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

Percentage Accurate: 65.9% → 99.8%

Time: 5.5s

Alternatives: 14

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -1200000000000:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;y \leq 13200:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \frac{-1}{y} - -1, 1 - x\right)}{y} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x 1.0) y)))
   (if (<= y -1200000000000.0)
     (- x t_0)
     (if (<= y 13200.0)
       (fma (/ y (- y -1.0)) (+ -1.0 x) 1.0)
       (+ (/ (fma t_0 (- (/ -1.0 y) -1.0) (- 1.0 x)) y) x)))))

C

double code(double x, double y) {
	double t_0 = (x - 1.0) / y;
	double tmp;
	if (y <= -1200000000000.0) {
		tmp = x - t_0;
	} else if (y <= 13200.0) {
		tmp = fma((y / (y - -1.0)), (-1.0 + x), 1.0);
	} else {
		tmp = (fma(t_0, ((-1.0 / y) - -1.0), (1.0 - x)) / y) + x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (y <= -1200000000000.0)
		tmp = Float64(x - t_0);
	elseif (y <= 13200.0)
		tmp = fma(Float64(y / Float64(y - -1.0)), Float64(-1.0 + x), 1.0);
	else
		tmp = Float64(Float64(fma(t_0, Float64(Float64(-1.0 / y) - -1.0), Float64(1.0 - x)) / y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1200000000000.0], N[(x - t$95$0), $MachinePrecision], If[LessEqual[y, 13200.0], N[(N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[(-1.0 / y), $MachinePrecision] - -1.0), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e12

   1. Initial program 23.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. 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%

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

   if -1.2e12 < y < 13200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 100.0%

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

   if 13200 < y

   1. Initial program 32.8%

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

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1200000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 13200:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y - -1}, -1 + x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x - 1}{y}, \frac{-1}{y} - -1, 1 - x\right)}{y} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+153}:\\ \;\;\;\;1 - \left(-x\right)\\ \mathbf{elif}\;t\_0 \leq -20:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+68} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+201}\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (- y -1.0))))
   (if (<= t_0 -4e+153)
     (- 1.0 (- x))
     (if (<= t_0 -20.0)
       (* y x)
       (if (<= t_0 0.5)
         (fma (- y 1.0) y 1.0)
         (if (or (<= t_0 2e+68) (not (<= t_0 2e+201)))
           (- 1.0 (- 1.0 x))
           (* y x)))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y - -1.0);
	double tmp;
	if (t_0 <= -4e+153) {
		tmp = 1.0 - -x;
	} else if (t_0 <= -20.0) {
		tmp = y * x;
	} else if (t_0 <= 0.5) {
		tmp = fma((y - 1.0), y, 1.0);
	} else if ((t_0 <= 2e+68) || !(t_0 <= 2e+201)) {
		tmp = 1.0 - (1.0 - x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -4e+153)
		tmp = Float64(1.0 - Float64(-x));
	elseif (t_0 <= -20.0)
		tmp = Float64(y * x);
	elseif (t_0 <= 0.5)
		tmp = fma(Float64(y - 1.0), y, 1.0);
	elseif ((t_0 <= 2e+68) || !(t_0 <= 2e+201))
		tmp = Float64(1.0 - Float64(1.0 - x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+153], N[(1.0 - (-x)), $MachinePrecision], If[LessEqual[t$95$0, -20.0], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(y - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], If[Or[LessEqual[t$95$0, 2e+68], N[Not[LessEqual[t$95$0, 2e+201]], $MachinePrecision]], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -4e153

   1. Initial program 39.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.1%

      \[\leadsto 1 - \left(-x\right) \]

   if -4e153 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -20 or 1.99999999999999991e68 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 2.00000000000000008e201

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.3%

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

   if -20 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.5

   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

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.8%

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

   if 0.5 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.99999999999999991e68 or 2.00000000000000008e201 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 25.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 32.8

