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

Percentage Accurate: 66.3% → 99.8%

Time: 16.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10800:\\ \;\;\;\;x - \left(\frac{1 - x}{{y}^{2}} + \left(\frac{x + -1}{y} - \frac{1 - x}{{y}^{3}}\right)\right)\\ \mathbf{elif}\;y \leq 8200000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -10800.0)
   (-
    x
    (+
     (/ (- 1.0 x) (pow y 2.0))
     (- (/ (+ x -1.0) y) (/ (- 1.0 x) (pow y 3.0)))))
   (if (<= y 8200000000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))
     (- x (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -10800.0) {
		tmp = x - (((1.0 - x) / pow(y, 2.0)) + (((x + -1.0) / y) - ((1.0 - x) / pow(y, 3.0))));
	} else if (y <= 8200000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-10800.0d0)) then
        tmp = x - (((1.0d0 - x) / (y ** 2.0d0)) + (((x + (-1.0d0)) / y) - ((1.0d0 - x) / (y ** 3.0d0))))
    else if (y <= 8200000000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    else
        tmp = x - ((-1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -10800.0) {
		tmp = x - (((1.0 - x) / Math.pow(y, 2.0)) + (((x + -1.0) / y) - ((1.0 - x) / Math.pow(y, 3.0))));
	} else if (y <= 8200000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -10800.0:
		tmp = x - (((1.0 - x) / math.pow(y, 2.0)) + (((x + -1.0) / y) - ((1.0 - x) / math.pow(y, 3.0))))
	elif y <= 8200000000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	else:
		tmp = x - (-1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -10800.0)
		tmp = Float64(x - Float64(Float64(Float64(1.0 - x) / (y ^ 2.0)) + Float64(Float64(Float64(x + -1.0) / y) - Float64(Float64(1.0 - x) / (y ^ 3.0)))));
	elseif (y <= 8200000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	else
		tmp = Float64(x - Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -10800.0)
		tmp = x - (((1.0 - x) / (y ^ 2.0)) + (((x + -1.0) / y) - ((1.0 - x) / (y ^ 3.0))));
	elseif (y <= 8200000000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	else
		tmp = x - (-1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -10800.0], N[(x - N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8200000000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -10800

   1. Initial program 21.6%

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

      1. associate-*l/ 41.1%

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

      2. +-commutative 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in y around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-+r+ 100.0%

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

      3. associate--l+ 100.0%

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

   7. Simplified 100.0%

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

   if -10800 < y < 8.2e9

   1. Initial program 99.9%

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

   if 8.2e9 < y

   1. Initial program 35.5%

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

      1. associate-*l/ 64.1%

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

      2. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10800:\\ \;\;\;\;x - \left(\frac{1 - x}{{y}^{2}} + \left(\frac{x + -1}{y} - \frac{1 - x}{{y}^{3}}\right)\right)\\ \mathbf{elif}\;y \leq 8200000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260000:\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \mathbf{elif}\;y \leq 9500000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -260000.0)
   (+ (/ (+ x -1.0) (pow y 2.0)) (+ x (/ (- 1.0 x) y)))
   (if (<= y 9500000000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))
     (- x (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -260000.0) {
		tmp = ((x + -1.0) / pow(y, 2.0)) + (x + ((1.0 - x) / y));
	} else if (y <= 9500000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-260000.0d0)) then
        tmp = ((x + (-1.0d0)) / (y ** 2.0d0)) + (x + ((1.0d0 - x) / y))
    else if (y <= 9500000000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    else
        tmp = x - ((-1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -260000.0) {
		tmp = ((x + -1.0) / Math.pow(y, 2.0)) + (x + ((1.0 - x) / y));
	} else if (y <= 9500000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -260000.0:
		tmp = ((x + -1.0) / math.pow(y, 2.0)) + (x + ((1.0 - x) / y))
	elif y <= 9500000000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	else:
		tmp = x - (-1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -260000.0)
		tmp = Float64(Float64(Float64(x + -1.0) / (y ^ 2.0)) + Float64(x + Float64(Float64(1.0 - x) / y)));
	elseif (y <= 9500000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	else
		tmp = Float64(x - Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -260000.0)
		tmp = ((x + -1.0) / (y ^ 2.0)) + (x + ((1.0 - x) / y));
	elseif (y <= 9500000000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	else
		tmp = x - (-1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -260000.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9500000000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6e5

