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

Percentage Accurate: 66.3% → 99.9%

Time: 9.1s

Alternatives: 14

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: 66.3% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16000:\\ \;\;\;\;x + \frac{\left(\frac{\left(x + -1\right) + \frac{1 - x}{y}}{y} - -1\right) - x}{y}\\ \mathbf{elif}\;y \leq 118000:\\ \;\;\;\;1 + \left(y + -1\right) \cdot \frac{y \cdot \left(x + -1\right)}{\mathsf{fma}\left(y, y, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -16000.0)
   (+ x (/ (- (- (/ (+ (+ x -1.0) (/ (- 1.0 x) y)) y) -1.0) x) y))
   (if (<= y 118000.0)
     (+ 1.0 (* (+ y -1.0) (/ (* y (+ x -1.0)) (fma y y -1.0))))
     (+ x (/ (+ (- 1.0 x) (/ (+ x -1.0) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -16000.0) {
		tmp = x + ((((((x + -1.0) + ((1.0 - x) / y)) / y) - -1.0) - x) / y);
	} else if (y <= 118000.0) {
		tmp = 1.0 + ((y + -1.0) * ((y * (x + -1.0)) / fma(y, y, -1.0)));
	} else {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -16000.0)
		tmp = Float64(x + Float64(Float64(Float64(Float64(Float64(Float64(x + -1.0) + Float64(Float64(1.0 - x) / y)) / y) - -1.0) - x) / y));
	elseif (y <= 118000.0)
		tmp = Float64(1.0 + Float64(Float64(y + -1.0) * Float64(Float64(y * Float64(x + -1.0)) / fma(y, y, -1.0))));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + -1.0) / y)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -16000.0], N[(x + N[(N[(N[(N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 118000.0], N[(1.0 + N[(N[(y + -1.0), $MachinePrecision] * N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -16000

   1. Initial program 36.9%

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

      1. associate-/l* 55.6%

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

      2. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in y around -inf 100.0%

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

   7. Simplified 100.0%

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

   if -16000 < y < 118000

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 118000 < y

   1. Initial program 25.9%

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

      1. associate-/l* 42.0%

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

      2. +-commutative 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in y around inf 99.9%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16000:\\ \;\;\;\;x + \frac{\left(\frac{\left(x + -1\right) + \frac{1 - x}{y}}{y} - -1\right) - x}{y}\\ \mathbf{elif}\;y \leq 118000:\\ \;\;\;\;1 + \left(y + -1\right) \cdot \frac{y \cdot \left(x + -1\right)}{\mathsf{fma}\left(y, y, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.75 \cdot 10^{-61}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.7:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y -2.75e-61)
     (* y x)
     (if (<= y 3.5e-62)
       1.0
       (if (<= y 3.7) (* y x) (if (<= y 1.16e+101) (/ 1.0 y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -2.75e-61) {
		tmp = y * x;
	} else if (y <= 3.5e-62) {
		tmp = 1.0;
	} else if (y <= 3.7) {
		tmp = y * x;
	} else if (y <= 1.16e+101) {
		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 <= (-2.75d-61)) then
        tmp = y * x
    else if (y <= 3.5d-62) then
        tmp = 1.0d0
    else if (y <= 3.7d0) then
        tmp = y * x
    else if (y <= 1.16d+101) 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 <= -2.75e-61) {
		tmp = y * x;
	} else if (y <= 3.5e-62) {
		tmp = 1.0;
	} else if (y <= 3.7) {
		tmp = y * x;
	} else if (y <= 1.16e+101) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -2.75e-61:
		tmp = y * x
	elif y <= 3.5e-62:
		tmp = 1.0
	elif y <= 3.7:
		tmp = y * x
	elif y <= 1.16e+101:
		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 <= -2.75e-61)
		tmp = Float64(y * x);
	elseif (y <= 3.5e-62)
		tmp = 1.0;
	elseif (y <= 3.7)
		tmp = Float64(y * x);
	elseif (y <= 1.16e+101)
		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 <= -2.75e-61)
		tmp = y * x;
	elseif (y <= 3.5e-62)
		tmp = 1.0;
	elseif (y <= 3.7)
		tmp = y * x;
	elseif (y <= 1.16e+101)
		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, -2.75e-61], N[(y * x), $MachinePrecision], If[LessEqual[y, 3.5e-62], 1.0, If[LessEqual[y, 3.7], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.16e+101], N[(1.0 / y), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1 or 1.16e101 < y

