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

Percentage Accurate: 66.6% → 99.9%

Time: 7.4s

Alternatives: 14

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{\mathsf{fma}\left(\frac{1 - x}{y}, 1 - \frac{1}{y}, x - 1\right)}{y}\\ \mathbf{if}\;y \leq -11000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 15000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (fma (/ (- 1.0 x) y) (- 1.0 (/ 1.0 y)) (- x 1.0)) y))))
   (if (<= y -11000.0)
     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 - (fma(((1.0 - x) / y), (1.0 - (1.0 / y)), (x - 1.0)) / y);
	double tmp;
	if (y <= -11000.0) {
		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(fma(Float64(Float64(1.0 - x) / y), Float64(1.0 - Float64(1.0 / y)), Float64(x - 1.0)) / y))
	tmp = 0.0
	if (y <= -11000.0)
		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[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -11000.0], 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 < -11000 or 15000 < y

   1. Initial program 29.4%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

   if -11000 < y < 15000

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 100.0

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

   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 2: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;t\_0 \leq 10^{+215}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y)))))
   (if (<= t_0 2e-6)
     x
     (if (<= t_0 2.0) (- 1.0 y) (if (<= t_0 1e+215) (* x y) x)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else if (t_0 <= 1e+215) {
		tmp = x * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    if (t_0 <= 2d-6) then
        tmp = x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - y
    else if (t_0 <= 1d+215) then
        tmp = x * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - y;
	} else if (t_0 <= 1e+215) {
		tmp = x * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	tmp = 0
	if t_0 <= 2e-6:
		tmp = x
	elif t_0 <= 2.0:
		tmp = 1.0 - y
	elif t_0 <= 1e+215:
		tmp = x * y
	else:
		tmp = x
	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 <= 2e-6)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - y);
	elseif (t_0 <= 1e+215)
		tmp = Float64(x * y);
	else
		tmp = x;
	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 <= 2e-6)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - y;
	elseif (t_0 <= 1e+215)
		tmp = x * y;
	else
		tmp = x;
	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, 2e-6], x, If[LessEqual[t$95$0, 2.0], N[(1.0 - y), $MachinePrecision], If[LessEqual[t$95$0, 1e+215], N[(x * y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 1.99999999999999991e-6 or 9.99999999999999907e214 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 32.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 52.8

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

   4. Applied rewrites 52.8%

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

   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 1.99999999999999991e-6 < (-.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

      \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.8%

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

   if 2 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 9.99999999999999907e214

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 2:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 10^{+215}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 1.0000000000002:\\ \;\;\;\;x - \frac{\frac{\frac{y - 1}{y}}{y} - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - -1} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x 1.0) y) (- -1.0 y))))
   (if (<= t_0 0.01)
     (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0)
     (if (<= t_0 1.0000000000002)
       (- x (/ (- (/ (/ (- y 1.0) y) y) 1.0) y))
       (* (/ y (- 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 <= 0.01) {
		tmp = fma((y / (-1.0 - y)), (1.0 - x), 1.0);
	} else if (t_0 <= 1.0000000000002) {
		tmp = x - (((((y - 1.0) / y) / y) - 1.0) / y);
	} else {
		tmp = (y / (y - -1.0)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 1.0);
	elseif (t_0 <= 1.0000000000002)
		tmp = Float64(x - Float64(Float64(Float64(Float64(Float64(y - 1.0) / y) / y) - 1.0) / y));
	else
		tmp = Float64(Float64(y / Float64(y - -1.0)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1.0000000000002], N[(x - N[(N[(N[(N[(N[(y - 1.0), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] * 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))) < 0.0100000000000000002

   1. Initial program 92.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.00000000000020006

   1. Initial program 5.9%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   if 1.00000000000020006 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 68.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 1.0000000000002:\\ \;\;\;\;x - \frac{\frac{\frac{y - 1}{y}}{y} - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - -1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq -50:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.9999988892766603:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x 1.0) y) (- -1.0 y))))
   (if (<= t_0 (- INFINITY))
     x
     (if (<= t_0 -50.0)
       (* x y)
       (if (<= t_0 0.9999988892766603) (fma (- y 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 = x;
	} else if (t_0 <= -50.0) {
		tmp = x * y;
	} else if (t_0 <= 0.9999988892766603) {
		tmp = fma((y - 1.0), y, 1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = x;
	elseif (t_0 <= -50.0)
		tmp = Float64(x * y);
	elseif (t_0 <= 0.9999988892766603)
		tmp = fma(Float64(y - 1.0), y, 1.0);
	else
		tmp = x;
	end
	return 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)], x, If[LessEqual[t$95$0, -50.0], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 0.9999988892766603], N[(N[(y - 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -inf.0 or 0.99999888927666025 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 31.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 52.7