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

   5. Applied rewrites 32.8%

      \[\leadsto 1 - \color{blue}{\left(1 - x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq -4 \cdot 10^{+153}:\\ \;\;\;\;1 - \left(-x\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq -20:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2 \cdot 10^{+68} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2 \cdot 10^{+201}\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+153}:\\ \;\;\;\;1 - \left(-x\right)\\ \mathbf{elif}\;t\_0 \leq -20:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+68} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+201}\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (- y -1.0))))
   (if (<= t_0 -4e+153)
     (- 1.0 (- x))
     (if (<= t_0 -20.0)
       (* y x)
       (if (<= t_0 0.5)
         (- 1.0 y)
         (if (or (<= t_0 2e+68) (not (<= t_0 2e+201)))
           (- 1.0 (- 1.0 x))
           (* y x)))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y - -1.0);
	double tmp;
	if (t_0 <= -4e+153) {
		tmp = 1.0 - -x;
	} else if (t_0 <= -20.0) {
		tmp = y * x;
	} else if (t_0 <= 0.5) {
		tmp = 1.0 - y;
	} else if ((t_0 <= 2e+68) || !(t_0 <= 2e+201)) {
		tmp = 1.0 - (1.0 - x);
	} else {
		tmp = y * 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 - x) * y) / (y - (-1.0d0))
    if (t_0 <= (-4d+153)) then
        tmp = 1.0d0 - -x
    else if (t_0 <= (-20.0d0)) then
        tmp = y * x
    else if (t_0 <= 0.5d0) then
        tmp = 1.0d0 - y
    else if ((t_0 <= 2d+68) .or. (.not. (t_0 <= 2d+201))) then
        tmp = 1.0d0 - (1.0d0 - x)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y - -1.0);
	double tmp;
	if (t_0 <= -4e+153) {
		tmp = 1.0 - -x;
	} else if (t_0 <= -20.0) {
		tmp = y * x;
	} else if (t_0 <= 0.5) {
		tmp = 1.0 - y;
	} else if ((t_0 <= 2e+68) || !(t_0 <= 2e+201)) {
		tmp = 1.0 - (1.0 - x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (y - -1.0)
	tmp = 0
	if t_0 <= -4e+153:
		tmp = 1.0 - -x
	elif t_0 <= -20.0:
		tmp = y * x
	elif t_0 <= 0.5:
		tmp = 1.0 - y
	elif (t_0 <= 2e+68) or not (t_0 <= 2e+201):
		tmp = 1.0 - (1.0 - x)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -4e+153)
		tmp = Float64(1.0 - Float64(-x));
	elseif (t_0 <= -20.0)
		tmp = Float64(y * x);
	elseif (t_0 <= 0.5)
		tmp = Float64(1.0 - y);
	elseif ((t_0 <= 2e+68) || !(t_0 <= 2e+201))
		tmp = Float64(1.0 - Float64(1.0 - x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (y - -1.0);
	tmp = 0.0;
	if (t_0 <= -4e+153)
		tmp = 1.0 - -x;
	elseif (t_0 <= -20.0)
		tmp = y * x;
	elseif (t_0 <= 0.5)
		tmp = 1.0 - y;
	elseif ((t_0 <= 2e+68) || ~((t_0 <= 2e+201)))
		tmp = 1.0 - (1.0 - x);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+153], N[(1.0 - (-x)), $MachinePrecision], If[LessEqual[t$95$0, -20.0], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(1.0 - y), $MachinePrecision], If[Or[LessEqual[t$95$0, 2e+68], N[Not[LessEqual[t$95$0, 2e+201]], $MachinePrecision]], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -4e153

   1. Initial program 39.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.1%

      \[\leadsto 1 - \left(-x\right) \]

   if -4e153 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -20 or 1.99999999999999991e68 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 2.00000000000000008e201

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.3%

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

   if -20 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.5

   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 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.8%

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

   if 0.5 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.99999999999999991e68 or 2.00000000000000008e201 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 25.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 32.8

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

   5. Applied rewrites 32.8%

      \[\leadsto 1 - \color{blue}{\left(1 - x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq -4 \cdot 10^{+153}:\\ \;\;\;\;1 - \left(-x\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq -20:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 0.5:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2 \cdot 10^{+68} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2 \cdot 10^{+201}\right):\\ \;\;\;\;1 - \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1200000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 240000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{1 + y} \cdot \left(1 - x\right)}{y - 1}, -\left(y - 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(1 - {x}^{-1}\right) \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1200000000000.0)
   (- x (/ (- x 1.0) y))
   (if (<= y 240000000.0)
     (fma (/ (* (/ y (+ 1.0 y)) (- 1.0 x)) (- y 1.0)) (- (- y 1.0)) 1.0)
     (- x (/ (* (- 1.0 (pow x -1.0)) x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1200000000000.0) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 240000000.0) {
		tmp = fma((((y / (1.0 + y)) * (1.0 - x)) / (y - 1.0)), -(y - 1.0), 1.0);
	} else {
		tmp = x - (((1.0 - pow(x, -1.0)) * x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1200000000000.0)
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	elseif (y <= 240000000.0)
		tmp = fma(Float64(Float64(Float64(y / Float64(1.0 + y)) * Float64(1.0 - x)) / Float64(y - 1.0)), Float64(-Float64(y - 1.0)), 1.0);
	else
		tmp = Float64(x - Float64(Float64(Float64(1.0 - (x ^ -1.0)) * x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1200000000000.0], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 240000000.0], N[(N[(N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision] * (-N[(y - 1.0), $MachinePrecision]) + 1.0), $MachinePrecision], N[(x - N[(N[(N[(1.0 - N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e12

