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

Percentage Accurate: 64.5% → 99.9%

Time: 9.5s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -13500:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \left(1 + \left(t\_0 - x\right)\right) - \frac{x}{y \cdot y}, \frac{1}{y \cdot \left(y \cdot y\right)}\right)\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{1}{y} + -1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -13500.0)
     (+
      x
      (fma
       (/ 1.0 y)
       (- (+ 1.0 (- t_0 x)) (/ x (* y y)))
       (/ 1.0 (* y (* y y)))))
     (if (<= y 280000.0)
       (fma y (/ (- 1.0 x) (- -1.0 y)) 1.0)
       (fma t_0 (+ (/ 1.0 y) -1.0) x)))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -13500.0) {
		tmp = x + fma((1.0 / y), ((1.0 + (t_0 - x)) - (x / (y * y))), (1.0 / (y * (y * y))));
	} else if (y <= 280000.0) {
		tmp = fma(y, ((1.0 - x) / (-1.0 - y)), 1.0);
	} else {
		tmp = fma(t_0, ((1.0 / y) + -1.0), x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -13500.0)
		tmp = Float64(x + fma(Float64(1.0 / y), Float64(Float64(1.0 + Float64(t_0 - x)) - Float64(x / Float64(y * y))), Float64(1.0 / Float64(y * Float64(y * y)))));
	elseif (y <= 280000.0)
		tmp = fma(y, Float64(Float64(1.0 - x) / Float64(-1.0 - y)), 1.0);
	else
		tmp = fma(t_0, Float64(Float64(1.0 / y) + -1.0), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -13500.0], N[(x + N[(N[(1.0 / y), $MachinePrecision] * N[(N[(1.0 + N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision] - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 280000.0], N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -13500

   1. Initial program 26.4%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

   if -13500 < 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

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

      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-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 2.8e5 < y

   1. Initial program 35.6%

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

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

      2. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

Alternative 2: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;1 - y \cdot \frac{x}{-1 - y}\\ \mathbf{elif}\;t\_0 \leq 1.00000005:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0))))
   (if (<= t_0 0.01)
     (- 1.0 (* y (/ x (- -1.0 y))))
     (if (<= t_0 1.00000005) (+ x (/ (- 1.0 x) y)) (* x (/ y (+ y 1.0)))))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = 1.0 - (y * (x / (-1.0 - y)));
	} else if (t_0 <= 1.00000005) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = x * (y / (y + 1.0));
	}
	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 = (y * (1.0d0 - x)) / (y + 1.0d0)
    if (t_0 <= 0.01d0) then
        tmp = 1.0d0 - (y * (x / ((-1.0d0) - y)))
    else if (t_0 <= 1.00000005d0) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = x * (y / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = 1.0 - (y * (x / (-1.0 - y)));
	} else if (t_0 <= 1.00000005) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = x * (y / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (y + 1.0)
	tmp = 0
	if t_0 <= 0.01:
		tmp = 1.0 - (y * (x / (-1.0 - y)))
	elif t_0 <= 1.00000005:
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = x * (y / (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(1.0 - Float64(y * Float64(x / Float64(-1.0 - y))));
	elseif (t_0 <= 1.00000005)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(x * Float64(y / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = 1.0 - (y * (x / (-1.0 - y)));
	elseif (t_0 <= 1.00000005)
		tmp = x + ((1.0 - x) / y);
	else
		tmp = x * (y / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(1.0 - N[(y * N[(x / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.00000005], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.0100000000000000002

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.00000004999999992

   1. Initial program 8.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 1.00000004999999992 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 71.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 0.01:\\ \;\;\;\;1 - y \cdot \frac{x}{-1 - y}\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 1.00000005:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 0.4:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))
   (if (<= t_0 0.4) x (if (<= t_0 5e+33) 1.0 x))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.4) {
		tmp = x;
	} else if (t_0 <= 5e+33) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    if (t_0 <= 0.4d0) then
        tmp = x
    else if (t_0 <= 5d+33) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.4) {
		tmp = x;
	} else if (t_0 <= 5e+33) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	tmp = 0
	if t_0 <= 0.4:
		tmp = x
	elif t_0 <= 5e+33:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= 0.4)
		tmp = x;
	elseif (t_0 <= 5e+33)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	tmp = 0.0;
	if (t_0 <= 0.4)
		tmp = x;
	elseif (t_0 <= 5e+33)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.4], x, If[LessEqual[t$95$0, 5e+33], 1.0, x]]]

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)))) < 0.40000000000000002 or 4.99999999999999973e33 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 44.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} - \left(\mathsf{neg}\left(x\right)\right) \]

      6. neg-sub0 N/A

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

      7. remove-double-neg 63.5

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

   7. Applied rewrites 63.5%

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

   if 0.40000000000000002 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 4.99999999999999973e33

