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

Percentage Accurate: 66.1% → 99.9%

Time: 18.3s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(y + 1\right)}^{2}\\ t_1 := \frac{{y}^{2}}{t\_0}\\ t_2 := \frac{y}{y + 1}\\ t_3 := 1 + \left(t\_1 + t\_2\right)\\ t_4 := {\left(y + 1\right)}^{3}\\ t_5 := -1 + \left(\frac{y}{-1 - y} - t\_1\right)\\ \mathbf{if}\;y \leq -19500 \lor \neg \left(y \leq 20000\right):\\ \;\;\;\;x + \left(\frac{1 - x}{{y}^{3}} - \left(\frac{x + -1}{y} - \frac{x + -1}{{y}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(3, \frac{{y}^{3}}{t\_4 \cdot t\_3}, \frac{\left(1 - \frac{{y}^{3}}{t\_4}\right) \cdot \left(t\_2 - \frac{{y}^{2} \cdot -2}{t\_0}\right)}{{t\_3}^{2}}\right), \frac{-1}{t\_5}\right) + \frac{{y}^{3}}{t\_4 \cdot t\_5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (+ y 1.0) 2.0))
        (t_1 (/ (pow y 2.0) t_0))
        (t_2 (/ y (+ y 1.0)))
        (t_3 (+ 1.0 (+ t_1 t_2)))
        (t_4 (pow (+ y 1.0) 3.0))
        (t_5 (+ -1.0 (- (/ y (- -1.0 y)) t_1))))
   (if (or (<= y -19500.0) (not (<= y 20000.0)))
     (+
      x
      (-
       (/ (- 1.0 x) (pow y 3.0))
       (- (/ (+ x -1.0) y) (/ (+ x -1.0) (pow y 2.0)))))
     (+
      (fma
       x
       (fma
        3.0
        (/ (pow y 3.0) (* t_4 t_3))
        (/
         (* (- 1.0 (/ (pow y 3.0) t_4)) (- t_2 (/ (* (pow y 2.0) -2.0) t_0)))
         (pow t_3 2.0)))
       (/ -1.0 t_5))
      (/ (pow y 3.0) (* t_4 t_5))))))

C

double code(double x, double y) {
	double t_0 = pow((y + 1.0), 2.0);
	double t_1 = pow(y, 2.0) / t_0;
	double t_2 = y / (y + 1.0);
	double t_3 = 1.0 + (t_1 + t_2);
	double t_4 = pow((y + 1.0), 3.0);
	double t_5 = -1.0 + ((y / (-1.0 - y)) - t_1);
	double tmp;
	if ((y <= -19500.0) || !(y <= 20000.0)) {
		tmp = x + (((1.0 - x) / pow(y, 3.0)) - (((x + -1.0) / y) - ((x + -1.0) / pow(y, 2.0))));
	} else {
		tmp = fma(x, fma(3.0, (pow(y, 3.0) / (t_4 * t_3)), (((1.0 - (pow(y, 3.0) / t_4)) * (t_2 - ((pow(y, 2.0) * -2.0) / t_0))) / pow(t_3, 2.0))), (-1.0 / t_5)) + (pow(y, 3.0) / (t_4 * t_5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + 1.0) ^ 2.0
	t_1 = Float64((y ^ 2.0) / t_0)
	t_2 = Float64(y / Float64(y + 1.0))
	t_3 = Float64(1.0 + Float64(t_1 + t_2))
	t_4 = Float64(y + 1.0) ^ 3.0
	t_5 = Float64(-1.0 + Float64(Float64(y / Float64(-1.0 - y)) - t_1))
	tmp = 0.0
	if ((y <= -19500.0) || !(y <= 20000.0))
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) / (y ^ 3.0)) - Float64(Float64(Float64(x + -1.0) / y) - Float64(Float64(x + -1.0) / (y ^ 2.0)))));
	else
		tmp = Float64(fma(x, fma(3.0, Float64((y ^ 3.0) / Float64(t_4 * t_3)), Float64(Float64(Float64(1.0 - Float64((y ^ 3.0) / t_4)) * Float64(t_2 - Float64(Float64((y ^ 2.0) * -2.0) / t_0))) / (t_3 ^ 2.0))), Float64(-1.0 / t_5)) + Float64((y ^ 3.0) / Float64(t_4 * t_5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(y + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[y, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(y + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 + N[(N[(y / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -19500.0], N[Not[LessEqual[y, 20000.0]], $MachinePrecision]], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(3.0 * N[(N[Power[y, 3.0], $MachinePrecision] / N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - N[(N[Power[y, 3.0], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - N[(N[(N[Power[y, 2.0], $MachinePrecision] * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[Power[y, 3.0], $MachinePrecision] / N[(t$95$4 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -19500 or 2e4 < y

