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

Percentage Accurate: 66.1% → 99.9%

Time: 9.2s

Alternatives: 16

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -15500:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \left(1 + \left(t\_0 - x\right)\right) - \frac{x}{y \cdot y}, \frac{1}{y \cdot \left(y \cdot y\right)}\right)\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(t\_0, \frac{1}{y} + -1, x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -15500.0)
     (+
      x
      (fma
       (/ 1.0 y)
       (- (+ 1.0 (- t_0 x)) (/ x (* y y)))
       (/ 1.0 (* y (* y y)))))
     (if (<= y 17000.0)
       (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
       (+ x (/ (- 1.0 (fma t_0 (+ (/ 1.0 y) -1.0) x)) y))))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -15500.0) {
		tmp = x + fma((1.0 / y), ((1.0 + (t_0 - x)) - (x / (y * y))), (1.0 / (y * (y * y))));
	} else if (y <= 17000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + ((1.0 - fma(t_0, ((1.0 / y) + -1.0), x)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -15500.0)
		tmp = Float64(x + fma(Float64(1.0 / y), Float64(Float64(1.0 + Float64(t_0 - x)) - Float64(x / Float64(y * y))), Float64(1.0 / Float64(y * Float64(y * y)))));
	elseif (y <= 17000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(Float64(1.0 - fma(t_0, Float64(Float64(1.0 / y) + -1.0), x)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -15500.0], N[(x + N[(N[(1.0 / y), $MachinePrecision] * N[(N[(1.0 + N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision] - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 17000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - N[(t$95$0 * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -15500

   1. Initial program 36.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 99.8%

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

   if -15500 < y < 17000

   1. Initial program 100.0%

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

   if 17000 < y

   1. Initial program 27.9%

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

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \left(1 + \left(\frac{x + -1}{y} - x\right)\right) - \frac{x}{y \cdot y}, \frac{1}{y \cdot \left(y \cdot y\right)}\right)\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+144}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+20}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(y, y, 1\right) - y\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+161}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))
   (if (<= t_0 -4e+144)
     x
     (if (<= t_0 -5e+20)
       (* y x)
       (if (<= t_0 0.0)
         x
         (if (<= t_0 200.0)
           (- (fma y y 1.0) y)
           (if (<= t_0 4e+161) (* y x) x)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= -4e+144) {
		tmp = x;
	} else if (t_0 <= -5e+20) {
		tmp = y * x;
	} else if (t_0 <= 0.0) {
		tmp = x;
	} else if (t_0 <= 200.0) {
		tmp = fma(y, y, 1.0) - y;
	} else if (t_0 <= 4e+161) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= -4e+144)
		tmp = x;
	elseif (t_0 <= -5e+20)
		tmp = Float64(y * x);
	elseif (t_0 <= 0.0)
		tmp = x;
	elseif (t_0 <= 200.0)
		tmp = Float64(fma(y, y, 1.0) - y);
	elseif (t_0 <= 4e+161)
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+144], x, If[LessEqual[t$95$0, -5e+20], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], x, If[LessEqual[t$95$0, 200.0], N[(N[(y * y + 1.0), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[t$95$0, 4e+161], N[(y * x), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -4.00000000000000009e144 or -5e20 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.0 or 4.0000000000000002e161 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 26.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. associate-/l/ N/A

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

      17. associate-/r/ N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-lft-identity 65.7

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

   7. Applied rewrites 65.7%

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

   if -4.00000000000000009e144 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -5e20 or 200 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 4.0000000000000002e161

