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

Percentage Accurate: 65.8% → 99.4%

Time: 8.2s

Alternatives: 13

Speedup: 1.4×

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: 65.8% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 12000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x - 1\right) \cdot y}{\mathsf{fma}\left(y, y, -1\right)}, y - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+26)
   (- x (/ -1.0 y))
   (if (<= y 12000.0)
     (fma (/ (* (- x 1.0) y) (fma y y -1.0)) (- y 1.0) 1.0)
     (- x (/ (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) (- x 1.0)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+26) {
		tmp = x - (-1.0 / y);
	} else if (y <= 12000.0) {
		tmp = fma((((x - 1.0) * y) / fma(y, y, -1.0)), (y - 1.0), 1.0);
	} else {
		tmp = x - (fma(((1.0 - x) / y), (1.0 - (1.0 / y)), (x - 1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+26)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 12000.0)
		tmp = fma(Float64(Float64(Float64(x - 1.0) * y) / fma(y, y, -1.0)), Float64(y - 1.0), 1.0);
	else
		tmp = Float64(x - Float64(fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), Float64(x - 1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+26], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 12000.0], N[(N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y - 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e26

   1. Initial program 30.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \frac{-1}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   if -1.05e26 < y < 12000

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 12000 < y

   1. Initial program 35.3%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 12000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x - 1\right) \cdot y}{\mathsf{fma}\left(y, y, -1\right)}, y - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (<= t_0 1e-11) (* x y) (if (<= t_0 2.0) (- 1.0 y) (* x y)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if (t_0 <= 1e-11) {
		tmp = x * y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((1.0d0 - x) * y) / (y - (-1.0d0)))
    if (t_0 <= 1d-11) then
        tmp = x * y
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if (t_0 <= 1e-11) {
		tmp = x * y;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0))
	tmp = 0
	if t_0 <= 1e-11:
		tmp = x * y
	elif t_0 <= 2.0:
		tmp = 1.0 - y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)))
	tmp = 0.0
	if (t_0 <= 1e-11)
		tmp = Float64(x * y);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	tmp = 0.0;
	if (t_0 <= 1e-11)
		tmp = x * y;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-11], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - y), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 9.99999999999999939e-12 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 44.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 17.4

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

   5. Applied rewrites 17.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.0%

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

   if 9.99999999999999939e-12 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 10^{-11}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 92000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x - 1\right) \cdot y}{\mathsf{fma}\left(y, y, -1\right)}, y - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+26)
   (- x (/ -1.0 y))
   (if (<= y 92000.0)
     (fma (/ (* (- x 1.0) y) (fma y y -1.0)) (- y 1.0) 1.0)
     (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+26) {
		tmp = x - (-1.0 / y);
	} else if (y <= 92000.0) {
		tmp = fma((((x - 1.0) * y) / fma(y, y, -1.0)), (y - 1.0), 1.0);
	} else {
		tmp = fma(((1.0 - x) / y), (1.0 - (1.0 / y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+26)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 92000.0)
		tmp = fma(Float64(Float64(Float64(x - 1.0) * y) / fma(y, y, -1.0)), Float64(y - 1.0), 1.0);
	else
		tmp = fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+26], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 92000.0], N[(N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y - 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e26

   1. Initial program 30.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \frac{-1}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   if -1.05e26 < y < 92000

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 N/A

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

      20. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 92000 < y

   1. Initial program 35.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. neg-sub0 N/A

         \[\leadsto \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) + \left(\color{blue}{\left(0 - \frac{x}{y}\right)} + \frac{1}{y}\right) \]

      7. associate--r- N/A

         \[\leadsto \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) + \color{blue}{\left(0 - \left(\frac{x}{y} - \frac{1}{y}\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) + \left(0 - \color{blue}{\frac{x - 1}{y}}\right) \]

      9. neg-sub0 N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 92000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x - 1\right) \cdot y}{\mathsf{fma}\left(y, y, -1\right)}, y - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 370000:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+26)
   (- x (/ -1.0 y))
   (if (<= y 370000.0)
     (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))
     (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+26) {
		tmp = x - (-1.0 / y);
	} else if (y <= 370000.0) {
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	} else {
		tmp = fma(((1.0 - x) / y), (1.0 - (1.0 / y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+26)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 370000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)));
	else
		tmp = fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+26], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 370000.0], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e26