   1. Initial program 30.1%

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

      1. associate-/l* 50.8%

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

      2. +-commutative 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in y around inf 83.2%

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

   if -1 < y < -2.7499999999999998e-61 or 3.5000000000000001e-62 < y < 3.7000000000000002

   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. +-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. Taylor expanded in x around inf 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-/l* 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in y around 0 64.4%

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

      1. *-commutative 64.4%

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

   10. Simplified 64.4%

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

   if -2.7499999999999998e-61 < y < 3.5000000000000001e-62

   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. +-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 75.4%

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

   if 3.7000000000000002 < y < 1.16e101

   1. Initial program 32.5%

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

      1. associate-/l* 35.6%

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

      2. +-commutative 35.6%

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

   3. Simplified 35.6%

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

      1. clear-num 35.4%

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

      2. un-div-inv 35.4%

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

   6. Applied egg-rr 35.4%

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

   7. Taylor expanded in y around -inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in x around 0 65.8%

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

Alternative 3: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-61}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.0051:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y -2.35e-61)
       (* y x)
       (if (<= y 4.7e-62) 1.0 (if (<= y 0.0051) (* y x) 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 <= -2.35e-61) {
		tmp = y * x;
	} else if (y <= 4.7e-62) {
		tmp = 1.0;
	} else if (y <= 0.0051) {
		tmp = y * x;
	} 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 <= (-2.35d-61)) then
        tmp = y * x
    else if (y <= 4.7d-62) then
        tmp = 1.0d0
    else if (y <= 0.0051d0) then
        tmp = y * x
    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 <= -2.35e-61) {
		tmp = y * x;
	} else if (y <= 4.7e-62) {
		tmp = 1.0;
	} else if (y <= 0.0051) {
		tmp = y * x;
	} 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 <= -2.35e-61:
		tmp = y * x
	elif y <= 4.7e-62:
		tmp = 1.0
	elif y <= 0.0051:
		tmp = y * x
	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 <= -2.35e-61)
		tmp = Float64(y * x);
	elseif (y <= 4.7e-62)
		tmp = 1.0;
	elseif (y <= 0.0051)
		tmp = Float64(y * x);
	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 <= -2.35e-61)
		tmp = y * x;
	elseif (y <= 4.7e-62)
		tmp = 1.0;
	elseif (y <= 0.0051)
		tmp = y * x;
	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, -2.35e-61], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.7e-62], 1.0, If[LessEqual[y, 0.0051], N[(y * x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.0051000000000000004 < y

   1. Initial program 30.6%

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

      1. associate-/l* 47.8%

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

      2. +-commutative 47.8%

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

   3. Simplified 47.8%

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

      1. clear-num 47.7%

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

      2. un-div-inv 47.7%

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

   6. Applied egg-rr 47.7%

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

   7. Taylor expanded in y around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in x around 0 99.3%

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

   if -1 < y < -2.3499999999999998e-61 or 4.69999999999999984e-62 < y < 0.0051000000000000004

   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. +-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. Taylor expanded in x around inf 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-/l* 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in y around 0 64.4%

