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

Percentage Accurate: 65.6% → 99.4%

Time: 7.9s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.6% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.32 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 115000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.32e+21)
   (- x (/ -1.0 y))
   (if (<= y 115000.0)
     (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))
     (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.32e+21) {
		tmp = x - (-1.0 / y);
	} else if (y <= 115000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = fma(((1.0 - x) / y), (1.0 - (1.0 / y)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.32e+21)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 115000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.32e+21], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 115000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.32e21

   1. Initial program 32.4%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if -2.32e21 < y < 115000

   1. Initial program 100.0%

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

   if 115000 < y

   1. Initial program 34.0%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. div-sub N/A

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

      12. neg-sub0 N/A

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

      13. mul-1-neg N/A

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

      14. associate-+l+ N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.32 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 115000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq -1000:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x 1.0) y) (- -1.0 y))))
   (if (<= t_0 (- INFINITY))
     (* 1.0 x)
     (if (<= t_0 -5e+98)
       (* x y)
       (if (<= t_0 -1000.0)
         (* 1.0 x)
         (if (<= t_0 5e-10) (- 1.0 y) (* 1.0 x)))))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) * y) / (-1.0 - y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 1.0 * x;
	} else if (t_0 <= -5e+98) {
		tmp = x * y;
	} else if (t_0 <= -1000.0) {
		tmp = 1.0 * x;
	} else if (t_0 <= 5e-10) {
		tmp = 1.0 - y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = ((x - 1.0) * y) / (-1.0 - y);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 * x;
	} else if (t_0 <= -5e+98) {
		tmp = x * y;
	} else if (t_0 <= -1000.0) {
		tmp = 1.0 * x;
	} else if (t_0 <= 5e-10) {
		tmp = 1.0 - y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - 1.0) * y) / (-1.0 - y)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 1.0 * x
	elif t_0 <= -5e+98:
		tmp = x * y
	elif t_0 <= -1000.0:
		tmp = 1.0 * x
	elif t_0 <= 5e-10:
		tmp = 1.0 - y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(1.0 * x);
	elseif (t_0 <= -5e+98)
		tmp = Float64(x * y);
	elseif (t_0 <= -1000.0)
		tmp = Float64(1.0 * x);
	elseif (t_0 <= 5e-10)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - 1.0) * y) / (-1.0 - y);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 1.0 * x;
	elseif (t_0 <= -5e+98)
		tmp = x * y;
	elseif (t_0 <= -1000.0)
		tmp = 1.0 * x;
	elseif (t_0 <= 5e-10)
		tmp = 1.0 - y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$0, -5e+98], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, -1000.0], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$0, 5e-10], N[(1.0 - y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -inf.0 or -4.9999999999999998e98 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -1e3 or 5.00000000000000031e-10 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 41.4%

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

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

      6. lower-+.f64 80.7

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 67.1%

      \[\leadsto 1 \cdot x \]

   if -inf.0 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -4.9999999999999998e98

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.7%

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

   if -1e3 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 5.00000000000000031e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -\infty:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -5 \cdot 10^{+98}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -1000:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 40000000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))))
   (if (<= t_0 0.02)
     (* x y)
     (if (<= t_0 40000000000000.0) (- 1.0 y) (* x y)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.02) {
		tmp = x * y;
	} else if (t_0 <= 40000000000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    if (t_0 <= 0.02d0) then
        tmp = x * y
    else if (t_0 <= 40000000000000.0d0) then
        tmp = 1.0d0 - y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= 0.02) {
		tmp = x * y;
	} else if (t_0 <= 40000000000000.0) {
		tmp = 1.0 - y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	tmp = 0
	if t_0 <= 0.02:
		tmp = x * y
	elif t_0 <= 40000000000000.0:
		tmp = 1.0 - y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= 0.02)
		tmp = Float64(x * y);
	elseif (t_0 <= 40000000000000.0)
		tmp = Float64(1.0 - y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	tmp = 0.0;
	if (t_0 <= 0.02)
		tmp = x * y;
	elseif (t_0 <= 40000000000000.0)
		tmp = 1.0 - y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.02], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 40000000000000.0], N[(1.0 - y), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.0200000000000000004 or 4e13 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 48.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 23.0

