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

Percentage Accurate: 66.6% → 99.8%

Time: 10.7s

Alternatives: 11

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -300000:\\ \;\;\;\;x - \left(\frac{1 - x}{y \cdot y} - t_0\right)\\ \mathbf{elif}\;y \leq 310000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -300000.0)
     (- x (- (/ (- 1.0 x) (* y y)) t_0))
     (if (<= y 310000000.0) (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0) (+ x t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -300000.0) {
		tmp = x - (((1.0 - x) / (y * y)) - t_0);
	} else if (y <= 310000000.0) {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -300000.0)
		tmp = Float64(x - Float64(Float64(Float64(1.0 - x) / Float64(y * y)) - t_0));
	elseif (y <= 310000000.0)
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -300000.0], N[(x - N[(N[(N[(1.0 - x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 310000000.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3e5

   1. Initial program 32.1%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. div-sub 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

      1. unpow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -3e5 < y < 3.1e8

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*l/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. neg-sub0 99.9%

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

      8. associate--r- 99.9%

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

      9. metadata-eval 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 3.1e8 < y

   1. Initial program 29.8%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000:\\ \;\;\;\;x - \left(\frac{1 - x}{y \cdot y} - \frac{1 - x}{y}\right)\\ \mathbf{elif}\;y \leq 310000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -500000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 310000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -500000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 310000000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -500000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 310000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 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 <= (-500000000000.0d0)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 310000000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (y + 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 <= -500000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 310000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -500000000000.0:
		tmp = x - (-1.0 / y)
	elif y <= 310000000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -500000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 310000000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 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 <= -500000000000.0)
		tmp = x - (-1.0 / y);
	elseif (y <= 310000000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -500000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 310000000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 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 < -5e11

   1. Initial program 30.9%

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

   2. Taylor expanded in y around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

   if -5e11 < y < 3.1e8

   1. Initial program 99.9%

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

   if 3.1e8 < y

   1. Initial program 29.8%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -500000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 310000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8000000000:\\ \;\;\;\;x + \left(\frac{x + -1}{y \cdot y} + \frac{1}{y}\right)\\ \mathbf{elif}\;y \leq 100000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8000000000.0)
   (+ x (+ (/ (+ x -1.0) (* y y)) (/ 1.0 y)))
   (if (<= y 100000000.0)
     (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8000000000.0) {
		tmp = x + (((x + -1.0) / (y * y)) + (1.0 / y));
	} else if (y <= 100000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 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 <= (-8000000000.0d0)) then
        tmp = x + (((x + (-1.0d0)) / (y * y)) + (1.0d0 / y))
    else if (y <= 100000000.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 - x)) / (y + 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 <= -8000000000.0) {
		tmp = x + (((x + -1.0) / (y * y)) + (1.0 / y));
	} else if (y <= 100000000.0) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8000000000.0:
		tmp = x + (((x + -1.0) / (y * y)) + (1.0 / y))
	elif y <= 100000000.0:
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8000000000.0)
		tmp = Float64(x + Float64(Float64(Float64(x + -1.0) / Float64(y * y)) + Float64(1.0 / y)));
	elseif (y <= 100000000.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 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 <= -8000000000.0)
		tmp = x + (((x + -1.0) / (y * y)) + (1.0 / y));
	elseif (y <= 100000000.0)
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8000000000.0], N[(x + N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 100000000.0], N[(1.0 - N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 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 < -8e9

   1. Initial program 30.9%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. div-sub 100.0%

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

      12. sub-neg 100.0%

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

      13. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

      1. unpow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -8e9 < y < 1e8

   1. Initial program 99.9%

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

   if 1e8 < y

   1. Initial program 29.8%

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

   2. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8000000000:\\ \;\;\;\;x + \left(\frac{x + -1}{y \cdot y} + \frac{1}{y}\right)\\ \mathbf{elif}\;y \leq 100000000:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-60}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y -2e-60) (* y x) (if (<= y 1.12e-10) (- 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 <= -2e-60) {
		tmp = y * x;
	} else if (y <= 1.12e-10) {
		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 <= (-2d-60)) then
        tmp = y * x
    else if (y <= 1.12d-10) 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 <= -2e-60) {
		tmp = y * x;
	} else if (y <= 1.12e-10) {
		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 <= -2e-60:
		tmp = y * x
	elif y <= 1.12e-10:
		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 <= -2e-60)
		tmp = Float64(y * x);
	elseif (y <= 1.12e-10)
		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 <= -2e-60)
		tmp = y * x;
	elseif (y <= 1.12e-10)
		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, -2e-60], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.12e-10], N[(1.0 - y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 1.12e-10 < y

   1. Initial program 34.0%

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

   2. Taylor expanded in y around -inf 96.0%

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

      1. mul-1-neg 96.0%

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

      2. unsub-neg 96.0%

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

      3. sub-neg 96.0%

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

      4. metadata-eval 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in x around 0 95.7%