   1. Initial program 30.6%

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

      1. associate-/l* 51.8%

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

      2. remove-double-neg 51.8%

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

      3. remove-double-neg 51.8%

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

      4. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in y around -inf 99.9%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right)\right) - \frac{1}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+r+ 99.9%

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

      3. associate--l+ 99.9%

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

      4. div-sub 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

      8. metadata-eval 99.9%

         \[\leadsto x + \left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \frac{\color{blue}{\left(0 - 1\right)} + x}{{y}^{2}}\right) \]

      9. associate--r- 99.9%

         \[\leadsto x + \left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \frac{\color{blue}{0 - \left(1 - x\right)}}{{y}^{2}}\right) \]

      10. neg-sub0 99.9%

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

      11. distribute-frac-neg 99.9%

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

      12. mul-1-neg 99.9%

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

      13. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   if -19500 < y < 2e4

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-*r/ 99.8%

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

      4. distribute-neg-frac2 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. flip3-+ 93.1%

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

      11. metadata-eval 93.1%

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

      12. metadata-eval 93.1%

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

   6. Applied egg-rr 93.1%

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

   7. Taylor expanded in x around 0 99.8%

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

   9. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -19500 \lor \neg \left(y \leq 20000\right):\\ \;\;\;\;x + \left(\frac{1 - x}{{y}^{3}} - \left(\frac{x + -1}{y} - \frac{x + -1}{{y}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(3, \frac{{y}^{3}}{{\left(y + 1\right)}^{3} \cdot \left(1 + \left(\frac{{y}^{2}}{{\left(y + 1\right)}^{2}} + \frac{y}{y + 1}\right)\right)}, \frac{\left(1 - \frac{{y}^{3}}{{\left(y + 1\right)}^{3}}\right) \cdot \left(\frac{y}{y + 1} - \frac{{y}^{2} \cdot -2}{{\left(y + 1\right)}^{2}}\right)}{{\left(1 + \left(\frac{{y}^{2}}{{\left(y + 1\right)}^{2}} + \frac{y}{y + 1}\right)\right)}^{2}}\right), \frac{-1}{-1 + \left(\frac{y}{-1 - y} - \frac{{y}^{2}}{{\left(y + 1\right)}^{2}}\right)}\right) + \frac{{y}^{3}}{{\left(y + 1\right)}^{3} \cdot \left(-1 + \left(\frac{y}{-1 - y} - \frac{{y}^{2}}{{\left(y + 1\right)}^{2}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -19500 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;x + \left(\frac{1 - x}{{y}^{3}} - \left(\frac{x + -1}{y} - \frac{x + -1}{{y}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -19500.0) (not (<= y 14000.0)))
   (+
    x
    (-
     (/ (- 1.0 x) (pow y 3.0))
     (- (/ (+ x -1.0) y) (/ (+ x -1.0) (pow y 2.0)))))
   (fma y (/ (- 1.0 x) (- -1.0 y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -19500.0) || !(y <= 14000.0)) {
		tmp = x + (((1.0 - x) / pow(y, 3.0)) - (((x + -1.0) / y) - ((x + -1.0) / pow(y, 2.0))));
	} else {
		tmp = fma(y, ((1.0 - x) / (-1.0 - y)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -19500.0) || !(y <= 14000.0))
		tmp = Float64(x + Float64(Float64(Float64(1.0 - x) / (y ^ 3.0)) - Float64(Float64(Float64(x + -1.0) / y) - Float64(Float64(x + -1.0) / (y ^ 2.0)))));
	else
		tmp = fma(y, Float64(Float64(1.0 - x) / Float64(-1.0 - y)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -19500.0], N[Not[LessEqual[y, 14000.0]], $MachinePrecision]], N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -19500 or 14000 < y

