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

Percentage Accurate: 64.6% → 99.9%

Time: 8.2s

Alternatives: 9

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: 64.6% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -10000:\\ \;\;\;\;x + \frac{t\_0 \cdot \left(\frac{-1}{y} + 1\right) + \left(1 - x\right)}{y}\\ \mathbf{elif}\;y \leq 240000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + t\_0 \cdot \left(-1 - \frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -10000.0)
     (+ x (/ (+ (* t_0 (+ (/ -1.0 y) 1.0)) (- 1.0 x)) y))
     (if (<= y 240000.0)
       (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
       (+ x (* t_0 (- -1.0 (/ -1.0 y))))))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -10000.0) {
		tmp = x + (((t_0 * ((-1.0 / y) + 1.0)) + (1.0 - x)) / y);
	} else if (y <= 240000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (t_0 * (-1.0 - (-1.0 / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + (-1.0d0)) / y
    if (y <= (-10000.0d0)) then
        tmp = x + (((t_0 * (((-1.0d0) / y) + 1.0d0)) + (1.0d0 - x)) / y)
    else if (y <= 240000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x + (t_0 * ((-1.0d0) - ((-1.0d0) / y)))
    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 <= -10000.0) {
		tmp = x + (((t_0 * ((-1.0 / y) + 1.0)) + (1.0 - x)) / y);
	} else if (y <= 240000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (t_0 * (-1.0 - (-1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + -1.0) / y
	tmp = 0
	if y <= -10000.0:
		tmp = x + (((t_0 * ((-1.0 / y) + 1.0)) + (1.0 - x)) / y)
	elif y <= 240000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x + (t_0 * (-1.0 - (-1.0 / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -10000.0)
		tmp = Float64(x + Float64(Float64(Float64(t_0 * Float64(Float64(-1.0 / y) + 1.0)) + Float64(1.0 - x)) / y));
	elseif (y <= 240000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(t_0 * Float64(-1.0 - Float64(-1.0 / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + -1.0) / y;
	tmp = 0.0;
	if (y <= -10000.0)
		tmp = x + (((t_0 * ((-1.0 / y) + 1.0)) + (1.0 - x)) / y);
	elseif (y <= 240000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x + (t_0 * (-1.0 - (-1.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -10000.0], N[(x + N[(N[(N[(t$95$0 * N[(N[(-1.0 / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 240000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 * N[(-1.0 - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e4

   1. Initial program 27.4%

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

   3. Taylor expanded in y around -inf

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

   5. Simplified 100.0%

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

   if -1e4 < y < 2.4e5

   1. Initial program 100.0%

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

   if 2.4e5 < y

   1. Initial program 24.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000:\\ \;\;\;\;x + \frac{\frac{x + -1}{y} \cdot \left(\frac{-1}{y} + 1\right) + \left(1 - x\right)}{y}\\ \mathbf{elif}\;y \leq 240000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + -1}{y} \cdot \left(-1 - \frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{x + -1}{y} \cdot \left(-1 - \frac{-1}{y}\right)\\ \mathbf{if}\;y \leq -110000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 160000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* (/ (+ x -1.0) y) (- -1.0 (/ -1.0 y))))))
   (if (<= y -110000.0)
     t_0
     (if (<= y 160000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (((x + -1.0) / y) * (-1.0 - (-1.0 / y)));
	double tmp;
	if (y <= -110000.0) {
		tmp = t_0;
	} else if (y <= 160000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-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 + (((x + (-1.0d0)) / y) * ((-1.0d0) - ((-1.0d0) / y)))
    if (y <= (-110000.0d0)) then
        tmp = t_0
    else if (y <= 160000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-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 + (((x + -1.0) / y) * (-1.0 - (-1.0 / y)));
	double tmp;
	if (y <= -110000.0) {
		tmp = t_0;
	} else if (y <= 160000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (((x + -1.0) / y) * (-1.0 - (-1.0 / y)))
	tmp = 0
	if y <= -110000.0:
		tmp = t_0
	elif y <= 160000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(Float64(x + -1.0) / y) * Float64(-1.0 - Float64(-1.0 / y))))
	tmp = 0.0
	if (y <= -110000.0)
		tmp = t_0;
	elseif (y <= 160000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (((x + -1.0) / y) * (-1.0 - (-1.0 / y)));
	tmp = 0.0;
	if (y <= -110000.0)
		tmp = t_0;
	elseif (y <= 160000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(-1.0 - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -110000.0], t$95$0, If[LessEqual[y, 160000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e5 or 1.6e5 < y