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

   5. Applied rewrites 23.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 23.7%

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

   if 0.0200000000000000004 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 4e13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.5%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 0.02:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 40000000000000:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.32 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 13600000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.32e+21)
   (- x (/ -1.0 y))
   (if (<= y 13600000.0)
     (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))
     (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.32e+21) {
		tmp = x - (-1.0 / y);
	} else if (y <= 13600000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.32d+21)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 13600000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = x - ((x - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.32e+21) {
		tmp = x - (-1.0 / y);
	} else if (y <= 13600000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.32e+21:
		tmp = x - (-1.0 / y)
	elif y <= 13600000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = x - ((x - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.32e+21)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 13600000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.32e+21)
		tmp = x - (-1.0 / y);
	elseif (y <= 13600000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = x - ((x - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.32e+21], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 13600000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.32e21

   1. Initial program 32.4%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if -2.32e21 < y < 1.36e7

   1. Initial program 100.0%

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

   if 1.36e7 < y

   1. Initial program 34.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.32 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 13600000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.32 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 330000:\\ \;\;\;\;1 - \frac{x}{-1 - y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.32e+21)
   (- x (/ -1.0 y))
   (if (<= y 330000.0) (- 1.0 (* (/ x (- -1.0 y)) y)) (- x (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.32e+21) {
		tmp = x - (-1.0 / y);
	} else if (y <= 330000.0) {
		tmp = 1.0 - ((x / (-1.0 - y)) * y);
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.32d+21)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 330000.0d0) then
        tmp = 1.0d0 - ((x / ((-1.0d0) - y)) * y)
    else
        tmp = x - ((x - 1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.32e+21) {
		tmp = x - (-1.0 / y);
	} else if (y <= 330000.0) {
		tmp = 1.0 - ((x / (-1.0 - y)) * y);
	} else {
		tmp = x - ((x - 1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.32e+21:
		tmp = x - (-1.0 / y)
	elif y <= 330000.0:
		tmp = 1.0 - ((x / (-1.0 - y)) * y)
	else:
		tmp = x - ((x - 1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.32e+21)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 330000.0)
		tmp = Float64(1.0 - Float64(Float64(x / Float64(-1.0 - y)) * y));
	else
		tmp = Float64(x - Float64(Float64(x - 1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.32e+21)
		tmp = x - (-1.0 / y);
	elseif (y <= 330000.0)
		tmp = 1.0 - ((x / (-1.0 - y)) * y);
	else
		tmp = x - ((x - 1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.32e+21], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 330000.0], N[(1.0 - N[(N[(x / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.32e21

   1. Initial program 32.4%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if -2.32e21 < y < 3.3e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 3.3e5 < y

   1. Initial program 34.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 6: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 36.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.1%

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

Alternative 7: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 36.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.6%

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

Alternative 8: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, x, x\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 0.82) (fma (fma (- y) x x) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.82) {
		tmp = fma(fma(-y, x, x), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.82)
		tmp = fma(fma(Float64(-y), x, x), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.82], N[(N[((-y) * x + x), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.819999999999999951 < y

   1. Initial program 36.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.6%

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

   if -1 < y < 0.819999999999999951

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.6%

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

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.80000000000000004 < y

   1. Initial program 36.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.6%

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

   if -1 < y < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 10: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 1.15) (fma (- x 1.0) y 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.15) {
		tmp = fma((x - 1.0), y, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.15)
		tmp = fma(Float64(x - 1.0), y, 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.15], N[(N[(x - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1499999999999999 < y

   1. Initial program 36.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 77.5%

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

   if -1 < y < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 11: 86.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 36.0%

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

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

      6. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 77.0%

      \[\leadsto 1 \cdot x \]

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 12: 39.0% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 69.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 42.0%

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

Developer Target 1: 99.7% 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 2024257 
(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))))