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

Percentage Accurate: 65.0% → 99.9%

Time: 10.1s

Alternatives: 17

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: 65.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9500:\\ \;\;\;\;x + \mathsf{fma}\left(1 + \frac{-1}{y}, \frac{x + -1}{y \cdot y}, \frac{1 - x}{y}\right)\\ \mathbf{elif}\;y \leq 18000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9500.0)
   (+ x (fma (+ 1.0 (/ -1.0 y)) (/ (+ x -1.0) (* y y)) (/ (- 1.0 x) y)))
   (if (<= y 18000.0)
     (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))
     (+ x (/ (- 1.0 (fma (/ (+ x -1.0) y) (+ -1.0 (/ 1.0 y)) x)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9500.0) {
		tmp = x + fma((1.0 + (-1.0 / y)), ((x + -1.0) / (y * y)), ((1.0 - x) / y));
	} else if (y <= 18000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = x + ((1.0 - fma(((x + -1.0) / y), (-1.0 + (1.0 / y)), x)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9500.0)
		tmp = Float64(x + fma(Float64(1.0 + Float64(-1.0 / y)), Float64(Float64(x + -1.0) / Float64(y * y)), Float64(Float64(1.0 - x) / y)));
	elseif (y <= 18000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = Float64(x + Float64(Float64(1.0 - fma(Float64(Float64(x + -1.0) / y), Float64(-1.0 + Float64(1.0 / y)), x)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9500.0], N[(x + N[(N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 18000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9500

   1. Initial program 27.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 50.0

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 100.0%

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

   if -9500 < y < 18000

   1. Initial program 100.0%

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

   if 18000 < y

   1. Initial program 24.8%

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

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9500:\\ \;\;\;\;x + \mathsf{fma}\left(1 + \frac{-1}{y}, \frac{x + -1}{y \cdot y}, \frac{1 - x}{y}\right)\\ \mathbf{elif}\;y \leq 18000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+232}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+55}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 5000:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+62}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))
   (if (<= t_0 -2e+232)
     x
     (if (<= t_0 -1e+55)
       (* y x)
       (if (<= t_0 0.1)
         x
         (if (<= t_0 5000.0) (- 1.0 y) (if (<= t_0 2e+62) (* y x) x)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= -2e+232) {
		tmp = x;
	} else if (t_0 <= -1e+55) {
		tmp = y * x;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else if (t_0 <= 5000.0) {
		tmp = 1.0 - y;
	} else if (t_0 <= 2e+62) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    if (t_0 <= (-2d+232)) then
        tmp = x
    else if (t_0 <= (-1d+55)) then
        tmp = y * x
    else if (t_0 <= 0.1d0) then
        tmp = x
    else if (t_0 <= 5000.0d0) then
        tmp = 1.0d0 - y
    else if (t_0 <= 2d+62) then
        tmp = y * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	double tmp;
	if (t_0 <= -2e+232) {
		tmp = x;
	} else if (t_0 <= -1e+55) {
		tmp = y * x;
	} else if (t_0 <= 0.1) {
		tmp = x;
	} else if (t_0 <= 5000.0) {
		tmp = 1.0 - y;
	} else if (t_0 <= 2e+62) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	tmp = 0
	if t_0 <= -2e+232:
		tmp = x
	elif t_0 <= -1e+55:
		tmp = y * x
	elif t_0 <= 0.1:
		tmp = x
	elif t_0 <= 5000.0:
		tmp = 1.0 - y
	elif t_0 <= 2e+62:
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= -2e+232)
		tmp = x;
	elseif (t_0 <= -1e+55)
		tmp = Float64(y * x);
	elseif (t_0 <= 0.1)
		tmp = x;
	elseif (t_0 <= 5000.0)
		tmp = Float64(1.0 - y);
	elseif (t_0 <= 2e+62)
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	tmp = 0.0;
	if (t_0 <= -2e+232)
		tmp = x;
	elseif (t_0 <= -1e+55)
		tmp = y * x;
	elseif (t_0 <= 0.1)
		tmp = x;
	elseif (t_0 <= 5000.0)
		tmp = 1.0 - y;
	elseif (t_0 <= 2e+62)
		tmp = y * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+232], x, If[LessEqual[t$95$0, -1e+55], N[(y * x), $MachinePrecision], If[LessEqual[t$95$0, 0.1], x, If[LessEqual[t$95$0, 5000.0], N[(1.0 - y), $MachinePrecision], If[LessEqual[t$95$0, 2e+62], N[(y * x), $MachinePrecision], x]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -2.00000000000000011e232 or -1.00000000000000001e55 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 0.10000000000000001 or 2.00000000000000007e62 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))))