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

      1. *-commutative 64.4%

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

   10. Simplified 64.4%

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

   if -2.3499999999999998e-61 < y < 4.69999999999999984e-62

   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. +-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 75.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-61}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.0051:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -16000:\\ \;\;\;\;x + \frac{\left(\frac{\left(x + -1\right) + \frac{1 - x}{y}}{y} - -1\right) - x}{y}\\ \mathbf{elif}\;y \leq 470000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -16000.0)
   (+ x (/ (- (- (/ (+ (+ x -1.0) (/ (- 1.0 x) y)) y) -1.0) x) y))
   (if (<= y 470000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))
     (+ x (/ (+ (- 1.0 x) (/ (+ x -1.0) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -16000.0) {
		tmp = x + ((((((x + -1.0) + ((1.0 - x) / y)) / y) - -1.0) - x) / y);
	} else if (y <= 470000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / 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 <= (-16000.0d0)) then
        tmp = x + ((((((x + (-1.0d0)) + ((1.0d0 - x) / y)) / y) - (-1.0d0)) - x) / y)
    else if (y <= 470000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    else
        tmp = x + (((1.0d0 - x) + ((x + (-1.0d0)) / y)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -16000.0) {
		tmp = x + ((((((x + -1.0) + ((1.0 - x) / y)) / y) - -1.0) - x) / y);
	} else if (y <= 470000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -16000.0:
		tmp = x + ((((((x + -1.0) + ((1.0 - x) / y)) / y) - -1.0) - x) / y)
	elif y <= 470000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	else:
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -16000.0)
		tmp = Float64(x + Float64(Float64(Float64(Float64(Float64(Float64(x + -1.0) + Float64(Float64(1.0 - x) / y)) / y) - -1.0) - x) / y));
	elseif (y <= 470000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + -1.0) / y)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -16000.0)
		tmp = x + ((((((x + -1.0) + ((1.0 - x) / y)) / y) - -1.0) - x) / y);
	elseif (y <= 470000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	else
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -16000.0], N[(x + N[(N[(N[(N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 470000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -16000

   1. Initial program 36.9%

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

      1. associate-/l* 55.6%

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

      2. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in y around -inf 100.0%

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

   7. Simplified 100.0%

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

   if -16000 < y < 4.7e5

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   if 4.7e5 < y

   1. Initial program 25.9%

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

      1. associate-/l* 42.0%

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

      2. +-commutative 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in y around inf 99.9%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16000:\\ \;\;\;\;x + \frac{\left(\frac{\left(x + -1\right) + \frac{1 - x}{y}}{y} - -1\right) - x}{y}\\ \mathbf{elif}\;y \leq 470000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{-61}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y -2.65e-61)
     (* y x)
     (if (<= y 4.4e-62) 1.0 (if (<= y 1.0) (* y x) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= -2.65e-61) {
		tmp = y * x;
	} else if (y <= 4.4e-62) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = y * x;
	} 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 <= (-2.65d-61)) then
        tmp = y * x
    else if (y <= 4.4d-62) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = y * x
    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 <= -2.65e-61) {
		tmp = y * x;
	} else if (y <= 4.4e-62) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= -2.65e-61:
		tmp = y * x
	elif y <= 4.4e-62:
		tmp = 1.0
	elif y <= 1.0:
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= -2.65e-61)
		tmp = Float64(y * x);
	elseif (y <= 4.4e-62)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(y * x);
	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 <= -2.65e-61)
		tmp = y * x;
	elseif (y <= 4.4e-62)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = y * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, -2.65e-61], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.4e-62], 1.0, If[LessEqual[y, 1.0], N[(y * x), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.6%

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

      1. associate-/l* 47.8%

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

      2. +-commutative 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in y around inf 74.0%

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

   if -1 < y < -2.65e-61 or 4.40000000000000035e-62 < y < 1

   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. +-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. Taylor expanded in x around inf 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-/l* 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in y around 0 64.4%

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

      1. *-commutative 64.4%

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

   10. Simplified 64.4%

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

   if -2.65e-61 < y < 4.40000000000000035e-62

   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. +-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 75.4%

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

Alternative 6: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -270000 \lor \neg \left(y \leq 330000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -270000.0) (not (<= y 330000.0)))
   (+ x (/ (+ (- 1.0 x) (/ (+ x -1.0) y)) y))
   (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -270000.0) || !(y <= 330000.0)) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / 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 <= (-270000.0d0)) .or. (.not. (y <= 330000.0d0))) then
        tmp = x + (((1.0d0 - x) + ((x + (-1.0d0)) / y)) / 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 <= -270000.0) || !(y <= 330000.0)) {
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y);
	} else {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -270000.0) or not (y <= 330000.0):
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / y)
	else:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -270000.0) || !(y <= 330000.0))
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) + Float64(Float64(x + -1.0) / y)) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -270000.0) || ~((y <= 330000.0)))
		tmp = x + (((1.0 - x) + ((x + -1.0) / y)) / 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, -270000.0], N[Not[LessEqual[y, 330000.0]], $MachinePrecision]], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e5 or 3.3e5 < y