   1. Initial program 23.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. 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%

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

   if -1.2e12 < y < 2.4e8

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. *-lft-identity N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. associate-/r/ N/A

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

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

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. difference-of-sqr--1 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 2.4e8 < 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 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 inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1200000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 240000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{y}{1 + y} \cdot \left(1 - x\right)}{y - 1}, -\left(y - 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(1 - {x}^{-1}\right) \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -125000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 410000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(1 - {x}^{-1}\right) \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -125000000000.0)
   (- x (/ (- x 1.0) y))
   (if (<= y 410000000.0)
     (fma y (/ (- x 1.0) (- y -1.0)) 1.0)
     (- x (/ (* (- 1.0 (pow x -1.0)) x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -125000000000.0) {
		tmp = x - ((x - 1.0) / y);
	} else if (y <= 410000000.0) {
		tmp = fma(y, ((x - 1.0) / (y - -1.0)), 1.0);
	} else {
		tmp = x - (((1.0 - pow(x, -1.0)) * x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -125000000000.0)
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	elseif (y <= 410000000.0)
		tmp = fma(y, Float64(Float64(x - 1.0) / Float64(y - -1.0)), 1.0);
	else
		tmp = Float64(x - Float64(Float64(Float64(1.0 - (x ^ -1.0)) * x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -125000000000.0], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 410000000.0], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x - N[(N[(N[(1.0 - N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25e11

   1. Initial program 24.8%

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

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

   if -1.25e11 < y < 4.1e8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   if 4.1e8 < 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 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 inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 410000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(1 - {x}^{-1}\right) \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{y - -1}\\ t_1 := 1 - \left(-x\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -10000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 1.02:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+201}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (- y -1.0))) (t_1 (- 1.0 (- x))))
   (if (<= t_0 -4e+153)
     t_1
     (if (<= t_0 -10000.0)
       (* y x)
       (if (<= t_0 1.02) 1.0 (if (<= t_0 2e+201) (* y x) t_1))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y - -1.0);
	double t_1 = 1.0 - -x;
	double tmp;
	if (t_0 <= -4e+153) {
		tmp = t_1;
	} else if (t_0 <= -10000.0) {
		tmp = y * x;
	} else if (t_0 <= 1.02) {
		tmp = 1.0;
	} else if (t_0 <= 2e+201) {
		tmp = y * x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 - x) * y) / (y - (-1.0d0))
    t_1 = 1.0d0 - -x
    if (t_0 <= (-4d+153)) then
        tmp = t_1
    else if (t_0 <= (-10000.0d0)) then
        tmp = y * x
    else if (t_0 <= 1.02d0) then
        tmp = 1.0d0
    else if (t_0 <= 2d+201) then
        tmp = y * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (y - -1.0);
	double t_1 = 1.0 - -x;
	double tmp;
	if (t_0 <= -4e+153) {
		tmp = t_1;
	} else if (t_0 <= -10000.0) {
		tmp = y * x;
	} else if (t_0 <= 1.02) {
		tmp = 1.0;
	} else if (t_0 <= 2e+201) {
		tmp = y * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((1.0 - x) * y) / (y - -1.0)
	t_1 = 1.0 - -x
	tmp = 0
	if t_0 <= -4e+153:
		tmp = t_1
	elif t_0 <= -10000.0:
		tmp = y * x
	elif t_0 <= 1.02:
		tmp = 1.0
	elif t_0 <= 2e+201:
		tmp = y * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(y - -1.0))
	t_1 = Float64(1.0 - Float64(-x))
	tmp = 0.0
	if (t_0 <= -4e+153)
		tmp = t_1;
	elseif (t_0 <= -10000.0)
		tmp = Float64(y * x);
	elseif (t_0 <= 1.02)
		tmp = 1.0;
	elseif (t_0 <= 2e+201)
		tmp = Float64(y * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((1.0 - x) * y) / (y - -1.0);
	t_1 = 1.0 - -x;
	tmp = 0.0;
	if (t_0 <= -4e+153)
		tmp = t_1;
	elseif (t_0 <= -10000.0)
		tmp = y * x;
	elseif (t_0 <= 1.02)
		tmp = 1.0;
	elseif (t_0 <= 2e+201)
		tmp = y * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - (-x)), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+153], t$95$1, If[LessEqual[t$95$0, -10000.0], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 1.02], 1.0, If[LessEqual[t$95$0, 2e+201], N[(y * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -4e153 or 2.00000000000000008e201 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 34.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 87.5%

      \[\leadsto 1 - \left(-x\right) \]

   if -4e153 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -1e4 or 1.02 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 2.00000000000000008e201

