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

Percentage Accurate: 66.3% → 99.9%

Time: 10.5s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_1 := \mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{if}\;t\_0 \leq 0.996:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y)))
        (t_1 (fma (/ y (- -1.0 y)) (- 1.0 x) 1.0)))
   (if (<= t_0 0.996)
     t_1
     (if (<= t_0 2.0)
       (+ x (/ (- 1.0 (fma (/ (+ x -1.0) y) (+ -1.0 (/ 1.0 y)) x)) y))
       t_1))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double t_1 = fma((y / (-1.0 - y)), (1.0 - x), 1.0);
	double tmp;
	if (t_0 <= 0.996) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x + ((1.0 - fma(((x + -1.0) / y), (-1.0 + (1.0 / y)), x)) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	t_1 = fma(Float64(y / Float64(-1.0 - y)), Float64(1.0 - x), 1.0)
	tmp = 0.0
	if (t_0 <= 0.996)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x + Float64(Float64(1.0 - fma(Float64(Float64(x + -1.0) / y), Float64(-1.0 + Float64(1.0 / y)), x)) / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.996], t$95$1, If[LessEqual[t$95$0, 2.0], 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], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 0.996 or 2 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 83.7%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 83.6

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 99.9

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 0.996 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 2

   1. Initial program 11.3%

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1} \]
   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 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - y}, 1 - x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\left(1 - x\right) \cdot y}{-1 - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+175}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+48}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* (- 1.0 x) y) (- -1.0 y)))))
   (if (<= t_0 -1e+175)
     x
     (if (<= t_0 -2e+48)
       (* x y)
       (if (<= t_0 0.002)
         x
         (if (<= t_0 2.0) (fma y (+ y -1.0) 1.0) (- 1.0 (- 1.0 x))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (((1.0 - x) * y) / (-1.0 - y));
	double tmp;
	if (t_0 <= -1e+175) {
		tmp = x;
	} else if (t_0 <= -2e+48) {
		tmp = x * y;
	} else if (t_0 <= 0.002) {
		tmp = x;
	} else if (t_0 <= 2.0) {
		tmp = fma(y, (y + -1.0), 1.0);
	} else {
		tmp = 1.0 - (1.0 - x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(Float64(1.0 - x) * y) / Float64(-1.0 - y)))
	tmp = 0.0
	if (t_0 <= -1e+175)
		tmp = x;
	elseif (t_0 <= -2e+48)
		tmp = Float64(x * y);
	elseif (t_0 <= 0.002)
		tmp = x;
	elseif (t_0 <= 2.0)
		tmp = fma(y, Float64(y + -1.0), 1.0);
	else
		tmp = Float64(1.0 - Float64(1.0 - x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+175], x, If[LessEqual[t$95$0, -2e+48], N[(x * y), $MachinePrecision], If[LessEqual[t$95$0, 0.002], x, If[LessEqual[t$95$0, 2.0], N[(y * N[(y + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 - N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -9.9999999999999994e174 or -2.00000000000000009e48 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < 2e-3

   1. Initial program 23.9%

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

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. *-rgt-identity N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in y around inf

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

      1. distribute-rgt-in N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

         \[\leadsto 0 + \color{blue}{1} \cdot x \]

      8. *-lft-identity N/A

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

      9. lower-+.f64 59.6

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

   7. Applied rewrites 59.6%

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

   if -9.9999999999999994e174 < (-.f64 #s(literal 1 binary64) (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))) < -2.00000000000000009e48

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 53.4

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.4%

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

   if 2e-3 < (-.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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. neg-sub0 N/A

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

      12. associate-+l- N/A

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

      13. neg-sub0 N/A

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

      14. +-commutative N/A

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

      15. sub-neg N/A

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

      16. distribute-lft-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.7%

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

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

   1. Initial program 68.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 65.3

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

   5. Applied rewrites 65.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{\left(1 - x\right) \cdot y}{-1 - y} \leq -1 \cdot 10^{+175}:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{\left(1 - x\right) \cdot y}{-1 - y} \leq -2 \cdot 10^{+48}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;1 + \frac{\left(1 - x\right) \cdot y}{-1 - y} \leq 0.002:\\ \;\;\;\;x\\ \mathbf{elif}\;1 + \frac{\left(1 - x\right) \cdot y}{-1 - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024223 
(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))))