   1. Initial program 30.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \frac{-1}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   if -1.05e26 < y < 3.7e5

   1. Initial program 99.9%

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

   if 3.7e5 < y

   1. Initial program 35.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. neg-sub0 N/A

         \[\leadsto \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) + \left(\color{blue}{\left(0 - \frac{x}{y}\right)} + \frac{1}{y}\right) \]

      7. associate--r- N/A

         \[\leadsto \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) + \color{blue}{\left(0 - \left(\frac{x}{y} - \frac{1}{y}\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}} + x\right) + \left(0 - \color{blue}{\frac{x - 1}{y}}\right) \]

      9. neg-sub0 N/A

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

      10. mul-1-neg N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 370000:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 130000000:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e+26)
   (- x (/ -1.0 y))
   (if (<= y 130000000.0)
     (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+26) {
		tmp = x - (-1.0 / y);
	} else if (y <= 130000000.0) {
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.05d+26)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 130000000.0d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y - (-1.0d0)))
    else
        tmp = x - ((x - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e+26) {
		tmp = x - (-1.0 / y);
	} else if (y <= 130000000.0) {
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e+26:
		tmp = x - (-1.0 / y)
	elif y <= 130000000.0:
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0))
	else:
		tmp = x - ((x - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e+26)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 130000000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)));
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e+26)
		tmp = x - (-1.0 / y);
	elseif (y <= 130000000.0)
		tmp = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	else
		tmp = x - ((x - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e+26], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 130000000.0], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e26

   1. Initial program 30.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \frac{-1}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   if -1.05e26 < y < 1.3e8

   1. Initial program 99.9%

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

   if 1.3e8 < y

   1. Initial program 35.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 130000000:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.3%

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

Alternative 7: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 8: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.78) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.78) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.78)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.78], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.78000000000000003 < y

   1. Initial program 34.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto x - \frac{-1}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.2%

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

   if -1 < y < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 9: 86.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 33.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if 1 < y

   1. Initial program 35.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 73.7%

      \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 34.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 74.8%

      \[\leadsto 1 \cdot x \]

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 11: 73.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* 1.0 x)
   (if (<= y 1.32e+20) (fma (- y 1.0) y 1.0) (* 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 * x;
	} else if (y <= 1.32e+20) {
		tmp = fma((y - 1.0), y, 1.0);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(1.0 * x);
	elseif (y <= 1.32e+20)
		tmp = fma(Float64(y - 1.0), y, 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(1.0 * x), $MachinePrecision], If[LessEqual[y, 1.32e+20], N[(N[(y - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.32e20 < y

   1. Initial program 34.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 77.2

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

      \[\leadsto 1 \cdot x \]

   if -1 < y < 1.32e20

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y - 1, y, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(y - 1, y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-12}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* 1.0 x) (if (<= y 2.6e-12) (- 1.0 y) (* 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 * x;
	} else if (y <= 2.6e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = 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)) then
        tmp = 1.0d0 * x
    else if (y <= 2.6d-12) then
        tmp = 1.0d0 - y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 * x;
	} else if (y <= 2.6e-12) {
		tmp = 1.0 - y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0 * x
	elif y <= 2.6e-12:
		tmp = 1.0 - y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(1.0 * x);
	elseif (y <= 2.6e-12)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0 * x;
	elseif (y <= 2.6e-12)
		tmp = 1.0 - y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(1.0 * x), $MachinePrecision], If[LessEqual[y, 2.6e-12], N[(1.0 - y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 2.59999999999999983e-12 < y

   1. Initial program 36.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 72.4%

      \[\leadsto 1 \cdot x \]

   if -1 < y < 2.59999999999999983e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.5%

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

Alternative 13: 38.1% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 38.6%

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

Developer Target 1: 99.6% accurate, 0.6× 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 2024249 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

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

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