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

Percentage Accurate: 66.0% → 99.9%

Time: 10.5s

Alternatives: 16

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: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -130000:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \left(1 + \left(t\_0 - x\right)\right) - \frac{x}{y \cdot y}, \frac{1}{y \cdot \left(y \cdot y\right)}\right)\\ \mathbf{elif}\;y \leq 15000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(t\_0, \frac{1}{y} + -1, x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -130000.0)
     (+
      x
      (fma
       (/ 1.0 y)
       (- (+ 1.0 (- t_0 x)) (/ x (* y y)))
       (/ 1.0 (* y (* y y)))))
     (if (<= y 15000.0)
       (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0)
       (+ x (/ (- 1.0 (fma t_0 (+ (/ 1.0 y) -1.0) x)) y))))))

C

double code(double x, double y) {
	double t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -130000.0) {
		tmp = x + fma((1.0 / y), ((1.0 + (t_0 - x)) - (x / (y * y))), (1.0 / (y * (y * y))));
	} else if (y <= 15000.0) {
		tmp = fma((y / (-1.0 - y)), (1.0 - x), 1.0);
	} else {
		tmp = x + ((1.0 - fma(t_0, ((1.0 / y) + -1.0), x)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -130000.0)
		tmp = Float64(x + fma(Float64(1.0 / y), Float64(Float64(1.0 + Float64(t_0 - x)) - Float64(x / Float64(y * y))), Float64(1.0 / Float64(y * Float64(y * y)))));
	elseif (y <= 15000.0)
		tmp = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 1.0);
	else
		tmp = Float64(x + Float64(Float64(1.0 - fma(t_0, Float64(Float64(1.0 / y) + -1.0), x)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -130000.0], N[(x + N[(N[(1.0 / y), $MachinePrecision] * N[(N[(1.0 + N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision] - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 15000.0], N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(N[(1.0 - N[(t$95$0 * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e5

   1. Initial program 35.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

   if -1.3e5 < y < 15000

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if 15000 < y

   1. Initial program 30.1%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(1 - x\right)}{y + 1}\\ t_1 := \mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{if}\;t\_0 \leq 0.004:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{y}, \frac{1}{y} + -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y (- 1.0 x)) (+ y 1.0)))
        (t_1 (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0)))
   (if (<= t_0 0.004)
     t_1
     (if (<= t_0 1.02) (fma (/ -1.0 y) (+ (/ 1.0 y) -1.0) x) t_1))))

C

double code(double x, double y) {
	double t_0 = (y * (1.0 - x)) / (y + 1.0);
	double t_1 = fma((y / (-1.0 - y)), (1.0 - x), 1.0);
	double tmp;
	if (t_0 <= 0.004) {
		tmp = t_1;
	} else if (t_0 <= 1.02) {
		tmp = fma((-1.0 / y), ((1.0 / y) + -1.0), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * Float64(1.0 - x)) / Float64(y + 1.0))
	t_1 = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 1.0)
	tmp = 0.0
	if (t_0 <= 0.004)
		tmp = t_1;
	elseif (t_0 <= 1.02)
		tmp = fma(Float64(-1.0 / y), Float64(Float64(1.0 / y) + -1.0), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.004], t$95$1, If[LessEqual[t$95$0, 1.02], N[(N[(-1.0 / y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.0040000000000000001 or 1.02 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 83.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if 0.0040000000000000001 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.02

   1. Initial program 12.8%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{elif}\;\frac{y \cdot \left(1 - x\right)}{y + 1} \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{y}, \frac{1}{y} + -1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 0.005:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))
   (if (<= t_0 0.005) x (if (<= t_0 2.0) (- 1.0 y) x))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.005) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    if (t_0 <= 0.005d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.005) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	tmp = 0
	if t_0 <= 0.005:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= 0.005)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	tmp = 0.0;
	if (t_0 <= 0.005)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.005], x, If[LessEqual[t$95$0, 2.0], N[(1.0 - y), $MachinePrecision], x]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.0050000000000000001 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 47.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. remove-double-neg 59.1