   1. Initial program 30.6%

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

      1. associate-/l* 51.8%

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

      2. remove-double-neg 51.8%

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

      3. remove-double-neg 51.8%

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

      4. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in y around -inf 99.9%

      \[\leadsto \color{blue}{\left(x + \left(-1 \cdot \frac{x - 1}{y} + \left(-1 \cdot \frac{x - 1}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right)\right) - \frac{1}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate-+r+ 99.9%

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

      3. associate--l+ 99.9%

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

      4. div-sub 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

      8. metadata-eval 99.9%

         \[\leadsto x + \left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \frac{\color{blue}{\left(0 - 1\right)} + x}{{y}^{2}}\right) \]

      9. associate--r- 99.9%

         \[\leadsto x + \left(\left(-1 \cdot \frac{x - 1}{y} + -1 \cdot \frac{x - 1}{{y}^{3}}\right) + \frac{\color{blue}{0 - \left(1 - x\right)}}{{y}^{2}}\right) \]

      10. neg-sub0 99.9%

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

      11. distribute-frac-neg 99.9%

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

      12. mul-1-neg 99.9%

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

      13. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   if -19500 < y < 14000

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

      7. distribute-frac-neg2 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. metadata-eval 99.8%

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

      11. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -19500 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;x + \left(\frac{1 - x}{{y}^{3}} - \left(\frac{x + -1}{y} - \frac{x + -1}{{y}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -280000 \lor \neg \left(y \leq 250000\right):\\ \;\;\;\;x + \left(\frac{x + -1}{{y}^{2}} + \frac{1 - x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -280000.0) (not (<= y 250000.0)))
   (+ x (+ (/ (+ x -1.0) (pow y 2.0)) (/ (- 1.0 x) y)))
   (fma y (/ (- 1.0 x) (- -1.0 y)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -280000.0) || !(y <= 250000.0)) {
		tmp = x + (((x + -1.0) / pow(y, 2.0)) + ((1.0 - x) / y));
	} else {
		tmp = fma(y, ((1.0 - x) / (-1.0 - y)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -280000.0) || !(y <= 250000.0))
		tmp = Float64(x + Float64(Float64(Float64(x + -1.0) / (y ^ 2.0)) + Float64(Float64(1.0 - x) / y)));
	else
		tmp = fma(y, Float64(Float64(1.0 - x) / Float64(-1.0 - y)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -280000.0], N[Not[LessEqual[y, 250000.0]], $MachinePrecision]], N[(x + N[(N[(N[(x + -1.0), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 - x), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e5 or 2.5e5 < y

   1. Initial program 30.3%

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

      1. associate-/l* 51.6%

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

      2. remove-double-neg 51.6%

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

      3. remove-double-neg 51.6%

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

      4. +-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in y around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. sub-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. div-sub 99.9%

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

      6. sub-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. neg-mul-1 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-lft-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. sub-neg 99.9%

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

      13. associate-*r/ 99.9%

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

      14. mul-1-neg 99.9%

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

      15. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -2.8e5 < y < 2.5e5

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-define 99.6%

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

      7. distribute-frac-neg2 99.6%

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

      8. +-commutative 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. metadata-eval 99.6%

