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

Percentage Accurate: 65.9% → 99.9%

Time: 6.9s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \mathbf{if}\;y \leq -10000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 12000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\left(-1 - y\right) - \left(-1 - y\right) \cdot x}{\left(-1 - y\right) \cdot \left(-1 - y\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) (- x 1.0)) y))))
   (if (<= y -10000.0)
     t_0
     (if (<= y 12000.0)
       (fma
        y
        (/ (- (- -1.0 y) (* (- -1.0 y) x)) (* (- -1.0 y) (- -1.0 y)))
        1.0)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x - (fma(((1.0 - x) / y), (1.0 - (1.0 / y)), (x - 1.0)) / y);
	double tmp;
	if (y <= -10000.0) {
		tmp = t_0;
	} else if (y <= 12000.0) {
		tmp = fma(y, (((-1.0 - y) - ((-1.0 - y) * x)) / ((-1.0 - y) * (-1.0 - y))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), Float64(x - 1.0)) / y))
	tmp = 0.0
	if (y <= -10000.0)
		tmp = t_0;
	elseif (y <= 12000.0)
		tmp = fma(y, Float64(Float64(Float64(-1.0 - y) - Float64(Float64(-1.0 - y) * x)) / Float64(Float64(-1.0 - y) * Float64(-1.0 - y))), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -10000.0], t$95$0, If[LessEqual[y, 12000.0], N[(y * N[(N[(N[(-1.0 - y), $MachinePrecision] - N[(N[(-1.0 - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - y), $MachinePrecision] * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e4 or 12000 < 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

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

   5. Applied rewrites 100.0%

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

   if -1e4 < y < 12000

   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. frac-sub N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \mathbf{elif}\;y \leq 12000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\left(-1 - y\right) - \left(-1 - y\right) \cdot x}{\left(-1 - y\right) \cdot \left(-1 - y\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x 1.0) y) (- -1.0 y))))
   (if (<= t_0 0.5)
     (- 1.0 (- x))
     (if (<= t_0 2e+68) x (if (<= t_0 2e+201) (* x y) (- 1.0 (- 1.0 x)))))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) * y) / (-1.0 - y);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = 1.0 - -x;
	} else if (t_0 <= 2e+68) {
		tmp = x;
	} else if (t_0 <= 2e+201) {
		tmp = x * y;
	} else {
		tmp = 1.0 - (1.0 - 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 = ((x - 1.0d0) * y) / ((-1.0d0) - y)
    if (t_0 <= 0.5d0) then
        tmp = 1.0d0 - -x
    else if (t_0 <= 2d+68) then
        tmp = x
    else if (t_0 <= 2d+201) then
        tmp = x * y
    else
        tmp = 1.0d0 - (1.0d0 - x)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = ((x - 1.0) * y) / (-1.0 - y)
	tmp = 0
	if t_0 <= 0.5:
		tmp = 1.0 - -x
	elif t_0 <= 2e+68:
		tmp = x
	elif t_0 <= 2e+201:
		tmp = x * y
	else:
		tmp = 1.0 - (1.0 - x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(1.0 - (-x)), $MachinePrecision], If[LessEqual[t$95$0, 2e+68], x, If[LessEqual[t$95$0, 2e+201], N[(x * y), $MachinePrecision], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $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))) < 0.5

   1. Initial program 88.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.1%

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

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

   1. Initial program 24.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 24.5

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

   4. Applied rewrites 24.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. frac-sub N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 21.9%

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

   7. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. neg-mul-1 N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity 60.1

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

   9. Applied rewrites 60.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.6%

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

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

   1. Initial program 27.8%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 68.3%

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

Alternative 3: 61.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x 1.0) y) (- -1.0 y))))
   (if (<= t_0 0.5)
     (- 1.0 (- x))
     (if (<= t_0 2e+68) x (if (<= t_0 2e+201) (* x y) x)))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) * y) / (-1.0 - y);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = 1.0 - -x;
	} else if (t_0 <= 2e+68) {
		tmp = x;
	} else if (t_0 <= 2e+201) {
		tmp = x * 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) :: t_0
    real(8) :: tmp
    t_0 = ((x - 1.0d0) * y) / ((-1.0d0) - y)
    if (t_0 <= 0.5d0) then
        tmp = 1.0d0 - -x
    else if (t_0 <= 2d+68) then
        tmp = x
    else if (t_0 <= 2d+201) then
        tmp = x * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = ((x - 1.0) * y) / (-1.0 - y)
	tmp = 0
	if t_0 <= 0.5:
		tmp = 1.0 - -x
	elif t_0 <= 2e+68:
		tmp = x
	elif t_0 <= 2e+201:
		tmp = x * y
	else:
		tmp = x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(1.0 - (-x)), $MachinePrecision], If[LessEqual[t$95$0, 2e+68], x, If[LessEqual[t$95$0, 2e+201], N[(x * y), $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))) < 0.5