   1. Initial program 30.6%

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

      1. associate-/l* 47.8%

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

      2. +-commutative 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in y around inf 99.8%

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

   7. Simplified 99.8%

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

   if -2.7e5 < y < 3.3e5

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000 \lor \neg \left(y \leq 330000\right):\\ \;\;\;\;x + \frac{\left(1 - x\right) + \frac{x + -1}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -135000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 7400000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -135000000.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 7400000000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -135000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 7400000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (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 <= (-135000000.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 7400000000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (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 <= -135000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 7400000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -135000000.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 7400000000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -135000000.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 7400000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / 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 <= -135000000.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 7400000000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -135000000.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7400000000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e8

   1. Initial program 36.3%

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

      1. associate-/l* 55.3%

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

      2. +-commutative 55.3%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
   6. 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%

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

   7. Simplified 99.9%

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

   if -1.35e8 < y < 7.4e9

   1. Initial program 99.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing

   if 7.4e9 < y

   1. Initial program 25.9%

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

      1. associate-/l* 42.0%

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

      2. +-commutative 42.0%

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

   3. Simplified 42.0%

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

      1. clear-num 42.0%

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

      2. un-div-inv 42.0%

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

   6. Applied egg-rr 42.0%

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

   7. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -135000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 7400000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 21000000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -115000000.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 21000000000.0)
     (+ 1.0 (* (- 1.0 x) (/ y (- -1.0 y))))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -115000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 21000000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} 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 <= (-115000000.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 21000000000.0d0) then
        tmp = 1.0d0 + ((1.0d0 - x) * (y / ((-1.0d0) - y)))
    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 <= -115000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 21000000000.0) {
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -115000000.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 21000000000.0:
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -115000000.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 21000000000.0)
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) * Float64(y / Float64(-1.0 - y))));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -115000000.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 21000000000.0)
		tmp = 1.0 + ((1.0 - x) * (y / (-1.0 - y)));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -115000000.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 21000000000.0], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] * N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15e8

   1. Initial program 36.3%

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

      1. associate-/l* 55.3%

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

      2. +-commutative 55.3%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{\left(x + \frac{1}{y}\right) - \frac{x}{y}} \]
   6. 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%

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

   7. Simplified 99.9%

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

   if -1.15e8 < y < 2.1e10

   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. +-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 2.1e10 < y

   1. Initial program 25.9%

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

      1. associate-/l* 42.0%

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

      2. +-commutative 42.0%

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

   3. Simplified 42.0%

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

      1. clear-num 42.0%

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

      2. un-div-inv 42.0%

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

   6. Applied egg-rr 42.0%

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

   7. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 21000000000:\\ \;\;\;\;1 + \left(1 - x\right) \cdot \frac{y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -25000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;1 + x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -25000.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 170000.0) (+ 1.0 (* x (/ y (+ y 1.0)))) (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -25000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 170000.0) {
		tmp = 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 <= (-25000.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 170000.0d0) then
        tmp = 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 <= -25000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 170000.0) {
		tmp = 1.0 + (x * (y / (y + 1.0)));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -25000.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 170000.0:
		tmp = 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 <= -25000.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 170000.0)
		tmp = Float64(1.0 + Float64(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 <= -25000.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 170000.0)
		tmp = 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, -25000.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 170000.0], N[(1.0 + N[(x * 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 < -25000

   1. Initial program 36.9%

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

      1. associate-/l* 55.6%

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

      2. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. associate--l+ 98.9%

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

      2. div-sub 98.9%

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

   7. Simplified 98.9%

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

   if -25000 < y < 1.7e5

   1. Initial program 100.0%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. associate-/l* 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