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

      11. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280000 \lor \neg \left(y \leq 250000\right):\\ \;\;\;\;x + \left(\frac{x + -1}{{y}^{2}} + \frac{1 - x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1 - x}{-1 - y}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6e+60)
   x
   (if (<= y -1.0)
     (/ 1.0 y)
     (if (<= y 5e-16)
       (- 1.0 y)
       (if (<= y 2.9e+105) x (if (<= y 2.6e+139) (/ 1.0 y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6e+60) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 5e-16) {
		tmp = 1.0 - y;
	} else if (y <= 2.9e+105) {
		tmp = x;
	} else if (y <= 2.6e+139) {
		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 <= (-6d+60)) then
        tmp = x
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0 / y
    else if (y <= 5d-16) then
        tmp = 1.0d0 - y
    else if (y <= 2.9d+105) then
        tmp = x
    else if (y <= 2.6d+139) 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 <= -6e+60) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 5e-16) {
		tmp = 1.0 - y;
	} else if (y <= 2.9e+105) {
		tmp = x;
	} else if (y <= 2.6e+139) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6e+60:
		tmp = x
	elif y <= -1.0:
		tmp = 1.0 / y
	elif y <= 5e-16:
		tmp = 1.0 - y
	elif y <= 2.9e+105:
		tmp = x
	elif y <= 2.6e+139:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6e+60)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 5e-16)
		tmp = Float64(1.0 - y);
	elseif (y <= 2.9e+105)
		tmp = x;
	elseif (y <= 2.6e+139)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6e+60)
		tmp = x;
	elseif (y <= -1.0)
		tmp = 1.0 / y;
	elseif (y <= 5e-16)
		tmp = 1.0 - y;
	elseif (y <= 2.9e+105)
		tmp = x;
	elseif (y <= 2.6e+139)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6e+60], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 5e-16], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 2.9e+105], x, If[LessEqual[y, 2.6e+139], N[(1.0 / y), $MachinePrecision], x]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.9999999999999997e60 or 5.0000000000000004e-16 < y < 2.9000000000000001e105 or 2.60000000000000022e139 < y

   1. Initial program 35.0%

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

      1. associate-/l* 59.4%

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

      2. remove-double-neg 59.4%

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

      3. remove-double-neg 59.4%

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

      4. +-commutative 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in y around inf 76.6%

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

   if -5.9999999999999997e60 < y < -1 or 2.9000000000000001e105 < y < 2.60000000000000022e139

   1. Initial program 29.6%

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

      1. sub-neg 29.6%

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

      2. +-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. associate-/l* 32.8%

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

      5. distribute-rgt-neg-in 32.8%

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

      6. fma-define 34.0%

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

      7. distribute-frac-neg2 34.0%

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

      8. +-commutative 34.0%

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

      9. distribute-neg-in 34.0%

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

      10. metadata-eval 34.0%

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

      11. unsub-neg 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in x around 0 17.5%

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

      1. metadata-eval 17.5%

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

      2. distribute-neg-frac 17.5%

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

      3. distribute-neg-frac2 17.5%

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

      4. distribute-neg-in 17.5%

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

      5. metadata-eval 17.5%

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

      6. sub-neg 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in y around inf 76.1%

      \[\leadsto \color{blue}{\frac{1}{y} - \frac{1}{{y}^{2}}} \]

   9. Taylor expanded in y around inf 72.4%

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

   if -1 < y < 5.0000000000000004e-16

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. metadata-eval 83.9%

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

      2. distribute-neg-frac 83.9%

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

      3. distribute-neg-frac2 83.9%

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

      4. distribute-neg-in 83.9%

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

      5. metadata-eval 83.9%

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

      6. sub-neg 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in y around 0 83.3%