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

   7. Applied rewrites 59.1%

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

   if 0.0050000000000000001 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 95.4

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

   8. Applied rewrites 95.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 0.005:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, \frac{1}{y} + -1, x\right)}{y}\\ \mathbf{if}\;y \leq -130000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 15000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 (fma (/ (+ x -1.0) y) (+ (/ 1.0 y) -1.0) x)) y))))
   (if (<= y -130000.0)
     t_0
     (if (<= y 15000.0) (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - fma(((x + -1.0) / y), ((1.0 / y) + -1.0), x)) / y);
	double tmp;
	if (y <= -130000.0) {
		tmp = t_0;
	} else if (y <= 15000.0) {
		tmp = fma((y / (-1.0 - y)), (1.0 - x), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - fma(Float64(Float64(x + -1.0) / y), Float64(Float64(1.0 / y) + -1.0), x)) / y))
	tmp = 0.0
	if (y <= -130000.0)
		tmp = t_0;
	elseif (y <= 15000.0)
		tmp = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -130000.0], t$95$0, If[LessEqual[y, 15000.0], N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e5 or 15000 < y

   1. Initial program 32.6%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

   if -1.3e5 < y < 15000

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

Alternative 5: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 0.005:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))
   (if (<= t_0 0.005) x (if (<= t_0 2.0) 1.0 x))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.005) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    if (t_0 <= 0.005d0) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.005) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	tmp = 0
	if t_0 <= 0.005:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= 0.005)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	tmp = 0.0;
	if (t_0 <= 0.005)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.005], x, If[LessEqual[t$95$0, 2.0], 1.0, x]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.0050000000000000001 or 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 47.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 64.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. remove-double-neg 59.1

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

   7. Applied rewrites 59.1%

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

   if 0.0050000000000000001 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 0.005:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (+ x -1.0) y) (+ (/ 1.0 y) -1.0) x)))
   (if (<= y -310000.0)
     t_0
     (if (<= y 310000.0) (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(((x + -1.0) / y), ((1.0 / y) + -1.0), x);
	double tmp;
	if (y <= -310000.0) {
		tmp = t_0;
	} else if (y <= 310000.0) {
		tmp = fma((y / (-1.0 - y)), (1.0 - x), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(x + -1.0) / y), Float64(Float64(1.0 / y) + -1.0), x)
	tmp = 0.0
	if (y <= -310000.0)
		tmp = t_0;
	elseif (y <= 310000.0)
		tmp = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -310000.0], t$95$0, If[LessEqual[y, 310000.0], N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1e5 or 3.1e5 < y

   1. Initial program 32.6%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 99.7%

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

   if -3.1e5 < y < 3.1e5

   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

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 100.0%

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

Alternative 7: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1}{y}\\ \mathbf{if}\;y \leq -6200000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 13500000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y))))
   (if (<= y -6200000000000.0)
     t_0
     (if (<= y 13500000000.0) (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 / y);
	double tmp;
	if (y <= -6200000000000.0) {
		tmp = t_0;
	} else if (y <= 13500000000.0) {
		tmp = fma((y / (-1.0 - y)), (1.0 - x), 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 <= -6200000000000.0)
		tmp = t_0;
	elseif (y <= 13500000000.0)
		tmp = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 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, -6200000000000.0], t$95$0, If[LessEqual[y, 13500000000.0], N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e12 or 1.35e10 < y

   1. Initial program 30.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.8%

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

   if -6.2e12 < y < 1.35e10

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

Alternative 8: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.8:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 3.8) (fma y (+ x -1.0) 1.0) (if (<= y 7.8e+40) (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 3.8) {
		tmp = fma(y, (x + -1.0), 1.0);
	} else if (y <= 7.8e+40) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 3.8)
		tmp = fma(y, Float64(x + -1.0), 1.0);
	elseif (y <= 7.8e+40)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 3.8], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 7.8e+40], N[(1.0 / y), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 7.8000000000000002e40 < y

