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

Percentage Accurate: 65.3% → 99.8%

Time: 9.1s

Alternatives: 11

Speedup: 2.0×

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: 65.3% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 + \left(\frac{x + -1}{y} - x\right)}{y}\\ \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (+ 1.0 (- (/ (+ x -1.0) y) x)) y))))
   (if (<= y -45000000000.0)
     t_0
     (if (<= y 280000.0) (fma y (/ (- 1.0 x) (- -1.0 y)) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 + (((x + -1.0) / y) - x)) / y);
	double tmp;
	if (y <= -45000000000.0) {
		tmp = t_0;
	} else if (y <= 280000.0) {
		tmp = fma(y, ((1.0 - x) / (-1.0 - y)), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 + Float64(Float64(Float64(x + -1.0) / y) - x)) / y))
	tmp = 0.0
	if (y <= -45000000000.0)
		tmp = t_0;
	elseif (y <= 280000.0)
		tmp = fma(y, Float64(Float64(1.0 - x) / Float64(-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[(1.0 + N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -45000000000.0], t$95$0, If[LessEqual[y, 280000.0], N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e10 or 2.8e5 < y

   1. Initial program 24.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 100.0%

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

   if -4.5e10 < y < 2.8e5

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;x + \frac{1 + \left(\frac{x + -1}{y} - x\right)}{y}\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 + \left(\frac{x + -1}{y} - x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -45000000000.0)
   (- x (/ (+ x -1.0) y))
   (if (<= y 48000000000000.0)
     (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -45000000000.0) {
		tmp = x - ((x + -1.0) / y);
	} else if (y <= 48000000000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-45000000000.0d0)) then
        tmp = x - ((x + (-1.0d0)) / y)
    else if (y <= 48000000000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -45000000000.0) {
		tmp = x - ((x + -1.0) / y);
	} else if (y <= 48000000000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -45000000000.0:
		tmp = x - ((x + -1.0) / y)
	elif y <= 48000000000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -45000000000.0)
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	elseif (y <= 48000000000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -45000000000.0)
		tmp = x - ((x + -1.0) / y);
	elseif (y <= 48000000000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -45000000000.0], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 48000000000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5e10

   1. Initial program 20.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -4.5e10 < y < 4.8e13

   1. Initial program 100.0%

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

   if 4.8e13 < y

   1. Initial program 25.5%

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

   3. Taylor expanded in y around inf

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

      1. +-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. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -45000000000.0)
   (- x (/ (+ x -1.0) y))
   (if (<= y 48000000000000.0)
     (fma y (/ (- 1.0 x) (- -1.0 y)) 1.0)
     (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -45000000000.0) {
		tmp = x - ((x + -1.0) / y);
	} else if (y <= 48000000000000.0) {
		tmp = fma(y, ((1.0 - x) / (-1.0 - y)), 1.0);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -45000000000.0)
		tmp = Float64(x - Float64(Float64(x + -1.0) / y));
	elseif (y <= 48000000000000.0)
		tmp = fma(y, Float64(Float64(1.0 - x) / Float64(-1.0 - y)), 1.0);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -45000000000.0], N[(x - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 48000000000000.0], N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5e10

   1. Initial program 20.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -4.5e10 < y < 4.8e13

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if 4.8e13 < y

   1. Initial program 25.5%

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

   3. Taylor expanded in y around inf

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

      1. +-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. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000000000:\\ \;\;\;\;x - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.9× speedup

Math

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

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 (* y x)) (+ y -1.0) 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 - (y * x)), (y + -1.0), 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(y - Float64(y * x)), Float64(y + -1.0), 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[(y - N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 25.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

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

      3. associate--l+ N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-in N/A

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

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

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

      10. distribute-rgt-neg-out N/A

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

      11. neg-sub0 N/A

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

      12. associate-+l- N/A

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

      13. neg-sub0 N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. distribute-lft-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.4%

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

Alternative 5: 98.5% accurate, 0.9× speedup

Math

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

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 (+ x -1.0) 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, (x + -1.0), 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(y, Float64(x + -1.0), 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[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 25.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. 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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 99.0%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.8) (fma y (+ x -1.0) 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 <= 0.8) {
		tmp = fma(y, (x + -1.0), 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 <= 0.8)
		tmp = fma(y, Float64(x + -1.0), 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, 0.8], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 25.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if -1 < y < 0.80000000000000004

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 7: 86.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 25.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. associate-+r+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      12. +-lft-identity 69.5

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

   7. Applied rewrites 69.5%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 8: 86.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 68:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 68.0) (fma x y 1.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 68.0) {
		tmp = fma(x, y, 1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 68.0)
		tmp = fma(x, y, 1.0);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 68.0], N[(x * y + 1.0), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 68 < y

   1. Initial program 25.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. associate-+r+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      12. +-lft-identity 69.5

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

   7. Applied rewrites 69.5%

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

   if -1 < y < 68

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.3

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

   10. Applied rewrites 97.3%

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

Alternative 9: 74.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.98:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.98) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.98) {
		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 <= 0.98d0) 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 <= 0.98) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.98:
		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 <= 0.98)
		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 <= 0.98)
		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, 0.98], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.97999999999999998 < y

   1. Initial program 25.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. associate-+r+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      12. +-lft-identity 69.5

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

   7. Applied rewrites 69.5%

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

   if -1 < y < 0.97999999999999998

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 76.4

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

   8. Applied rewrites 76.4%

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

Alternative 10: 74.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.26:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 1.26) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.26) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 1.26d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.26) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.26:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.26)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.26)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.26], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.26000000000000001 < y

   1. Initial program 25.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. associate-+r+ N/A

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

      11. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      12. +-lft-identity 69.5

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

   7. Applied rewrites 69.5%

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

   if -1 < y < 1.26000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 75.7%

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

Alternative 11: 38.4% 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 60.9%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 38.1%

   \[\leadsto \color{blue}{1} \]
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 2024214 
(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))))