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

   4. Applied rewrites 52.7%

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

   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 59.5

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

   7. Applied rewrites 59.5%

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < -50

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.2%

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

   if -50 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.99999888927666025

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(\left(x + y \cdot \left(1 - x\right)\right) - 1\right)} \]
   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 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq -50:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;\frac{\left(x - 1\right) \cdot y}{-1 - y} \leq 0.9999988892766603:\\ \;\;\;\;\mathsf{fma}\left(y - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \mathbf{if}\;y \leq -1850000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 190000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- (- x (/ (- x 1.0) y)) 1.0) y))))
   (if (<= y -1850000000000.0)
     t_0
     (if (<= y 190000.0) (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	double tmp;
	if (y <= -1850000000000.0) {
		tmp = t_0;
	} else if (y <= 190000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (((x - ((x - 1.0d0) / y)) - 1.0d0) / y)
    if (y <= (-1850000000000.0d0)) then
        tmp = t_0
    else if (y <= 190000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	double tmp;
	if (y <= -1850000000000.0) {
		tmp = t_0;
	} else if (y <= 190000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y)
	tmp = 0
	if y <= -1850000000000.0:
		tmp = t_0
	elif y <= 190000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(Float64(Float64(x - Float64(Float64(x - 1.0) / y)) - 1.0) / y))
	tmp = 0.0
	if (y <= -1850000000000.0)
		tmp = t_0;
	elseif (y <= 190000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (((x - ((x - 1.0) / y)) - 1.0) / y);
	tmp = 0.0;
	if (y <= -1850000000000.0)
		tmp = t_0;
	elseif (y <= 190000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1850000000000.0], t$95$0, If[LessEqual[y, 190000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85e12 or 1.9e5 < y

   1. Initial program 27.6%

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

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.8%

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

   if -1.85e12 < y < 1.9e5

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1850000000000:\\ \;\;\;\;x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \mathbf{elif}\;y \leq 190000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(x - \frac{x - 1}{y}\right) - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -1850000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 120000000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -1850000000000.0)
     t_0
     (if (<= y 120000000.0) (- 1.0 (/ (* (- x 1.0) y) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -1850000000000.0) {
		tmp = t_0;
	} else if (y <= 120000000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - ((x - 1.0d0) / y)
    if (y <= (-1850000000000.0d0)) then
        tmp = t_0
    else if (y <= 120000000.0d0) then
        tmp = 1.0d0 - (((x - 1.0d0) * y) / ((-1.0d0) - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -1850000000000.0) {
		tmp = t_0;
	} else if (y <= 120000000.0) {
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - ((x - 1.0) / y)
	tmp = 0
	if y <= -1850000000000.0:
		tmp = t_0
	elif y <= 120000000.0:
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y))
	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 <= -1850000000000.0)
		tmp = t_0;
	elseif (y <= 120000000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(x - 1.0) * y) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - ((x - 1.0) / y);
	tmp = 0.0;
	if (y <= -1850000000000.0)
		tmp = t_0;
	elseif (y <= 120000000.0)
		tmp = 1.0 - (((x - 1.0) * y) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1850000000000.0], t$95$0, If[LessEqual[y, 120000000.0], N[(1.0 - N[(N[(N[(x - 1.0), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85e12 or 1.2e8 < y

   1. Initial program 26.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.85e12 < y < 1.2e8

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1850000000000:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \mathbf{elif}\;y \leq 120000000:\\ \;\;\;\;1 - \frac{\left(x - 1\right) \cdot y}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x - 1}{y}\\ \mathbf{if}\;y \leq -155000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 140000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ (- x 1.0) y))))
   (if (<= y -155000000.0)
     t_0
     (if (<= y 140000000.0) (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x - ((x - 1.0) / y);
	double tmp;
	if (y <= -155000000.0) {
		tmp = t_0;
	} else if (y <= 140000000.0) {
		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(x - 1.0) / y))
	tmp = 0.0
	if (y <= -155000000.0)
		tmp = t_0;
	elseif (y <= 140000000.0)
		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[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -155000000.0], t$95$0, If[LessEqual[y, 140000000.0], 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 < -1.55e8 or 1.4e8 < y

   1. Initial program 27.2%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.55e8 < y < 1.4e8

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 8: 98.5% accurate, 0.9× speedup

Math

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

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 (* (- y 1.0) (- 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(((y - 1.0) * (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(y - 1.0) * 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[(y - 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   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 99.9%

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

4. Final simplification 99.3%

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

Alternative 9: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   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

      \[\leadsto \color{blue}{1 + y \cdot \left(x - 1\right)} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 10: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(x - 1, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 1.2) (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 <= 1.2) {
		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 <= 1.2)
		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, 1.2], 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 1.19999999999999996 < y

   1. Initial program 30.0%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 70.7%

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

   if -1 < y < 1.19999999999999996

   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 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 11: 86.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 30.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 51.3

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

   4. Applied rewrites 51.3%

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

   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 69.9

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

   7. Applied rewrites 69.9%

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

   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. *-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 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 12: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.86:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.86) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.86) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.86d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.86) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.86:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.86)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.86)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.86], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.859999999999999987 < y

   1. Initial program 30.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 51.3

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

   4. Applied rewrites 51.3%

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

   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 69.9

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

   7. Applied rewrites 69.9%

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

   if -1 < y < 0.859999999999999987

   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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.0%

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

Alternative 13: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0235:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.0235) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0235) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.0235d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0235) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.0235:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0235)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0235)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.0235], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0235 < y

   1. Initial program 30.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 51.3

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

   4. Applied rewrites 51.3%

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

   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 69.9

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

   7. Applied rewrites 69.9%

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

   if -1 < y < 0.0235

   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 70.6%

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

Alternative 14: 38.8% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 66.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. distribute-neg-frac2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. metadata-eval N/A

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

   16. unsub-neg N/A

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

   17. lower--.f64 76.6

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

4. Applied rewrites 76.6%

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

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 35.3

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

7. Applied rewrites 35.3%

   \[\leadsto \color{blue}{x} \]
8. 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 2024268 
(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))))