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

      1. neg-mul-1 83.3%

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

      2. unsub-neg 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.65e+59)
   x
   (if (<= y -1.0)
     (/ 1.0 y)
     (if (<= y 1.0)
       (- 1.0 y)
       (if (<= y 4.4e+102) (- x (/ x y)) (if (<= y 2.6e+139) (/ 1.0 y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.65e+59) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 1.0) {
		tmp = 1.0 - y;
	} else if (y <= 4.4e+102) {
		tmp = x - (x / y);
	} else if (y <= 2.6e+139) {
		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 <= (-2.65d+59)) then
        tmp = x
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0 / y
    else if (y <= 1.0d0) then
        tmp = 1.0d0 - y
    else if (y <= 4.4d+102) then
        tmp = x - (x / y)
    else if (y <= 2.6d+139) 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 <= -2.65e+59) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 1.0) {
		tmp = 1.0 - y;
	} else if (y <= 4.4e+102) {
		tmp = x - (x / y);
	} else if (y <= 2.6e+139) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.65e+59:
		tmp = x
	elif y <= -1.0:
		tmp = 1.0 / y
	elif y <= 1.0:
		tmp = 1.0 - y
	elif y <= 4.4e+102:
		tmp = x - (x / y)
	elif y <= 2.6e+139:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.65e+59)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 1.0)
		tmp = Float64(1.0 - y);
	elseif (y <= 4.4e+102)
		tmp = Float64(x - Float64(x / y));
	elseif (y <= 2.6e+139)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.65e+59)
		tmp = x;
	elseif (y <= -1.0)
		tmp = 1.0 / y;
	elseif (y <= 1.0)
		tmp = 1.0 - y;
	elseif (y <= 4.4e+102)
		tmp = x - (x / y);
	elseif (y <= 2.6e+139)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.65e+59], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 4.4e+102], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+139], N[(1.0 / y), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.6499999999999998e59 or 2.60000000000000022e139 < y

   1. Initial program 26.1%

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

      1. associate-/l* 56.8%

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

      2. remove-double-neg 56.8%

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

      3. remove-double-neg 56.8%

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

      4. +-commutative 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in y around inf 84.5%

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

   if -2.6499999999999998e59 < y < -1 or 4.40000000000000015e102 < y < 2.60000000000000022e139

   1. Initial program 29.6%

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

      1. sub-neg 29.6%

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

      2. +-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. associate-/l* 32.8%

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

      5. distribute-rgt-neg-in 32.8%

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

      6. fma-define 34.0%

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

      7. distribute-frac-neg2 34.0%

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

      8. +-commutative 34.0%

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

      9. distribute-neg-in 34.0%

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

      10. metadata-eval 34.0%

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

      11. unsub-neg 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in x around 0 17.5%

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

      1. metadata-eval 17.5%

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

      2. distribute-neg-frac 17.5%

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

      3. distribute-neg-frac2 17.5%

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

      4. distribute-neg-in 17.5%

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

      5. metadata-eval 17.5%

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

      6. sub-neg 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in y around inf 76.1%

      \[\leadsto \color{blue}{\frac{1}{y} - \frac{1}{{y}^{2}}} \]

   9. Taylor expanded in y around inf 72.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 81.8%

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

      1. metadata-eval 81.8%

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

      2. distribute-neg-frac 81.8%

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

      3. distribute-neg-frac2 81.8%

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

      4. distribute-neg-in 81.8%

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

      5. metadata-eval 81.8%

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

      6. sub-neg 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in y around 0 81.1%

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

      1. neg-mul-1 81.1%

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

      2. unsub-neg 81.1%

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

   10. Simplified 81.1%

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

   if 1 < y < 4.40000000000000015e102

   1. Initial program 56.6%

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

      1. associate-/l* 63.4%

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

      2. remove-double-neg 63.4%

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

      3. remove-double-neg 63.4%

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

      4. +-commutative 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around inf 57.1%

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

      1. *-commutative 57.1%

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

      2. associate-/l* 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in y around inf 62.5%

