <= 2.2e-89) {
		tmp = y * x;
	} else if (y <= 0.94) {
		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 <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 3.65d-127) then
        tmp = 1.0d0 - y
    else if (y <= 2.2d-89) then
        tmp = y * x
    else if (y <= 0.94d0) 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 <= -1.0) {
		tmp = t_0;
	} else if (y <= 3.65e-127) {
		tmp = 1.0 - y;
	} else if (y <= 2.2e-89) {
		tmp = y * x;
	} else if (y <= 0.94) {
		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 <= -1.0:
		tmp = t_0
	elif y <= 3.65e-127:
		tmp = 1.0 - y
	elif y <= 2.2e-89:
		tmp = y * x
	elif y <= 0.94:
		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 <= -1.0)
		tmp = t_0;
	elseif (y <= 3.65e-127)
		tmp = Float64(1.0 - y);
	elseif (y <= 2.2e-89)
		tmp = Float64(y * x);
	elseif (y <= 0.94)
		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 <= -1.0)
		tmp = t_0;
	elseif (y <= 3.65e-127)
		tmp = 1.0 - y;
	elseif (y <= 2.2e-89)
		tmp = y * x;
	elseif (y <= 0.94)
		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, -1.0], t$95$0, If[LessEqual[y, 3.65e-127], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 2.2e-89], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.94], N[(1.0 - y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.93999999999999995 < y

   1. Initial program 34.2%

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

      1. associate-*l/ 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. div-sub 99.5%

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

   6. Simplified 99.5%

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

   if -1 < y < 3.65000000000000017e-127 or 2.20000000000000012e-89 < y < 0.93999999999999995

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.5%

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

   5. Taylor expanded in x around 0 77.4%

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

   if 3.65000000000000017e-127 < y < 2.20000000000000012e-89

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-89}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.94:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 5: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -155000000 \lor \neg \left(y \leq 140000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -155000000.0) (not (<= y 140000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -155000000.0) || !(y <= 140000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 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 <= (-155000000.0d0)) .or. (.not. (y <= 140000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -155000000.0) || !(y <= 140000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -155000000.0) or not (y <= 140000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -155000000.0) || !(y <= 140000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -155000000.0) || ~((y <= 140000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -155000000.0], N[Not[LessEqual[y, 140000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e8 or 1.4e8 < y

   1. Initial program 33.7%

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

      1. associate-*l/ 52.1%

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

      2. +-commutative 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. div-sub 99.9%

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

   6. Simplified 99.9%

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

   if -1.55e8 < y < 1.4e8

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155000000 \lor \neg \left(y \leq 140000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \end{array} \]

Alternative 6: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.024:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 3.65e-127)
       (- 1.0 y)
       (if (<= y 1.32e-97) (* y x) (if (<= y 0.024) (- 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 <= 3.65e-127) {
		tmp = 1.0 - y;
	} else if (y <= 1.32e-97) {
		tmp = y * x;
	} else if (y <= 0.024) {
		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 <= 3.65d-127) then
        tmp = 1.0d0 - y
    else if (y <= 1.32d-97) then
        tmp = y * x
    else if (y <= 0.024d0) 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 <= 3.65e-127) {
		tmp = 1.0 - y;
	} else if (y <= 1.32e-97) {
		tmp = y * x;
	} else if (y <= 0.024) {
		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 <= 3.65e-127:
		tmp = 1.0 - y
	elif y <= 1.32e-97:
		tmp = y * x
	elif y <= 0.024:
		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 <= 3.65e-127)
		tmp = Float64(1.0 - y);
	elseif (y <= 1.32e-97)
		tmp = Float64(y * x);
	elseif (y <= 0.024)
		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 <= 3.65e-127)
		tmp = 1.0 - y;
	elseif (y <= 1.32e-97)
		tmp = y * x;
	elseif (y <= 0.024)
		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, 3.65e-127], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 1.32e-97], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.024], N[(1.0 - y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.024 < y

   1. Initial program 34.2%

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

      1. associate-*l/ 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. div-sub 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.1%