   1. Initial program 99.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.0%

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

   if -1e4 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.02

   1. Initial program 61.7%

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

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

      3. fp-cancel-sub-sign-inv N/A

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. associate-/r/ N/A

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

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

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. difference-of-sqr--1 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 61.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 61.5

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

   6. Applied rewrites 61.5%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 58.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq -4 \cdot 10^{+153}:\\ \;\;\;\;1 - \left(-x\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq -10000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 1.02:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{y - -1} \leq 2 \cdot 10^{+201}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(1 - x\right) \cdot y}{y - -1}\\ \mathbf{if}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 50\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- 1.0 x) y) (- y -1.0)))))
   (if (or (<= t_0 -0.02) (not (<= t_0 50.0))) (* y x) 1.0)))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0));
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 50.0)) {
		tmp = y * x;
	} else {
		tmp = 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 = 1.0d0 - (((1.0d0 - x) * y) / (y - (-1.0d0)))
    if ((t_0 <= (-0.02d0)) .or. (.not. (t_0 <= 50.0d0))) then
        tmp = y * x
    else
        tmp = 1.0d0
    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 <= -0.02) || !(t_0 <= 50.0)) {
		tmp = y * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((1.0 - x) * y) / (y - -1.0))
	tmp = 0
	if (t_0 <= -0.02) or not (t_0 <= 50.0):
		tmp = y * x
	else:
		tmp = 1.0
	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 <= -0.02) || !(t_0 <= 50.0))
		tmp = Float64(y * x);
	else
		tmp = 1.0;
	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 <= -0.02) || ~((t_0 <= 50.0)))
		tmp = y * x;
	else
		tmp = 1.0;
	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[Or[LessEqual[t$95$0, -0.02], N[Not[LessEqual[t$95$0, 50.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], 1.0]]

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)))) < -0.0200000000000000004 or 50 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 69.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)} \]
   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 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 38.5%

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

   if -0.0200000000000000004 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 50

   1. Initial program 61.7%

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

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

      3. fp-cancel-sub-sign-inv N/A

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

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

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

      5. *-lft-identity N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. associate-/r/ N/A

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

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

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. difference-of-sqr--1 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 61.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 61.5

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

   6. Applied rewrites 61.5%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 58.7%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq -0.02 \lor \neg \left(1 - \frac{\left(1 - x\right) \cdot y}{y - -1} \leq 50\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -125000000000 \lor \neg \left(y \leq 410000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -125000000000.0) (not (<= y 410000000.0)))
   (- x (/ (- x 1.0) y))
   (fma y (/ (- x 1.0) (- y -1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -125000000000.0) || !(y <= 410000000.0)) {
		tmp = x - ((x - 1.0) / y);
	} else {
		tmp = fma(y, ((x - 1.0) / (y - -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -125000000000.0) || !(y <= 410000000.0))
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	else
		tmp = fma(y, Float64(Float64(x - 1.0) / Float64(y - -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -125000000000.0], N[Not[LessEqual[y, 410000000.0]], $MachinePrecision]], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e11 or 4.1e8 < y

   1. Initial program 28.6%

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

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

   if -1.25e11 < y < 4.1e8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125000000000 \lor \neg \left(y \leq 410000000\right):\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   if -1 < 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

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

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 99.0%

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

Alternative 10: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   if -1 < 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. *-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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.7%

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

Alternative 11: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.83\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.83)))
   (- x (/ -1.0 y))
   (fma (- x 1.0) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.83)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = fma((x - 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.83))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = fma(Float64(x - 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.83]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.82999999999999996 < y

   1. Initial program 30.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.3%

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

   if -1 < y < 0.82999999999999996

   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. *-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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.83\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.08\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.08))) (- x (/ x y)) (fma (- x 1.0) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.08)) {
		tmp = x - (x / y);
	} else {
		tmp = fma((x - 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.08))
		tmp = Float64(x - Float64(x / y));
	else
		tmp = fma(Float64(x - 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.08]], $MachinePrecision]], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.0800000000000001 < y

   1. Initial program 30.2%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.1%

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

   if -1 < y < 1.0800000000000001

   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. *-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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.08\right):\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (- 1.0 (- 1.0 x))
   (fma (- x 1.0) y 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if -1 < 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. *-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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 75.9%

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

Alternative 14: 39.5% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 65.1%

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

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

   3. fp-cancel-sub-sign-inv N/A

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

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

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

   5. *-lft-identity N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. flip-+ N/A

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

   10. associate-/r/ N/A

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

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

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 64.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-fma.f64 N/A

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

   5. difference-of-sqr--1 N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. associate-/r* N/A

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate-/l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 77.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 77.5

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

6. Applied rewrites 77.5%

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

7. Taylor expanded in y around 0

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

9. Applied rewrites 35.6%

   \[\leadsto \color{blue}{1} \]
10. 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 2024332 
(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))))