   1. Initial program 33.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. remove-double-neg 77.2

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

   7. Applied rewrites 77.2%

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

   if -1 < y < 3.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 3.7999999999999998 < y < 7.8000000000000002e40

   1. Initial program 41.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 68.5

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

   8. Applied rewrites 68.5%

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

Alternative 9: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 37.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -1 < y < 0.849999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. neg-sub0 N/A

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

      12. associate-+l- N/A

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

      13. neg-sub0 N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. distribute-lft-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   if 0.849999999999999978 < y

   1. Initial program 30.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.2%

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

Alternative 10: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 37.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 0.80000000000000004 < y

   1. Initial program 30.1%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.2%

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

Alternative 11: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 33.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.2%

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

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 12: 73.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y, y, 1\right) - y\\ \mathbf{elif}\;y \leq 21500000:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 1.02e-76) (- (fma y y 1.0) y) (if (<= y 21500000.0) (* y x) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.02e-76) {
		tmp = fma(y, y, 1.0) - y;
	} else if (y <= 21500000.0) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.02e-76)
		tmp = Float64(fma(y, y, 1.0) - y);
	elseif (y <= 21500000.0)
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.02e-76], N[(N[(y * y + 1.0), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[y, 21500000.0], N[(y * x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 2.15e7 < y

   1. Initial program 33.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. remove-double-neg 74.2

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

   7. Applied rewrites 74.2%

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

   if -1 < y < 1.02000000000000006e-76

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 76.4

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. distribute-rgt-in N/A

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

       4. unpow2 N/A

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

       5. associate-+r+ N/A

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

       6. mul-1-neg N/A

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

       7. unsub-neg N/A

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

       8. lft-mult-inverse N/A

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

       9. distribute-lft1-in N/A

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

       10. +-commutative N/A

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

       11. lower--.f64 N/A

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

       12. +-commutative N/A

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

       13. distribute-rgt1-in N/A

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

       14. unpow2 N/A

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

       15. lft-mult-inverse N/A

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

       16. lower-fma.f64 76.4

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

   11. Applied rewrites 76.4%

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

   if 1.02000000000000006e-76 < y < 2.15e7

   1. Initial program 97.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.9

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

   8. Applied rewrites 54.9%

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

Alternative 13: 73.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-76}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 21500000:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 1.02e-76) (- 1.0 y) (if (<= y 21500000.0) (* y x) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.02e-76) {
		tmp = 1.0 - y;
	} else if (y <= 21500000.0) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 1.02d-76) then
        tmp = 1.0d0 - y
    else if (y <= 21500000.0d0) then
        tmp = y * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.02e-76) {
		tmp = 1.0 - y;
	} else if (y <= 21500000.0) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.02e-76:
		tmp = 1.0 - y
	elif y <= 21500000.0:
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.02e-76)
		tmp = Float64(1.0 - y);
	elseif (y <= 21500000.0)
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.02e-76)
		tmp = 1.0 - y;
	elseif (y <= 21500000.0)
		tmp = y * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.02e-76], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 21500000.0], N[(y * x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 2.15e7 < y

   1. Initial program 33.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. remove-double-neg 74.2

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

   7. Applied rewrites 74.2%

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

   if -1 < y < 1.02000000000000006e-76

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 76.3

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

   8. Applied rewrites 76.3%

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

   if 1.02000000000000006e-76 < y < 2.15e7

   1. Initial program 97.5%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.9

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

   8. Applied rewrites 54.9%

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

Alternative 14: 86.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 33.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. neg-sub0 N/A

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

      7. remove-double-neg 73.2

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

   7. Applied rewrites 73.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 15: 38.8% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 64.7%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 34.1%

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

Alternative 16: 3.1% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 64.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 35.7

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

5. Applied rewrites 35.7%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 3.1%

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

   1. metadata-eval 3.1

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

10. Applied rewrites 3.1%

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

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))))