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

      1. mul-1-neg 62.5%

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

      2. unsub-neg 62.5%

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

   10. Simplified 62.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 440:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.35e+62)
   x
   (if (<= y -1.0)
     (/ 1.0 y)
     (if (<= y 440.0)
       (+ 1.0 (* y x))
       (if (<= y 2e+104) (- x (/ x y)) (if (<= y 2.6e+139) (/ 1.0 y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.35e+62) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 440.0) {
		tmp = 1.0 + (y * x);
	} else if (y <= 2e+104) {
		tmp = x - (x / y);
	} else if (y <= 2.6e+139) {
		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 <= (-3.35d+62)) then
        tmp = x
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0 / y
    else if (y <= 440.0d0) then
        tmp = 1.0d0 + (y * x)
    else if (y <= 2d+104) then
        tmp = x - (x / y)
    else if (y <= 2.6d+139) 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 <= -3.35e+62) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 440.0) {
		tmp = 1.0 + (y * x);
	} else if (y <= 2e+104) {
		tmp = x - (x / y);
	} else if (y <= 2.6e+139) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.35e+62:
		tmp = x
	elif y <= -1.0:
		tmp = 1.0 / y
	elif y <= 440.0:
		tmp = 1.0 + (y * x)
	elif y <= 2e+104:
		tmp = x - (x / y)
	elif y <= 2.6e+139:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.35e+62)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 440.0)
		tmp = Float64(1.0 + Float64(y * x));
	elseif (y <= 2e+104)
		tmp = Float64(x - Float64(x / y));
	elseif (y <= 2.6e+139)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.35e+62)
		tmp = x;
	elseif (y <= -1.0)
		tmp = 1.0 / y;
	elseif (y <= 440.0)
		tmp = 1.0 + (y * x);
	elseif (y <= 2e+104)
		tmp = x - (x / y);
	elseif (y <= 2.6e+139)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.35e+62], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 440.0], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+104], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+139], N[(1.0 / y), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.3499999999999998e62 or 2.60000000000000022e139 < y

   1. Initial program 26.1%

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

      1. associate-/l* 56.8%

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

      2. remove-double-neg 56.8%

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

      3. remove-double-neg 56.8%

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

      4. +-commutative 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in y around inf 84.5%

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

   if -3.3499999999999998e62 < y < -1 or 2e104 < y < 2.60000000000000022e139

   1. Initial program 29.6%

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

      1. sub-neg 29.6%

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

      2. +-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. associate-/l* 32.8%

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

      5. distribute-rgt-neg-in 32.8%

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

      6. fma-define 34.0%

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

      7. distribute-frac-neg2 34.0%

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

      8. +-commutative 34.0%

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

      9. distribute-neg-in 34.0%

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

      10. metadata-eval 34.0%

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

      11. unsub-neg 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in x around 0 17.5%

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

      1. metadata-eval 17.5%

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

      2. distribute-neg-frac 17.5%

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

      3. distribute-neg-frac2 17.5%

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

      4. distribute-neg-in 17.5%

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

      5. metadata-eval 17.5%

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

      6. sub-neg 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in y around inf 76.1%

      \[\leadsto \color{blue}{\frac{1}{y} - \frac{1}{{y}^{2}}} \]

   9. Taylor expanded in y around inf 72.4%

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

   if -1 < y < 440

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. remove-double-neg 99.9%

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

      4. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 97.4%

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

   6. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. distribute-lft-neg-out 97.2%

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

      3. *-commutative 97.2%

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

   8. Simplified 97.2%

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

   if 440 < y < 2e104

   1. Initial program 55.3%

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

      1. associate-/l* 62.3%

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

      2. remove-double-neg 62.3%

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

      3. remove-double-neg 62.3%

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

      4. +-commutative 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in x around inf 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-/l* 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in y around inf 64.9%