   1. Initial program 28.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 59.7%

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

   if -2.00000000000000011e232 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -1.00000000000000001e55 or 5e3 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2.00000000000000007e62

   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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 76.8

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

   5. Simplified 76.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 73.8

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

   8. Simplified 73.8%

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

   if 0.10000000000000001 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 5e3

   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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 96.7

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

   5. Simplified 96.7%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 96.2

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

   8. Simplified 96.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq -2 \cdot 10^{+232}:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq -1 \cdot 10^{+55}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 0.1:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 5000:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{if}\;y \leq -10500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 18000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 (fma (/ (+ x -1.0) y) (+ -1.0 (/ 1.0 y)) x)) y))))
   (if (<= y -10500.0)
     t_0
     (if (<= y 18000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - fma(((x + -1.0) / y), (-1.0 + (1.0 / y)), x)) / y);
	double tmp;
	if (y <= -10500.0) {
		tmp = t_0;
	} else if (y <= 18000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - fma(Float64(Float64(x + -1.0) / y), Float64(-1.0 + Float64(1.0 / y)), x)) / y))
	tmp = 0.0
	if (y <= -10500.0)
		tmp = t_0;
	elseif (y <= 18000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -10500.0], t$95$0, If[LessEqual[y, 18000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -10500 or 18000 < y

   1. Initial program 26.1%

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

   3. Taylor expanded in y around -inf

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

   5. Simplified 100.0%

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

   if -10500 < y < 18000

   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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10500:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{elif}\;y \leq 18000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)\\ \mathbf{if}\;y \leq -350000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (/ (+ x -1.0) y) (+ -1.0 (/ 1.0 y)) x)))
   (if (<= y -350000.0)
     t_0
     (if (<= y 320000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(((x + -1.0) / y), (-1.0 + (1.0 / y)), x);
	double tmp;
	if (y <= -350000.0) {
		tmp = t_0;
	} else if (y <= 320000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(x + -1.0) / y), Float64(-1.0 + Float64(1.0 / y)), x)
	tmp = 0.0
	if (y <= -350000.0)
		tmp = t_0;
	elseif (y <= 320000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -350000.0], t$95$0, If[LessEqual[y, 320000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5e5 or 3.2e5 < y

   1. Initial program 26.1%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

   5. Simplified 99.7%

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

   if -3.5e5 < y < 3.2e5

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -350000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)\\ \mathbf{elif}\;y \leq 320000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -370000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 245000000:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(-y, x, y\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -370000000.0)
     t_0
     (if (<= y 245000000.0) (+ 1.0 (/ (fma (- y) x y) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -370000000.0) {
		tmp = t_0;
	} else if (y <= 245000000.0) {
		tmp = 1.0 + (fma(-y, x, y) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -370000000.0)
		tmp = t_0;
	elseif (y <= 245000000.0)
		tmp = Float64(1.0 + Float64(fma(Float64(-y), x, y) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -370000000.0], t$95$0, If[LessEqual[y, 245000000.0], N[(1.0 + N[(N[((-y) * x + y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e8 or 2.45e8 < y

   1. Initial program 24.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --lowering--.f64 99.6

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

   5. Simplified 99.6%

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

   if -3.7e8 < y < 2.45e8

   1. Initial program 99.6%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. neg-mul-1 N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-rgt-identity N/A

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

      10. *-rgt-identity N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. neg-lowering-neg.f64 99.6

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

   4. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -370000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 245000000:\\ \;\;\;\;1 + \frac{\mathsf{fma}\left(-y, x, y\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -370000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 125000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -370000000.0)
     t_0
     (if (<= y 125000000.0) (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -370000000.0) {
		tmp = t_0;
	} else if (y <= 125000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((1.0d0 - x) / y)
    if (y <= (-370000000.0d0)) then
        tmp = t_0
    else if (y <= 125000000.0d0) then
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -370000000.0) {
		tmp = t_0;
	} else if (y <= 125000000.0) {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + ((1.0 - x) / y)
	tmp = 0
	if y <= -370000000.0:
		tmp = t_0
	elif y <= 125000000.0:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -370000000.0)
		tmp = t_0;
	elseif (y <= 125000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -370000000.0)
		tmp = t_0;
	elseif (y <= 125000000.0)
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -370000000.0], t$95$0, If[LessEqual[y, 125000000.0], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e8 or 1.25e8 < y

   1. Initial program 24.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --lowering--.f64 99.6

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

   5. Simplified 99.6%

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

   if -3.7e8 < y < 1.25e8

   1. Initial program 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -370000000:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 125000000:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -370000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 255000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (/ (- 1.0 x) y))))
   (if (<= y -370000000.0)
     t_0
     (if (<= y 255000000.0) (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0) t_0))))