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

   if -1 < y < -1.9999999999999999e-60

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. *-commutative 72.3%

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

   4. Simplified 72.3%

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

   5. Taylor expanded in y around 0 71.8%

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

      1. *-commutative 71.8%

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

   7. Simplified 71.8%

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

   if -1.9999999999999999e-60 < y < 1.12e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in x around 0 77.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-60}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]

Alternative 5: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.82)))
   (- x (/ -1.0 y))
   (- 1.0 (* y (- 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 - (y * (1.0 - 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 <= 0.82d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 - (y * (1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.82)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 - (y * (1.0 - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.82):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 - (y * (1.0 - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.82))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 - Float64(y * Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.82)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 - (y * (1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.82]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.819999999999999951 < y

   1. Initial program 32.4%

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

   2. Taylor expanded in y around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 97.7%

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

   if -1 < y < 0.819999999999999951

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.82\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 32.4%

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

   2. Taylor expanded in y around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

   4. Simplified 98.3%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.4%

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

4. Final simplification 98.3%

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

Alternative 7: 73.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e-13)
   x
   (if (<= y -6.8e-59) (* y x) (if (<= y 1.12e-10) (- 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e-13) {
		tmp = x;
	} else if (y <= -6.8e-59) {
		tmp = y * x;
	} else if (y <= 1.12e-10) {
		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 <= (-7.5d-13)) then
        tmp = x
    else if (y <= (-6.8d-59)) then
        tmp = y * x
    else if (y <= 1.12d-10) 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 <= -7.5e-13) {
		tmp = x;
	} else if (y <= -6.8e-59) {
		tmp = y * x;
	} else if (y <= 1.12e-10) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e-13:
		tmp = x
	elif y <= -6.8e-59:
		tmp = y * x
	elif y <= 1.12e-10:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e-13)
		tmp = x;
	elseif (y <= -6.8e-59)
		tmp = Float64(y * x);
	elseif (y <= 1.12e-10)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.5e-13)
		tmp = x;
	elseif (y <= -6.8e-59)
		tmp = y * x;
	elseif (y <= 1.12e-10)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e-13], x, If[LessEqual[y, -6.8e-59], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.12e-10], N[(1.0 - y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5000000000000004e-13 or 1.12e-10 < y

   1. Initial program 35.0%

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

   2. Taylor expanded in y around inf 75.9%

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

   if -7.5000000000000004e-13 < y < -6.80000000000000035e-59

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.0%

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

      1. associate-*r/ 84.0%

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

      2. *-commutative 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in y around 0 83.5%

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

      1. *-commutative 83.5%

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

   7. Simplified 83.5%

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

   if -6.80000000000000035e-59 < y < 1.12e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in x around 0 77.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-59}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 97.8% accurate, 1.2× 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 32.4%

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

   2. Taylor expanded in y around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 97.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.4%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. distribute-lft-neg-out 97.9%

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

      3. *-commutative 97.9%

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

   5. Simplified 97.9%

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

4. Final simplification 97.8%

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

Alternative 9: 73.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-62}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e-13) x (if (<= y -7e-62) (* y x) (if (<= y 1.12e-10) 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e-13) {
		tmp = x;
	} else if (y <= -7e-62) {
		tmp = y * x;
	} else if (y <= 1.12e-10) {
		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 <= (-7.5d-13)) then
        tmp = x
    else if (y <= (-7d-62)) then
        tmp = y * x
    else if (y <= 1.12d-10) 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 <= -7.5e-13) {
		tmp = x;
	} else if (y <= -7e-62) {
		tmp = y * x;
	} else if (y <= 1.12e-10) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e-13:
		tmp = x
	elif y <= -7e-62:
		tmp = y * x
	elif y <= 1.12e-10:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e-13)
		tmp = x;
	elseif (y <= -7e-62)
		tmp = Float64(y * x);
	elseif (y <= 1.12e-10)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.5e-13)
		tmp = x;
	elseif (y <= -7e-62)
		tmp = y * x;
	elseif (y <= 1.12e-10)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e-13], x, If[LessEqual[y, -7e-62], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.12e-10], 1.0, x]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5000000000000004e-13 or 1.12e-10 < y

   1. Initial program 35.0%

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

   2. Taylor expanded in y around inf 75.9%

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

   if -7.5000000000000004e-13 < y < -7.0000000000000003e-62

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.0%

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

      1. associate-*r/ 84.0%

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

      2. *-commutative 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in y around 0 83.5%

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

      1. *-commutative 83.5%

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

   7. Simplified 83.5%

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

   if -7.0000000000000003e-62 < y < 1.12e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-62}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 73.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.12e-10) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.12e-10) {
		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 <= 1.12d-10) 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 <= 1.12e-10) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.12e-10:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.12e-10)
		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 <= 1.12e-10)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.12e-10], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.12e-10 < y

   1. Initial program 34.0%

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

   2. Taylor expanded in y around inf 77.0%

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

   if -1 < y < 1.12e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 71.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 38.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 67.0%

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

2. Taylor expanded in y around 0 37.4%

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

3. Final simplification 37.4%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023310 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

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

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