      4. distribute-neg-frac2 99.1%

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

      5. distribute-neg-in 99.1%

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

      6. metadata-eval 99.1%

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

      7. sub-neg 99.1%

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

   7. Simplified 99.1%

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

   if 1.7e5 < y

   1. Initial program 25.9%

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

      1. associate-/l* 42.0%

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

      2. +-commutative 42.0%

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

   3. Simplified 42.0%

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

      1. clear-num 42.0%

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

      2. un-div-inv 42.0%

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

   6. Applied egg-rr 42.0%

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

   7. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -25000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;1 + x \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 0.79:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 0.79) (+ 1.0 (* y (+ x -1.0))) (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 0.79) {
		tmp = 1.0 + (y * (x + -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 <= (-1.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 0.79d0) then
        tmp = 1.0d0 + (y * (x + (-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 <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 0.79) {
		tmp = 1.0 + (y * (x + -1.0));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 0.79:
		tmp = 1.0 + (y * (x + -1.0))
	else:
		tmp = x + (1.0 / 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 <= 0.79)
		tmp = Float64(1.0 + Float64(y * Float64(x + -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 <= -1.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 0.79)
		tmp = 1.0 + (y * (x + -1.0));
	else
		tmp = x + (1.0 / 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, 0.79], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 36.9%

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

      1. associate-/l* 55.6%

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

      2. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. associate--l+ 98.9%

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

      2. div-sub 98.9%

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

   7. Simplified 98.9%

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

   if -1 < y < 0.79000000000000004

   1. Initial program 100.0%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 97.5%

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

   if 0.79000000000000004 < y

   1. Initial program 25.9%

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

      1. associate-/l* 42.0%

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

      2. +-commutative 42.0%

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

   3. Simplified 42.0%

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

      1. clear-num 42.0%

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

      2. un-div-inv 42.0%

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

   6. Applied egg-rr 42.0%

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

   7. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 0.79:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(1.0 + Float64(y * x));
	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 / y);
	else
		tmp = 1.0 + (y * x);
	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[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.6%

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

      1. associate-/l* 47.8%

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

      2. +-commutative 47.8%

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

   3. Simplified 47.8%

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

      1. clear-num 47.7%

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

      2. un-div-inv 47.7%

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

   6. Applied egg-rr 47.7%

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

   7. Taylor expanded in y around -inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in x around 0 99.3%

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

   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* 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. associate-/l* 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

      4. distribute-neg-frac2 99.1%

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

      5. distribute-neg-in 99.1%

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

      6. metadata-eval 99.1%

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

      7. sub-neg 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y around 0 97.4%

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

      1. neg-mul-1 97.4%

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

      2. distribute-rgt-neg-in 97.4%

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

   10. Simplified 97.4%

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

4. Final simplification 98.4%

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

Alternative 12: 98.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 1.0) (+ 1.0 (* y x)) (+ x (/ 1.0 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);
	} 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 <= (-1.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 1.0d0) then
        tmp = 1.0d0 + (y * x)
    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 <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 1.0) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x + (1.0 / 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)
	else:
		tmp = x + (1.0 / 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 * x));
	else
		tmp = Float64(x + Float64(1.0 / 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);
	else
		tmp = x + (1.0 / 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 * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 36.9%

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

      1. associate-/l* 55.6%

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

      2. +-commutative 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. associate--l+ 98.9%

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

      2. div-sub 98.9%

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

   7. Simplified 98.9%

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

   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* 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. associate-/l* 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

      4. distribute-neg-frac2 99.1%

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

      5. distribute-neg-in 99.1%

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

      6. metadata-eval 99.1%

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

      7. sub-neg 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y around 0 97.4%

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

      1. neg-mul-1 97.4%

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

      2. distribute-rgt-neg-in 97.4%

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

   10. Simplified 97.4%

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

   if 1 < y

   1. Initial program 25.9%

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

      1. associate-/l* 42.0%

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

      2. +-commutative 42.0%

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

   3. Simplified 42.0%

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

      1. clear-num 42.0%

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

      2. un-div-inv 42.0%

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

   6. Applied egg-rr 42.0%

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

   7. Taylor expanded in y around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.4%

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

Alternative 13: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 1.8e-24) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.8e-24) {
		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 <= 1.8d-24) 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 <= 1.8e-24) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.8e-24:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.8e-24)
		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 <= 1.8e-24)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.8e-24], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.8e-24 < y

   1. Initial program 33.1%

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

      1. associate-/l* 49.7%

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

      2. +-commutative 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in y around inf 71.6%

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

   if -1 < y < 1.8e-24

   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. +-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 65.0%

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

Alternative 14: 38.7% 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 64.7%

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

   1. associate-/l* 73.4%

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

   2. +-commutative 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in y around 0 33.0%

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

Developer target: 99.7% 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 2024106 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :alt
  (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))))