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

      1. mul-1-neg 64.9%

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

      2. unsub-neg 64.9%

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

   10. Simplified 64.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 440:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.7e+60)
   x
   (if (<= y -1.0)
     (/ 1.0 y)
     (if (<= y 5e-16)
       (+ 1.0 (* y x))
       (if (<= y 1.3e+103)
         (* y (/ x (+ y 1.0)))
         (if (<= y 2.6e+139) (/ 1.0 y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.7e+60) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 5e-16) {
		tmp = 1.0 + (y * x);
	} else if (y <= 1.3e+103) {
		tmp = y * (x / (y + 1.0));
	} else if (y <= 2.6e+139) {
		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 <= (-5.7d+60)) then
        tmp = x
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0 / y
    else if (y <= 5d-16) then
        tmp = 1.0d0 + (y * x)
    else if (y <= 1.3d+103) then
        tmp = y * (x / (y + 1.0d0))
    else if (y <= 2.6d+139) 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 <= -5.7e+60) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 5e-16) {
		tmp = 1.0 + (y * x);
	} else if (y <= 1.3e+103) {
		tmp = y * (x / (y + 1.0));
	} else if (y <= 2.6e+139) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.7e+60:
		tmp = x
	elif y <= -1.0:
		tmp = 1.0 / y
	elif y <= 5e-16:
		tmp = 1.0 + (y * x)
	elif y <= 1.3e+103:
		tmp = y * (x / (y + 1.0))
	elif y <= 2.6e+139:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.7e+60)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 5e-16)
		tmp = Float64(1.0 + Float64(y * x));
	elseif (y <= 1.3e+103)
		tmp = Float64(y * Float64(x / Float64(y + 1.0)));
	elseif (y <= 2.6e+139)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.7e+60)
		tmp = x;
	elseif (y <= -1.0)
		tmp = 1.0 / y;
	elseif (y <= 5e-16)
		tmp = 1.0 + (y * x);
	elseif (y <= 1.3e+103)
		tmp = y * (x / (y + 1.0));
	elseif (y <= 2.6e+139)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.7e+60], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 5e-16], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+103], N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+139], N[(1.0 / y), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.69999999999999978e60 or 2.60000000000000022e139 < y

   1. Initial program 26.1%

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

      1. associate-/l* 56.8%

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

      2. remove-double-neg 56.8%

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

      3. remove-double-neg 56.8%

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

      4. +-commutative 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in y around inf 84.5%

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

   if -5.69999999999999978e60 < y < -1 or 1.3000000000000001e103 < y < 2.60000000000000022e139

   1. Initial program 29.6%

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

      1. sub-neg 29.6%

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

      2. +-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. associate-/l* 32.8%

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

      5. distribute-rgt-neg-in 32.8%

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

      6. fma-define 34.0%

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

      7. distribute-frac-neg2 34.0%

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

      8. +-commutative 34.0%

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

      9. distribute-neg-in 34.0%

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

      10. metadata-eval 34.0%

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

      11. unsub-neg 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in x around 0 17.5%

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

      1. metadata-eval 17.5%

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

      2. distribute-neg-frac 17.5%

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

      3. distribute-neg-frac2 17.5%

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

      4. distribute-neg-in 17.5%

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

      5. metadata-eval 17.5%

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

      6. sub-neg 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in y around inf 76.1%

      \[\leadsto \color{blue}{\frac{1}{y} - \frac{1}{{y}^{2}}} \]

   9. Taylor expanded in y around inf 72.4%

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

   if -1 < y < 5.0000000000000004e-16

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.4%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. distribute-lft-neg-out 99.0%

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

      3. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   if 5.0000000000000004e-16 < y < 1.3000000000000001e103

