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

Percentage Accurate: 65.4% → 98.7%

Time: 9.6s

Alternatives: 13

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: 65.4% 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: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1 + y}\\ t_1 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_2 := \left(1 - x\right) \cdot \left(-1 + \frac{1}{y}\right)\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, \left(1 + \frac{-1}{x}\right) \cdot t\_0, 1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{\frac{\frac{t\_2 \cdot t\_2}{y \cdot y} + \left(1 - x\right) \cdot \left(x + -1\right)}{\mathsf{fma}\left(\frac{x + -1}{y}, 1 + \frac{-1}{y}, x + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ 1.0 y)))
        (t_1 (/ (* (- 1.0 x) y) (+ 1.0 y)))
        (t_2 (* (- 1.0 x) (+ -1.0 (/ 1.0 y)))))
   (if (<= t_1 0.0001)
     (fma x (* (+ 1.0 (/ -1.0 x)) t_0) 1.0)
     (if (<= t_1 2e+17)
       (+
        x
        (/
         (/
          (+ (/ (* t_2 t_2) (* y y)) (* (- 1.0 x) (+ x -1.0)))
          (fma (/ (+ x -1.0) y) (+ 1.0 (/ -1.0 y)) (+ x -1.0)))
         y))
       (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = y / (1.0 + y);
	double t_1 = ((1.0 - x) * y) / (1.0 + y);
	double t_2 = (1.0 - x) * (-1.0 + (1.0 / y));
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = fma(x, ((1.0 + (-1.0 / x)) * t_0), 1.0);
	} else if (t_1 <= 2e+17) {
		tmp = x + (((((t_2 * t_2) / (y * y)) + ((1.0 - x) * (x + -1.0))) / fma(((x + -1.0) / y), (1.0 + (-1.0 / y)), (x + -1.0))) / y);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y / Float64(1.0 + y))
	t_1 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	t_2 = Float64(Float64(1.0 - x) * Float64(-1.0 + Float64(1.0 / y)))
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = fma(x, Float64(Float64(1.0 + Float64(-1.0 / x)) * t_0), 1.0);
	elseif (t_1 <= 2e+17)
		tmp = Float64(x + Float64(Float64(Float64(Float64(Float64(t_2 * t_2) / Float64(y * y)) + Float64(Float64(1.0 - x) * Float64(x + -1.0))) / fma(Float64(Float64(x + -1.0) / y), Float64(1.0 + Float64(-1.0 / y)), Float64(x + -1.0))) / y));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - x), $MachinePrecision] * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], N[(x * N[(N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+17], N[(x + N[(N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 89.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. rgt-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if 1.00000000000000005e-4 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 2e17

   1. Initial program 18.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. rgt-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 17.9%

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

   9. Taylor expanded in y around -inf

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

   11. Applied rewrites 99.8%

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

   13. Applied rewrites 99.8%

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

   if 2e17 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, \left(1 + \frac{-1}{x}\right) \cdot \frac{y}{1 + y}, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{\frac{\frac{\left(\left(1 - x\right) \cdot \left(-1 + \frac{1}{y}\right)\right) \cdot \left(\left(1 - x\right) \cdot \left(-1 + \frac{1}{y}\right)\right)}{y \cdot y} + \left(1 - x\right) \cdot \left(x + -1\right)}{\mathsf{fma}\left(\frac{x + -1}{y}, 1 + \frac{-1}{y}, x + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{1 + y}\\ t_1 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t\_1 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, \left(1 + \frac{-1}{x}\right) \cdot t\_0, 1\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ 1.0 y))) (t_1 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (<= t_1 0.0001)
     (fma x (* (+ 1.0 (/ -1.0 x)) t_0) 1.0)
     (if (<= t_1 2e+17)
       (+ x (/ (- 1.0 (fma (/ (+ x -1.0) y) (+ -1.0 (/ 1.0 y)) x)) y))
       (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = y / (1.0 + y);
	double t_1 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if (t_1 <= 0.0001) {
		tmp = fma(x, ((1.0 + (-1.0 / x)) * t_0), 1.0);
	} else if (t_1 <= 2e+17) {
		tmp = x + ((1.0 - fma(((x + -1.0) / y), (-1.0 + (1.0 / y)), x)) / y);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y / Float64(1.0 + y))
	t_1 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if (t_1 <= 0.0001)
		tmp = fma(x, Float64(Float64(1.0 + Float64(-1.0 / x)) * t_0), 1.0);
	elseif (t_1 <= 2e+17)
		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 = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0001], N[(x * N[(N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+17], 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], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 1.00000000000000005e-4

   1. Initial program 88.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. rgt-mult-inverse N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if 1.00000000000000005e-4 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64))) < 2e17

   1. Initial program 15.9%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 95.3%

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

   if 2e17 < (/.f64 (*.f64 (-.f64 #s(literal 1 binary64) x) y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 67.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, \left(1 + \frac{-1}{x}\right) \cdot \frac{y}{1 + y}, 1\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;x + \frac{1 - \mathsf{fma}\left(\frac{x + -1}{y}, -1 + \frac{1}{y}, x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{1 + y}\\ \end{array} \]
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 2024228 
(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))))