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.5

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

   8. Applied rewrites 72.5%

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

   if 0.0 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 200

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in y around 0

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

      1. distribute-rgt-out-- N/A

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

      2. unpow2 N/A

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

      3. *-lft-identity N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 93.8

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

   8. Applied rewrites 93.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq -4 \cdot 10^{+144}:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq -5 \cdot 10^{+20}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 0:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 200:\\ \;\;\;\;\mathsf{fma}\left(y, y, 1\right) - y\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 4 \cdot 10^{+161}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+165}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq -2000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.4:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 10^{+144}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0))))
   (if (<= t_0 -1e+165)
     x
     (if (<= t_0 -2000000000.0)
       (* y x)
       (if (<= t_0 0.4)
         (- 1.0 y)
         (if (<= t_0 2e+19) x (if (<= t_0 1e+144) (* y x) x)))))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= -1e+165) {
		tmp = x;
	} else if (t_0 <= -2000000000.0) {
		tmp = y * x;
	} else if (t_0 <= 0.4) {
		tmp = 1.0 - y;
	} else if (t_0 <= 2e+19) {
		tmp = x;
	} else if (t_0 <= 1e+144) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (1.0d0 - x)) / (y + 1.0d0)
    if (t_0 <= (-1d+165)) then
        tmp = x
    else if (t_0 <= (-2000000000.0d0)) then
        tmp = y * x
    else if (t_0 <= 0.4d0) then
        tmp = 1.0d0 - y
    else if (t_0 <= 2d+19) then
        tmp = x
    else if (t_0 <= 1d+144) then
        tmp = y * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double tmp;
	if (t_0 <= -1e+165) {
		tmp = x;
	} else if (t_0 <= -2000000000.0) {
		tmp = y * x;
	} else if (t_0 <= 0.4) {
		tmp = 1.0 - y;
	} else if (t_0 <= 2e+19) {
		tmp = x;
	} else if (t_0 <= 1e+144) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * (1.0 - x)) / (y + 1.0)
	tmp = 0
	if t_0 <= -1e+165:
		tmp = x
	elif t_0 <= -2000000000.0:
		tmp = y * x
	elif t_0 <= 0.4:
		tmp = 1.0 - y
	elif t_0 <= 2e+19:
		tmp = x
	elif t_0 <= 1e+144:
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+165)
		tmp = x;
	elseif (t_0 <= -2000000000.0)
		tmp = Float64(y * x);
	elseif (t_0 <= 0.4)
		tmp = Float64(1.0 - y);
	elseif (t_0 <= 2e+19)
		tmp = x;
	elseif (t_0 <= 1e+144)
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * (1.0 - x)) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= -1e+165)
		tmp = x;
	elseif (t_0 <= -2000000000.0)
		tmp = y * x;
	elseif (t_0 <= 0.4)
		tmp = 1.0 - y;
	elseif (t_0 <= 2e+19)
		tmp = x;
	elseif (t_0 <= 1e+144)
		tmp = y * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+165], x, If[LessEqual[t$95$0, -2000000000.0], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 0.4], N[(1.0 - y), $MachinePrecision], If[LessEqual[t$95$0, 2e+19], x, If[LessEqual[t$95$0, 1e+144], N[(y * x), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -9.99999999999999899e164 or 0.40000000000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 2e19 or 1.00000000000000002e144 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 27.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. associate-/l/ N/A

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

      17. associate-/r/ N/A

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

   4. Applied rewrites 17.2%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-lft-identity 64.6

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

   7. Applied rewrites 64.6%

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

   if -9.99999999999999899e164 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -2e9 or 2e19 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.00000000000000002e144

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.5

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

   8. Applied rewrites 72.5%

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

   if -2e9 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 95.3

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

   8. Applied rewrites 95.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq -1 \cdot 10^{+165}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq -2000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 0.4:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 2 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 10^{+144}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)}{y}\\ \mathbf{if}\;y \leq -15500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 (fma (/ (+ x -1.0) y) (+ (/ 1.0 y) -1.0) x)) y))))
   (if (<= y -15500.0)
     t_0
     (if (<= y 17000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - fma(((x + -1.0) / y), ((1.0 / y) + -1.0), x)) / y);
	double tmp;
	if (y <= -15500.0) {
		tmp = t_0;
	} else if (y <= 17000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - fma(Float64(Float64(x + -1.0) / y), Float64(Float64(1.0 / y) + -1.0), x)) / y))
	tmp = 0.0
	if (y <= -15500.0)
		tmp = t_0;
	elseif (y <= 17000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -15500.0], t$95$0, If[LessEqual[y, 17000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -15500 or 17000 < y

   1. Initial program 33.1%

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

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

   if -15500 < y < 17000

   1. Initial program 100.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)}{y}\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -270000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{1}{y} + -1, x\right)\\ \mathbf{elif}\;y \leq 290000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y \cdot y} - \left(t\_0 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -270000.0)
     (fma t_0 (+ (/ 1.0 y) -1.0) x)
     (if (<= y 290000.0)
       (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
       (- (/ (+ x -1.0) (* y y)) (- t_0 x))))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -270000.0) {
		tmp = fma(t_0, ((1.0 / y) + -1.0), x);
	} else if (y <= 290000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = ((x + -1.0) / (y * y)) - (t_0 - x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -270000.0)
		tmp = fma(t_0, Float64(Float64(1.0 / y) + -1.0), x);
	elseif (y <= 290000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(Float64(Float64(x + -1.0) / Float64(y * y)) - Float64(t_0 - x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -270000.0], N[(t$95$0 * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 290000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7e5