   1. Initial program 25.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 100.0%

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

   if -1.1e5 < y < 1.6e5

   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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -110000:\\ \;\;\;\;x + \frac{x + -1}{y} \cdot \left(-1 - \frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 160000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + -1}{y} \cdot \left(-1 - \frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3200000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 160000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3200000000.0)
   (- x (/ -1.0 y))
   (if (<= y 160000000.0)
     (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3200000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 160000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3200000000.0d0)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 160000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3200000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 160000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3200000000.0:
		tmp = x - (-1.0 / y)
	elif y <= 160000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3200000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 160000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3200000000.0)
		tmp = x - (-1.0 / y);
	elseif (y <= 160000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3200000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 160000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e9

   1. Initial program 27.4%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. div-sub N/A

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

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{x - 1}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x - 1\right), \color{blue}{y}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      12. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{-1}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{y}\right)\right) \]

   8. Simplified 99.5%

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

   if -3.2e9 < y < 1.6e8

   1. Initial program 100.0%

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

   if 1.6e8 < y

   1. Initial program 24.1%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. div-sub N/A

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

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{x - 1}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x - 1\right), \color{blue}{y}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      12. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3200000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 160000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 27.4%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. div-sub N/A

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

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{x - 1}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x - 1\right), \color{blue}{y}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      12. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{-1}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{y}\right)\right) \]

   8. Simplified 99.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(y \cdot \left(1 - x\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{1}\right)\right)\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(0 - x\right) + 1\right)\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(-1 \cdot \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(x - 1\right)\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(0 - x\right) + \color{blue}{1}\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 - \color{blue}{x}\right)\right)\right) \]

      14. --lowering--.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.8%

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

   if 1 < y

   1. Initial program 25.5%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. div-sub N/A

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

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{x - 1}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x - 1\right), \color{blue}{y}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      12. +-lowering-+.f64 99.4%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 99.4%

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

4. Final simplification 99.7%

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

Alternative 5: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.78) (+ 1.0 (* y (+ x -1.0))) 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 <= 0.78) {
		tmp = 1.0 + (y * (x + -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 = x - ((-1.0d0) / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 0.78d0) then
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    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 <= 0.78) {
		tmp = 1.0 + (y * (x + -1.0));
	} 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 <= 0.78:
		tmp = 1.0 + (y * (x + -1.0))
	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 <= 0.78)
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	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 <= 0.78)
		tmp = 1.0 + (y * (x + -1.0));
	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, 0.78], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.78000000000000003 < y

   1. Initial program 26.5%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. div-sub N/A

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

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{x - 1}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x - 1\right), \color{blue}{y}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      12. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{-1}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 99.1%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{y}\right)\right) \]

   8. Simplified 99.1%

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

   if -1 < y < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(y \cdot \left(1 - x\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{1}\right)\right)\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\left(0 - x\right) + 1\right)\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(y \cdot \left(-1 \cdot \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(x - 1\right)\right)}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{\left(x - 1\right)}\right)\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(0 - x\right) + \color{blue}{1}\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(x\right)\right) + 1\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \left(1 - \color{blue}{x}\right)\right)\right) \]

      14. --lowering--.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.078:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.078) (- 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 <= 0.078) {
		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 <= 0.078d0) 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 <= 0.078) {
		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 <= 0.078:
		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 <= 0.078)
		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 <= 0.078)
		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, 0.078], N[(1.0 - y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0779999999999999999 < y

   1. Initial program 26.5%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. sub-neg N/A

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

      6. associate--r- N/A

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

      7. div-sub N/A

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

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{x - 1}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x - 1\right), \color{blue}{y}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(x + -1\right), y\right)\right) \]

      12. +-lowering-+.f64 99.5%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, -1\right), y\right)\right) \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{-1}{y}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 99.1%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{y}\right)\right) \]

   8. Simplified 99.1%

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

   if -1 < y < 0.0779999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 75.7%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{y}\right) \]

   8. Simplified 75.6%

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

Alternative 7: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.08) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.08) {
		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 <= 0.08d0) 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 <= 0.08) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.08:
		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 <= 0.08)
		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 <= 0.08)
		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, 0.08], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0800000000000000017 < y

   1. Initial program 26.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 75.4%

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

   if -1 < y < 0.0800000000000000017

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 75.7%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(y\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{y}\right) \]

   8. Simplified 75.6%

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

Alternative 8: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 1.85) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.85) {
		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 <= 1.85d0) 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 <= 1.85) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.85:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.85], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.8500000000000001 < y

   1. Initial program 26.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 75.4%

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

   if -1 < y < 1.8500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 75.1%

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

Alternative 9: 37.5% 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 65.8%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 42.0%

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

Developer Target 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

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

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