   1. Initial program 88.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.1%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. frac-sub N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. neg-mul-1 N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity 66.4

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

   9. Applied rewrites 66.4%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.6%

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

4. Final simplification 68.3%

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

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -125000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 240000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\left(-1 - y\right) - \left(-1 - y\right) \cdot x}{\left(-1 - y\right) \cdot \left(-1 - y\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -125000000000.0)
     t_0
     (if (<= y 240000000.0)
       (fma
        y
        (/ (- (- -1.0 y) (* (- -1.0 y) x)) (* (- -1.0 y) (- -1.0 y)))
        1.0)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -125000000000.0) {
		tmp = t_0;
	} else if (y <= 240000000.0) {
		tmp = fma(y, (((-1.0 - y) - ((-1.0 - y) * x)) / ((-1.0 - y) * (-1.0 - y))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(x - 1.0) / y))
	tmp = 0.0
	if (y <= -125000000000.0)
		tmp = t_0;
	elseif (y <= 240000000.0)
		tmp = fma(y, Float64(Float64(Float64(-1.0 - y) - Float64(Float64(-1.0 - y) * x)) / Float64(Float64(-1.0 - y) * Float64(-1.0 - y))), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -125000000000.0], t$95$0, If[LessEqual[y, 240000000.0], N[(y * N[(N[(N[(-1.0 - y), $MachinePrecision] - N[(N[(-1.0 - y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - y), $MachinePrecision] * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e11 or 2.4e8 < 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 < 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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. frac-sub N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 240000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\left(-1 - y\right) - \left(-1 - y\right) \cdot x}{\left(-1 - y\right) \cdot \left(-1 - y\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -125000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -125000000000.0)
     t_0
     (if (<= y 250000000.0) (fma y (/ (- x 1.0) (- y -1.0)) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -125000000000.0) {
		tmp = t_0;
	} else if (y <= 250000000.0) {
		tmp = fma(y, ((x - 1.0) / (y - -1.0)), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(x - 1.0) / y))
	tmp = 0.0
	if (y <= -125000000000.0)
		tmp = t_0;
	elseif (y <= 250000000.0)
		tmp = fma(y, Float64(Float64(x - 1.0) / Float64(y - -1.0)), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -125000000000.0], t$95$0, If[LessEqual[y, 250000000.0], N[(y * N[(N[(x - 1.0), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e11 or 2.5e8 < 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 < 2.5e8

   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -125000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x - 1}{y - -1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 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

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

   5. Applied rewrites 98.6%

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

4. Final simplification 99.0%

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

Alternative 7: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 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. Add Preprocessing

Alternative 8: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.83:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.83) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.83) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.83)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.83], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.82999999999999996 < 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. Add Preprocessing

Alternative 9: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 1.1) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.1) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.1)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.1], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1000000000000001 < 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 < 1.1000000000000001

   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. Add Preprocessing

Alternative 10: 86.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 55.0

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

   4. Applied rewrites 55.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. frac-sub N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 23.7%

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

   7. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. neg-mul-1 N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity 79.2

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

   9. Applied rewrites 79.2%

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

   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. Add Preprocessing

Alternative 11: 49.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.2e-10) x (if (<= y 1.0) (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.2e-10) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = x * 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 <= (-4.2d-10)) then
        tmp = x
    else if (y <= 1.0d0) then
        tmp = x * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.2e-10) {
		tmp = x;
	} else if (y <= 1.0) {
		tmp = x * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.2e-10:
		tmp = x
	elif y <= 1.0:
		tmp = x * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.2e-10)
		tmp = x;
	elseif (y <= 1.0)
		tmp = Float64(x * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.2e-10)
		tmp = x;
	elseif (y <= 1.0)
		tmp = x * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.2e-10], x, If[LessEqual[y, 1.0], N[(x * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e-10 or 1 < y

   1. Initial program 30.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. frac-sub N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. lower-/.f64 N/A

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

   6. Applied rewrites 24.3%

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

   7. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. neg-mul-1 N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity 78.6

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

   9. Applied rewrites 78.6%

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

   if -4.2e-10 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 33.5

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

   5. Applied rewrites 33.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 32.7%

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

Alternative 12: 38.5% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 65.1%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 77.5

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

4. Applied rewrites 77.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-in N/A

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

   9. metadata-eval N/A

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

   10. frac-2neg N/A

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

   11. frac-sub N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-in N/A

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

   17. lower-/.f64 N/A

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

6. Applied rewrites 61.8%

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

7. Taylor expanded in y around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. neg-mul-1 N/A

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

   5. remove-double-neg N/A

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

   6. associate-+r+ N/A

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

   7. metadata-eval N/A

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

   8. +-lft-identity 41.4

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

9. Applied rewrites 41.4%

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