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

   if -1 < y < 3.65000000000000017e-127 or 1.32e-97 < y < 0.024

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.5%

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

   5. Taylor expanded in x around 0 77.4%

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

   if 3.65000000000000017e-127 < y < 1.32e-97

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.024:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 7: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-127}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.73:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 3.6e-127)
     (- 1.0 y)
     (if (<= y 1.3e-97) (* y x) (if (<= y 0.73) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 3.6e-127) {
		tmp = 1.0 - y;
	} else if (y <= 1.3e-97) {
		tmp = y * x;
	} else if (y <= 0.73) {
		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 <= (-1.0d0)) then
        tmp = x
    else if (y <= 3.6d-127) then
        tmp = 1.0d0 - y
    else if (y <= 1.3d-97) then
        tmp = y * x
    else if (y <= 0.73d0) 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 <= -1.0) {
		tmp = x;
	} else if (y <= 3.6e-127) {
		tmp = 1.0 - y;
	} else if (y <= 1.3e-97) {
		tmp = y * x;
	} else if (y <= 0.73) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 3.6e-127:
		tmp = 1.0 - y
	elif y <= 1.3e-97:
		tmp = y * x
	elif y <= 0.73:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 3.6e-127)
		tmp = Float64(1.0 - y);
	elseif (y <= 1.3e-97)
		tmp = Float64(y * x);
	elseif (y <= 0.73)
		tmp = Float64(1.0 - y);
	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 <= 3.6e-127)
		tmp = 1.0 - y;
	elseif (y <= 1.3e-97)
		tmp = y * x;
	elseif (y <= 0.73)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 3.6e-127], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 1.3e-97], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.73], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.72999999999999998 < y

   1. Initial program 34.2%

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

      1. associate-*l/ 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 70.4%

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

   if -1 < y < 3.5999999999999999e-127 or 1.30000000000000003e-97 < y < 0.72999999999999998

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.5%

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

   5. Taylor expanded in x around 0 77.4%

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

   if 3.5999999999999999e-127 < y < 1.30000000000000003e-97

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-127}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.73:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 98.4% accurate, 1.0× 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 34.2%

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

      1. associate-*l/ 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. div-sub 99.5%

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

   6. Simplified 99.5%

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

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 9: 73.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 3.65e-127)
     1.0
     (if (<= y 1.3e-97) (* y x) (if (<= y 1.05) 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 3.65e-127) {
		tmp = 1.0;
	} else if (y <= 1.3e-97) {
		tmp = y * x;
	} else if (y <= 1.05) {
		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 <= 3.65d-127) then
        tmp = 1.0d0
    else if (y <= 1.3d-97) then
        tmp = y * x
    else if (y <= 1.05d0) 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 <= 3.65e-127) {
		tmp = 1.0;
	} else if (y <= 1.3e-97) {
		tmp = y * x;
	} else if (y <= 1.05) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 3.65e-127:
		tmp = 1.0
	elif y <= 1.3e-97:
		tmp = y * x
	elif y <= 1.05:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 3.65e-127)
		tmp = 1.0;
	elseif (y <= 1.3e-97)
		tmp = Float64(y * x);
	elseif (y <= 1.05)
		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 <= 3.65e-127)
		tmp = 1.0;
	elseif (y <= 1.3e-97)
		tmp = y * x;
	elseif (y <= 1.05)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 3.65e-127], 1.0, If[LessEqual[y, 1.3e-97], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.05], 1.0, x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.05000000000000004 < y

   1. Initial program 34.2%

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

      1. associate-*l/ 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 70.4%

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

   if -1 < y < 3.65000000000000017e-127 or 1.30000000000000003e-97 < y < 1.05000000000000004

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 75.8%

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

   if 3.65000000000000017e-127 < y < 1.30000000000000003e-97

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-97}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 73.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.43:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.43) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.43) {
		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 <= 0.43d0) 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 <= 0.43) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.43:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.43)
		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 <= 0.43)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.43], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.429999999999999993 < y

   1. Initial program 34.2%

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

      1. associate-*l/ 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in y around inf 70.4%

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

   if -1 < y < 0.429999999999999993

   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%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 73.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.43:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 39.1% 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 67.1%

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

   1. associate-*l/ 76.2%

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

   2. +-commutative 76.2%

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

3. Simplified 76.2%

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

4. Taylor expanded in y around 0 38.6%

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

5. Final simplification 38.6%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 0.7× 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 2023309 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

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