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

Percentage Accurate: 65.9% → 99.8%

Time: 6.8s

Alternatives: 11

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: 65.9% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -155000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 12200:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -155000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 12200.0)
     (fma y (/ (- x 1.0) (- y -1.0)) 1.0)
     (- x (/ (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) (- x 1.0)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -155000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 12200.0) {
		tmp = fma(y, ((x - 1.0) / (y - -1.0)), 1.0);
	} else {
		tmp = x - (fma(((1.0 - x) / y), (1.0 - (1.0 / y)), (x - 1.0)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -155000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 12200.0)
		tmp = fma(y, Float64(Float64(x - 1.0) / Float64(y - -1.0)), 1.0);
	else
		tmp = Float64(x - Float64(fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), Float64(x - 1.0)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -155000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 12200.0], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e11

   1. Initial program 35.0%

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

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if -1.55e11 < y < 12200

   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 12200 < y

   1. Initial program 29.7%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 12200:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(y, -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (<= t_0 0.0005) x (if (<= t_0 1.5e+27) (fma y -1.0 1.0) x))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if (t_0 <= 0.0005) {
		tmp = x;
	} else if (t_0 <= 1.5e+27) {
		tmp = fma(y, -1.0, 1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)))
	tmp = 0.0
	if (t_0 <= 0.0005)
		tmp = x;
	elseif (t_0 <= 1.5e+27)
		tmp = fma(y, -1.0, 1.0);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0005], x, If[LessEqual[t$95$0, 1.5e+27], N[(y * -1.0 + 1.0), $MachinePrecision], 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)))) < 5.0000000000000001e-4 or 1.49999999999999988e27 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 46.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 67.1

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

   4. Applied rewrites 67.1%

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

   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. mul-1-neg N/A

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

      5. associate--r+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \color{blue}{0} - -1 \cdot x \]

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg 57.0

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

   7. Applied rewrites 57.0%

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

   if 5.0000000000000001e-4 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 1.49999999999999988e27

   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%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 93.3%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{elif}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(y, -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (<= t_0 0.0005) x (if (<= t_0 1.5e+27) 1.0 x))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if (t_0 <= 0.0005) {
		tmp = x;
	} else if (t_0 <= 1.5e+27) {
		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 - (((1.0d0 - x) * y) / (y - (-1.0d0)))
    if (t_0 <= 0.0005d0) then
        tmp = x
    else if (t_0 <= 1.5d+27) 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 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if (t_0 <= 0.0005) {
		tmp = x;
	} else if (t_0 <= 1.5e+27) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0))
	tmp = 0
	if t_0 <= 0.0005:
		tmp = x
	elif t_0 <= 1.5e+27:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0)))
	tmp = 0.0
	if (t_0 <= 0.0005)
		tmp = x;
	elseif (t_0 <= 1.5e+27)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	tmp = 0.0;
	if (t_0 <= 0.0005)
		tmp = x;
	elseif (t_0 <= 1.5e+27)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0005], x, If[LessEqual[t$95$0, 1.5e+27], 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)))) < 5.0000000000000001e-4 or 1.49999999999999988e27 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 46.6%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 67.1

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

   4. Applied rewrites 67.1%

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

   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. mul-1-neg N/A

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

      5. associate--r+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \color{blue}{0} - -1 \cdot x \]

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg 57.0

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

   7. Applied rewrites 57.0%

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

   if 5.0000000000000001e-4 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 1.49999999999999988e27

   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%

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

   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 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 92.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{elif}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -155000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 255000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -155000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 255000.0)
     (fma y (/ (- x 1.0) (- y -1.0)) 1.0)
     (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -155000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 255000.0) {
		tmp = fma(y, ((x - 1.0) / (y - -1.0)), 1.0);
	} else {
		tmp = fma(((1.0 - x) / y), (1.0 - (1.0 / y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -155000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 255000.0)
		tmp = fma(y, Float64(Float64(x - 1.0) / Float64(y - -1.0)), 1.0);
	else
		tmp = fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -155000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 255000.0], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e11