   1. Initial program 36.2%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   if -2.7e5 < y < 2.9e5

   1. Initial program 99.7%

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

   if 2.9e5 < y

   1. Initial program 26.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. associate-/l/ N/A

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

      17. associate-/r/ N/A

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

   4. Applied rewrites 10.7%

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

   5. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. associate-+r+ N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)\\ \mathbf{elif}\;y \leq 290000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y \cdot y} - \left(\frac{x + -1}{y} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)\\ \mathbf{if}\;y \leq -270000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 290000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (+ x -1.0) y) (+ (/ 1.0 y) -1.0) x)))
   (if (<= y -270000.0)
     t_0
     (if (<= y 290000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(((x + -1.0) / y), ((1.0 / y) + -1.0), x);
	double tmp;
	if (y <= -270000.0) {
		tmp = t_0;
	} else if (y <= 290000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(x + -1.0) / y), Float64(Float64(1.0 / y) + -1.0), x)
	tmp = 0.0
	if (y <= -270000.0)
		tmp = t_0;
	elseif (y <= 290000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -270000.0], t$95$0, If[LessEqual[y, 290000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e5 or 2.9e5 < y

   1. Initial program 32.3%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   if -2.7e5 < y < 2.9e5

   1. Initial program 99.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)\\ \mathbf{elif}\;y \leq 290000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -56000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -56000000.0)
     t_0
     (if (<= y 155000000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -56000000.0) {
		tmp = t_0;
	} else if (y <= 155000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) / y)
    if (y <= (-56000000.0d0)) then
        tmp = t_0
    else if (y <= 155000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -56000000.0) {
		tmp = t_0;
	} else if (y <= 155000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) / y)
	tmp = 0
	if y <= -56000000.0:
		tmp = t_0
	elif y <= 155000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -56000000.0)
		tmp = t_0;
	elseif (y <= 155000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -56000000.0)
		tmp = t_0;
	elseif (y <= 155000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -56000000.0], t$95$0, If[LessEqual[y, 155000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6e7 or 1.55e8 < y

   1. Initial program 32.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -5.6e7 < y < 1.55e8

   1. Initial program 99.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -56000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 155000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8.5:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 8.5) (fma y (+ x -1.0) 1.0) (if (<= y 6.6e+100) (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 8.5) {
		tmp = fma(y, (x + -1.0), 1.0);
	} else if (y <= 6.6e+100) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 8.5)
		tmp = fma(y, Float64(x + -1.0), 1.0);
	elseif (y <= 6.6e+100)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 8.5], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 6.6e+100], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 6.6000000000000002e100 < y

   1. Initial program 32.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. associate-/l/ N/A

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

      17. associate-/r/ N/A

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

   4. Applied rewrites 15.0%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-lft-identity 77.7

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

   7. Applied rewrites 77.7%

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

   if -1 < y < 8.5

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if 8.5 < y < 6.6000000000000002e100

   1. Initial program 36.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 10.2

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 61.2

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

   8. Applied rewrites 61.2%

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

Alternative 9: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. neg-sub0 N/A

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

      12. associate-+l- N/A

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

      13. neg-sub0 N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. distribute-lft-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 98.2%

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

Alternative 10: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 11: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 97.9

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

   8. Applied rewrites 97.9%

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

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 12: 86.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. associate-/l/ N/A

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

      17. associate-/r/ N/A

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

   4. Applied rewrites 17.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-lft-identity 71.2

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

   7. Applied rewrites 71.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 13: 74.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.98d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.98:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.98)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.98)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.98], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.97999999999999998 < y

   1. Initial program 33.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. associate-/l/ N/A

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

      17. associate-/r/ N/A

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

   4. Applied rewrites 17.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-lft-identity 71.2

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

   7. Applied rewrites 71.2%

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

   if -1 < y < 0.97999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 65.4

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

   8. Applied rewrites 65.4%

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

Alternative 14: 74.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 225000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 225000.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 225000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 225000.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 225000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 225000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 225000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 225000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 225000.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 225000 < y

   1. Initial program 32.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. associate-/l/ N/A

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

      17. associate-/r/ N/A

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-lft-identity 71.7

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

   7. Applied rewrites 71.7%

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

   if -1 < y < 225000

   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

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

   5. Applied rewrites 64.4%

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

Alternative 15: 39.0% 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 68.1%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 35.6%

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

Alternative 16: 3.1% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 68.1%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 37.7

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

5. Applied rewrites 37.7%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 3.1%

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

   1. metadata-eval 3.1

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

10. Applied rewrites 3.1%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 y) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = 1.0 - (((1.0 - x) * y) / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024216 
(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))))