   1. Initial program 21.6%

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

      1. associate-*l/ 41.1%

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

      2. +-commutative 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in y around -inf 99.9%

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

      1. associate-+r+ 99.9%

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

      2. associate--l+ 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. div-sub 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   if -2.6e5 < y < 9.5e9

   1. Initial program 99.9%

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

   if 9.5e9 < y

   1. Initial program 35.5%

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

      1. associate-*l/ 64.1%

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

      2. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260000:\\ \;\;\;\;\frac{x + -1}{{y}^{2}} + \left(x + \frac{1 - x}{y}\right)\\ \mathbf{elif}\;y \leq 9500000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+79}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e+79)
   x
   (if (<= y -1.15e+57)
     (/ 1.0 y)
     (if (<= y -1.0) x (if (<= y 1.0) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+79) {
		tmp = x;
	} else if (y <= -1.15e+57) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.7d+79)) then
        tmp = x
    else if (y <= (-1.15d+57)) then
        tmp = 1.0d0 / y
    else if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 1.0d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+79) {
		tmp = x;
	} else if (y <= -1.15e+57) {
		tmp = 1.0 / y;
	} else if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.7e+79:
		tmp = x
	elif y <= -1.15e+57:
		tmp = 1.0 / y
	elif y <= -1.0:
		tmp = x
	elif y <= 1.0:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e+79)
		tmp = x;
	elseif (y <= -1.15e+57)
		tmp = Float64(1.0 / y);
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.7e+79)
		tmp = x;
	elseif (y <= -1.15e+57)
		tmp = 1.0 / y;
	elseif (y <= -1.0)
		tmp = x;
	elseif (y <= 1.0)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e+79], x, If[LessEqual[y, -1.15e+57], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.0], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.70000000000000023e79 or -1.1499999999999999e57 < y < -1 or 1 < y

   1. Initial program 31.2%

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

      1. associate-*l/ 56.5%

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

      2. +-commutative 56.5%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in y around inf 74.8%

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

   if -4.70000000000000023e79 < y < -1.1499999999999999e57

   1. Initial program 13.3%

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

      1. associate-*l/ 12.8%

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

      2. +-commutative 12.8%

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

   3. Simplified 12.8%

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

   5. Taylor expanded in x around 0 3.5%

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

   6. Taylor expanded in y around inf 71.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.2%

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

   6. Taylor expanded in y around 0 82.1%

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

      1. neg-mul-1 82.1%

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

      2. sub-neg 82.1%

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

   8. Simplified 82.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+79}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -128000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 6000000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -128000000.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 6000000000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (- x (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -128000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 6000000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-128000000.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 6000000000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x - ((-1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -128000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 6000000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -128000000.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 6000000000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x - (-1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -128000000.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 6000000000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x - Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -128000000.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 6000000000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x - (-1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -128000000.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6000000000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.28e8

   1. Initial program 20.9%

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

      1. associate-*l/ 40.7%

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

      2. +-commutative 40.7%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. div-sub 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. distribute-neg-frac 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. unsub-neg 99.8%

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

      11. sub-neg 99.8%

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

      12. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   if -1.28e8 < y < 6e9

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   if 6e9 < y

   1. Initial program 35.5%

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

      1. associate-*l/ 64.1%

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

      2. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -128000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 6000000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -124000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 11800000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -124000000.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 11800000000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ y 1.0)))
     (- x (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -124000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 11800000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-124000000.0d0)) then
        tmp = x + ((1.0d0 - x) / y)
    else if (y <= 11800000000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (y + 1.0d0))
    else
        tmp = x - ((-1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -124000000.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 11800000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	} else {
		tmp = x - (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -124000000.0:
		tmp = x + ((1.0 - x) / y)
	elif y <= 11800000000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0))
	else:
		tmp = x - (-1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -124000000.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 11800000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(y + 1.0)));
	else
		tmp = Float64(x - Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -124000000.0)
		tmp = x + ((1.0 - x) / y);
	elseif (y <= 11800000000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (y + 1.0));
	else
		tmp = x - (-1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -124000000.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 11800000000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.24e8

   1. Initial program 20.9%

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

      1. associate-*l/ 40.7%

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

      2. +-commutative 40.7%

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

   3. Simplified 40.7%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. div-sub 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. metadata-eval 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. distribute-neg-frac 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. unsub-neg 99.8%

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

      11. sub-neg 99.8%

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

      12. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   if -1.24e8 < y < 1.18e10