   1. Initial program 35.0%

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

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if -1.55e11 < y < 255000

   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 255000 < y

   1. Initial program 29.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. div-sub N/A

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

      12. neg-sub0 N/A

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

      13. mul-1-neg N/A

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

      14. associate-+l+ N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 255000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -155000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 62000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -155000000000.0)
     t_0
     (if (<= y 62000000000.0) (fma y (/ (- x 1.0) (- y -1.0)) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -155000000000.0) {
		tmp = t_0;
	} else if (y <= 62000000000.0) {
		tmp = fma(y, ((x - 1.0) / (y - -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 <= -155000000000.0)
		tmp = t_0;
	elseif (y <= 62000000000.0)
		tmp = fma(y, Float64(Float64(x - 1.0) / 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[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -155000000000.0], t$95$0, If[LessEqual[y, 62000000000.0], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e11 or 6.2e10 < y

   1. Initial program 31.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.8%

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

   if -1.55e11 < y < 6.2e10

   1. Initial program 99.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 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. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 62000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 760:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 760.0) (fma (- x 1.0) y 1.0) (if (<= y 3.7e+60) (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 760.0) {
		tmp = fma((x - 1.0), y, 1.0);
	} else if (y <= 3.7e+60) {
		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 <= 760.0)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	elseif (y <= 3.7e+60)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 760.0], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 3.7e+60], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 3.69999999999999988e60 < 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 60.6

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

   4. Applied rewrites 60.6%

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

   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. mul-1-neg N/A

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

      5. associate--r+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \color{blue}{0} - -1 \cdot x \]

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg 76.5

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

   7. Applied rewrites 76.5%

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

   if -1 < y < 760

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if 760 < y < 3.69999999999999988e60

   1. Initial program 38.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.3%

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

Alternative 7: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.84:\\ \;\;\;\;\mathsf{fma}\left(\left(y - 1\right) \cdot \left(1 - x\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 0.84) (fma (* (- y 1.0) (- 1.0 x)) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.84) {
		tmp = fma(((y - 1.0) * (1.0 - x)), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.84)
		tmp = fma(Float64(Float64(y - 1.0) * Float64(1.0 - x)), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.84], N[(N[(N[(y - 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.839999999999999969 < y

   1. Initial program 32.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.5%

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

   if -1 < y < 0.839999999999999969

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 98.5%

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

4. Final simplification 99.0%

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

Alternative 8: 98.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.8) (- 1.0 (* (* (- y 1.0) x) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.8) {
		tmp = 1.0 - (((y - 1.0) * x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - ((-1.0d0) / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 0.8d0) then
        tmp = 1.0d0 - (((y - 1.0d0) * x) * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.8) {
		tmp = 1.0 - (((y - 1.0) * x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (-1.0 / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 0.8:
		tmp = 1.0 - (((y - 1.0) * x) * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.8)
		tmp = Float64(1.0 - Float64(Float64(Float64(y - 1.0) * x) * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (-1.0 / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.8)
		tmp = 1.0 - (((y - 1.0) * x) * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.8], N[(1.0 - N[(N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 32.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.5%

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

   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 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. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 - \left(-1 \cdot x + x \cdot y\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 97.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;1 - \left(\left(y - 1\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.78000000000000003 < y

   1. Initial program 32.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.5%

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

   if -1 < y < 0.78000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 10: 87.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.0) (fma (- x 1.0) y 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((x - 1.0), y, 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(Float64(x - 1.0), y, 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[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 32.2%

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

      1. lift--.f64 N/A

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

      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 58.2

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

   4. Applied rewrites 58.2%

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

   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. mul-1-neg N/A

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

      5. associate--r+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \color{blue}{0} - -1 \cdot x \]

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg 71.4

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

   7. Applied rewrites 71.4%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 11: 38.8% 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 67.7%

   \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
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 80.1

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

4. Applied rewrites 80.1%

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

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 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 38.6%

    \[\leadsto \color{blue}{1} \]
11. 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 2024285 
(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))))