C

double code(double x, double y) {
	double t_0 = x + ((1.0 - x) / y);
	double tmp;
	if (y <= -370000000.0) {
		tmp = t_0;
	} else if (y <= 255000000.0) {
		tmp = fma((y / (-1.0 - y)), (1.0 - x), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x + Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -370000000.0)
		tmp = t_0;
	elseif (y <= 255000000.0)
		tmp = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -370000000.0], t$95$0, If[LessEqual[y, 255000000.0], N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e8 or 2.55e8 < y

   1. Initial program 24.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --lowering--.f64 99.6

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

   5. Simplified 99.6%

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

   if -3.7e8 < y < 2.55e8

   1. Initial program 99.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.6

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

   4. Applied egg-rr 99.6%

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

Alternative 8: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.8e+62)
   x
   (if (<= y -1.0)
     (/ 1.0 y)
     (if (<= y 1.22) (fma y (+ x -1.0) 1.0) (- x (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.8e+62) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 1.22) {
		tmp = fma(y, (x + -1.0), 1.0);
	} else {
		tmp = x - (x / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.8e+62)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 1.22)
		tmp = fma(y, Float64(x + -1.0), 1.0);
	else
		tmp = Float64(x - Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.8e+62], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 1.22], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.7999999999999994e62

   1. Initial program 24.2%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 74.6%

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

   if -9.7999999999999994e62 < y < -1

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 10.3%

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

   6. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 63.4

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

   8. Simplified 63.4%

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

   if -1 < y < 1.21999999999999997

   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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

   if 1.21999999999999997 < y

   1. Initial program 27.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 47.6

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

   4. Applied egg-rr 47.6%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-lowering-+.f64 50.8

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

   7. Simplified 50.8%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 70.3

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

   10. Simplified 70.3%

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

Alternative 9: 98.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 27.4%

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

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

      7. mul-1-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --lowering--.f64 97.6

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

   5. Simplified 97.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. neg-sub0 N/A

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

      12. associate-+l- N/A

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

      13. neg-sub0 N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. distribute-lft-out N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.7%

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

Alternative 10: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 27.4%

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

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

      7. mul-1-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. +-commutative N/A

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

      16. sub-neg N/A

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

      17. --lowering--.f64 97.6

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

   5. Simplified 97.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

Alternative 11: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+64}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2e+64)
   x
   (if (<= y -1.0) (/ 1.0 y) (if (<= y 1.0) (fma y (+ x -1.0) 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.2e+64) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 1.0) {
		tmp = fma(y, (x + -1.0), 1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.2e+64)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 1.0)
		tmp = fma(y, Float64(x + -1.0), 1.0);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.2e+64], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 1.0], N[(y * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.1999999999999998e64 or 1 < y

   1. Initial program 26.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 71.5%

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

   if -6.1999999999999998e64 < y < -1

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 10.3%

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

   6. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 63.4

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

   8. Simplified 63.4%

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

Alternative 12: 86.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 27.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 66.4%

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

Alternative 13: 85.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 430:\\ \;\;\;\;\mathsf{fma}\left(y, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 430.0) (fma y x 1.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 430.0) {
		tmp = fma(y, x, 1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 430.0)
		tmp = fma(y, x, 1.0);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 430.0], N[(y * x + 1.0), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 430 < y

   1. Initial program 26.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 66.9%

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

   if -1 < y < 430

   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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 96.2%

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

Alternative 14: 74.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.4:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.4) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.4) {
		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 <= 0.4d0) 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 <= 0.4) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.4:
		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 <= 0.4)
		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 <= 0.4)
		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, 0.4], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.40000000000000002 < y

   1. Initial program 27.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 66.4%

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

   if -1 < y < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.5

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

   8. Simplified 75.5%

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

Alternative 15: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 18:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 18.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 18.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 18.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 18.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 18.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 18.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 18.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 18.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 18 < y

   1. Initial program 26.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 66.9%

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

   if -1 < y < 18

   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. Simplified 74.0%

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

Alternative 16: 38.1% 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 67.7%

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

3. Taylor expanded in y around 0

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

5. Simplified 43.1%

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

Alternative 17: 3.1% accurate, 26.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in y around inf

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

   1. --lowering--.f64 21.8

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

5. Simplified 21.8%

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

6. Taylor expanded in x around 0

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

8. Simplified 3.1%

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

   1. metadata-eval 3.1

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

10. Applied egg-rr 3.1%

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

Developer Target 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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))))