   1. Initial program 99.6%

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

   if 1.18e10 < y

   1. Initial program 35.5%

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

      1. associate-*l/ 64.1%

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

      2. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -124000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 11800000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.84\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.84)))
   (- x (/ -1.0 y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.84)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.84d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.84)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.84):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.84))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.84)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.84]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.839999999999999969 < y

   1. Initial program 30.4%

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

      1. associate-*l/ 53.5%

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

      2. +-commutative 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in y around inf 97.5%

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

      1. associate--l+ 97.5%

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

      2. div-sub 97.5%

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

      3. sub-neg 97.5%

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

      4. +-commutative 97.5%

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

      5. metadata-eval 97.5%

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

      6. distribute-neg-in 97.5%

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

      7. distribute-neg-frac 97.5%

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

      8. metadata-eval 97.5%

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

      9. sub-neg 97.5%

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

      10. unsub-neg 97.5%

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

      11. sub-neg 97.5%

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

      12. metadata-eval 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in x around 0 97.2%

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

   if -1 < y < 0.839999999999999969

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.84\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.9\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.9)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.9)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.9d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.9)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.9):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.9))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.9)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.9]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.900000000000000022 < y

   1. Initial program 30.4%

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

      1. associate-*l/ 53.5%

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

      2. +-commutative 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in y around inf 97.5%

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

      1. associate--l+ 97.5%

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

      2. div-sub 97.5%

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

      3. sub-neg 97.5%

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

      4. +-commutative 97.5%

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

      5. metadata-eval 97.5%

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

      6. distribute-neg-in 97.5%

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

      7. distribute-neg-frac 97.5%

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

      8. metadata-eval 97.5%

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

      9. sub-neg 97.5%

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

      10. unsub-neg 97.5%

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

      11. sub-neg 97.5%

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

      12. metadata-eval 97.5%

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

   7. Simplified 97.5%

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

   if -1 < y < 0.900000000000000022

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.9\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 3.3 \cdot 10^{-18}\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 3.3e-18))) (- x (/ -1.0 y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 3.3e-18)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 3.3d-18))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 3.3e-18)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 3.3e-18):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 3.3e-18))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 3.3e-18)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 3.3e-18]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 3.3000000000000002e-18 < y

   1. Initial program 30.9%

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

      1. associate-*l/ 53.8%

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

      2. +-commutative 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in y around inf 96.8%

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

      1. associate--l+ 96.8%

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

      2. div-sub 96.8%

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

      3. sub-neg 96.8%

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

      4. +-commutative 96.8%

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

      5. metadata-eval 96.8%

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

      6. distribute-neg-in 96.8%

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

      7. distribute-neg-frac 96.8%

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

      8. metadata-eval 96.8%

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

      9. sub-neg 96.8%

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

      10. unsub-neg 96.8%

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

      11. sub-neg 96.8%

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

      12. metadata-eval 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in x around 0 96.5%

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

   if -1 < y < 3.3000000000000002e-18

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.7%

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

   6. Taylor expanded in y around 0 83.3%

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

      1. neg-mul-1 83.3%

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

      2. sub-neg 83.3%

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

   8. Simplified 83.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 3.3 \cdot 10^{-18}\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.99:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.99) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.99) {
		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 <= 0.99d0) 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 <= 0.99) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.99:
		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 <= 0.99)
		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 <= 0.99)
		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, 0.99], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.98999999999999999 < y

   1. Initial program 29.8%

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

      1. associate-*l/ 53.1%

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

      2. +-commutative 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in y around inf 71.1%

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

   if -1 < y < 0.98999999999999999

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.2%

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

   6. Taylor expanded in y around 0 82.1%

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

      1. neg-mul-1 82.1%

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

      2. sub-neg 82.1%

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

   8. Simplified 82.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.99:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 680:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 680.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 680.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 680.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 680.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 680.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 680.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 680.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 680.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 680 < y

   1. Initial program 29.4%

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

      1. associate-*l/ 52.9%

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

      2. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in y around inf 71.6%

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

   if -1 < y < 680

   1. Initial program 99.9%

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

      1. associate-*l/ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 81.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 680:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.0% 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 63.8%

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

   1. associate-*l/ 75.8%

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

   2. +-commutative 75.8%

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

3. Simplified 75.8%

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

5. Taylor expanded in y around 0 41.4%

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

6. Final simplification 41.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 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 2024031 
(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))))