   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 92.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 0.4:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 5 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)\\ \mathbf{if}\;y \leq -13500:\\ \;\;\;\;x + \frac{1 - t\_0}{y}\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (+ x -1.0) y) (+ (/ 1.0 y) -1.0) x)))
   (if (<= y -13500.0)
     (+ x (/ (- 1.0 t_0) y))
     (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 = fma(((x + -1.0) / y), ((1.0 / y) + -1.0), x);
	double tmp;
	if (y <= -13500.0) {
		tmp = x + ((1.0 - t_0) / y);
	} 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 = fma(Float64(Float64(x + -1.0) / y), Float64(Float64(1.0 / y) + -1.0), x)
	tmp = 0.0
	if (y <= -13500.0)
		tmp = Float64(x + Float64(Float64(1.0 - t_0) / y));
	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[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -13500.0], N[(x + N[(N[(1.0 - t$95$0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 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 3 regimes

2. if y < -13500

   1. Initial program 26.4%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

   if -13500 < 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

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

      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-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 2.8e5 < y

   1. Initial program 35.6%

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

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

      2. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

Alternative 5: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)\\ \mathbf{if}\;y \leq -310000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (+ x -1.0) y) (+ (/ 1.0 y) -1.0) x)))
   (if (<= y -310000.0)
     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 = fma(((x + -1.0) / y), ((1.0 / y) + -1.0), x);
	double tmp;
	if (y <= -310000.0) {
		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 = fma(Float64(Float64(x + -1.0) / y), Float64(Float64(1.0 / y) + -1.0), x)
	tmp = 0.0
	if (y <= -310000.0)
		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[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -310000.0], 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 < -3.1e5 or 2.8e5 < y

   1. Initial program 31.6%

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

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

      2. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   if -3.1e5 < 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

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

      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-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 100.0

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

   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. Add Preprocessing

Alternative 6: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -68000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 27000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -68000000000.0)
     t_0
     (if (<= y 27000000000.0) (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 / y);
	double tmp;
	if (y <= -68000000000.0) {
		tmp = t_0;
	} else if (y <= 27000000000.0) {
		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(1.0 / y))
	tmp = 0.0
	if (y <= -68000000000.0)
		tmp = t_0;
	elseif (y <= 27000000000.0)
		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[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -68000000000.0], t$95$0, If[LessEqual[y, 27000000000.0], 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 < -6.8e10 or 2.7e10 < y

   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. 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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   if -6.8e10 < y < 2.7e10

   1. Initial program 99.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 7: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(y - y \cdot x, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 0.85) (fma (- y (* y x)) (+ y -1.0) 1.0) (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 0.85) {
		tmp = fma((y - (y * x)), (y + -1.0), 1.0);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 0.85)
		tmp = fma(Float64(y - Float64(y * x)), Float64(y + -1.0), 1.0);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.85], N[(N[(y - N[(y * x), $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 27.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   if -1 < y < 0.849999999999999978

   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 98.5%

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

   if 0.849999999999999978 < y

   1. Initial program 35.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto x + \frac{1}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2400000:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.35e+59)
   x
   (if (<= y -2400000.0) (/ 1.0 y) (if (<= y 1.0) (fma y (+ x -1.0) 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.35e+59) {
		tmp = x;
	} else if (y <= -2400000.0) {
		tmp = 1.0 / y;
	} 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 <= -3.35e+59)
		tmp = x;
	elseif (y <= -2400000.0)
		tmp = Float64(1.0 / y);
	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, -3.35e+59], x, If[LessEqual[y, -2400000.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 1.0], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3500000000000002e59 or 1 < y

   1. Initial program 31.4%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 55.2

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} - \left(\mathsf{neg}\left(x\right)\right) \]

      6. neg-sub0 N/A

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

      7. remove-double-neg 81.7

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

   7. Applied rewrites 81.7%

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

   if -3.3500000000000002e59 < y < -2.4e6

   1. Initial program 26.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 93.6

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.2%

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

   if -2.4e6 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)} \]
   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 95.9

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

   5. Applied rewrites 95.9%

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

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (+ x (/ (- 1.0 x) y))
   (if (<= y 0.8) (fma y (+ x -1.0) 1.0) (+ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x + ((1.0 - x) / y);
	} else if (y <= 0.8) {
		tmp = fma(y, (x + -1.0), 1.0);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	elseif (y <= 0.8)
		tmp = fma(y, Float64(x + -1.0), 1.0);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.8], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 27.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   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 97.3

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

   5. Applied rewrites 97.3%

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

   if 0.80000000000000004 < y

   1. Initial program 35.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

      \[\leadsto x + \frac{1}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 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 32.1%

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

   3. Taylor expanded in y around inf

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

      1. +-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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   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 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 11: 85.8% 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 32.1%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 54.3

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

   4. Applied rewrites 54.3%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} - \left(\mathsf{neg}\left(x\right)\right) \]

      6. neg-sub0 N/A

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

      7. remove-double-neg 77.0

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

   7. Applied rewrites 77.0%

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

   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 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 12: 39.7% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 65.0%

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

   1. lift--.f64 N/A

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

   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-frac2 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. sub-neg N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

      \[\leadsto \color{blue}{0} - \left(\mathsf{neg}\left(x\right)\right) \]

   6. neg-sub0 N/A

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

   7. remove-double-neg 41.5

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

7. Applied rewrites 41.5%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 99.7% 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 2024238 
(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))))