   1. Initial program 61.0%

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

      1. associate-/l* 67.0%

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

      2. remove-double-neg 67.0%

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

      3. remove-double-neg 67.0%

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

      4. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in x around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.5e+58)
   x
   (if (<= y -1.0)
     (/ 1.0 y)
     (if (<= y 5e-16)
       (+ 1.0 (* y (+ x -1.0)))
       (if (<= y 3e+105)
         (* y (/ x (+ y 1.0)))
         (if (<= y 2.6e+139) (/ 1.0 y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.5e+58) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 5e-16) {
		tmp = 1.0 + (y * (x + -1.0));
	} else if (y <= 3e+105) {
		tmp = y * (x / (y + 1.0));
	} else if (y <= 2.6e+139) {
		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 <= (-7.5d+58)) then
        tmp = x
    else if (y <= (-1.0d0)) then
        tmp = 1.0d0 / y
    else if (y <= 5d-16) then
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    else if (y <= 3d+105) then
        tmp = y * (x / (y + 1.0d0))
    else if (y <= 2.6d+139) 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 <= -7.5e+58) {
		tmp = x;
	} else if (y <= -1.0) {
		tmp = 1.0 / y;
	} else if (y <= 5e-16) {
		tmp = 1.0 + (y * (x + -1.0));
	} else if (y <= 3e+105) {
		tmp = y * (x / (y + 1.0));
	} else if (y <= 2.6e+139) {
		tmp = 1.0 / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.5e+58:
		tmp = x
	elif y <= -1.0:
		tmp = 1.0 / y
	elif y <= 5e-16:
		tmp = 1.0 + (y * (x + -1.0))
	elif y <= 3e+105:
		tmp = y * (x / (y + 1.0))
	elif y <= 2.6e+139:
		tmp = 1.0 / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.5e+58)
		tmp = x;
	elseif (y <= -1.0)
		tmp = Float64(1.0 / y);
	elseif (y <= 5e-16)
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	elseif (y <= 3e+105)
		tmp = Float64(y * Float64(x / Float64(y + 1.0)));
	elseif (y <= 2.6e+139)
		tmp = Float64(1.0 / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.5e+58)
		tmp = x;
	elseif (y <= -1.0)
		tmp = 1.0 / y;
	elseif (y <= 5e-16)
		tmp = 1.0 + (y * (x + -1.0));
	elseif (y <= 3e+105)
		tmp = y * (x / (y + 1.0));
	elseif (y <= 2.6e+139)
		tmp = 1.0 / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.5e+58], x, If[LessEqual[y, -1.0], N[(1.0 / y), $MachinePrecision], If[LessEqual[y, 5e-16], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+105], N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+139], N[(1.0 / y), $MachinePrecision], x]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.5000000000000001e58 or 2.60000000000000022e139 < y

   1. Initial program 26.1%

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

      1. associate-/l* 56.8%

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

      2. remove-double-neg 56.8%

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

      3. remove-double-neg 56.8%

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

      4. +-commutative 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in y around inf 84.5%

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

   if -7.5000000000000001e58 < y < -1 or 3.0000000000000001e105 < y < 2.60000000000000022e139

   1. Initial program 29.6%

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

      1. sub-neg 29.6%

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

      2. +-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. associate-/l* 32.8%

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

      5. distribute-rgt-neg-in 32.8%

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

      6. fma-define 34.0%

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

      7. distribute-frac-neg2 34.0%

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

      8. +-commutative 34.0%

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

      9. distribute-neg-in 34.0%

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

      10. metadata-eval 34.0%

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

      11. unsub-neg 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in x around 0 17.5%

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

      1. metadata-eval 17.5%

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

      2. distribute-neg-frac 17.5%

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

      3. distribute-neg-frac2 17.5%

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

      4. distribute-neg-in 17.5%

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

      5. metadata-eval 17.5%

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

      6. sub-neg 17.5%

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

   7. Simplified 17.5%

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

   8. Taylor expanded in y around inf 76.1%

      \[\leadsto \color{blue}{\frac{1}{y} - \frac{1}{{y}^{2}}} \]

   9. Taylor expanded in y around inf 72.4%

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

   if -1 < y < 5.0000000000000004e-16

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.4%

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

   if 5.0000000000000004e-16 < y < 3.0000000000000001e105

   1. Initial program 61.0%

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

      1. associate-/l* 67.0%

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

      2. remove-double-neg 67.0%

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

      3. remove-double-neg 67.0%

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

      4. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in x around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -135000000 \lor \neg \left(y \leq 135000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{y + 1} \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -135000000.0) (not (<= y 135000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* (/ y (+ y 1.0)) (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -135000000.0) || !(y <= 135000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-135000000.0d0)) .or. (.not. (y <= 135000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((y / (y + 1.0d0)) * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -135000000.0) || !(y <= 135000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -135000000.0) or not (y <= 135000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -135000000.0) || !(y <= 135000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y / Float64(y + 1.0)) * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -135000000.0) || ~((y <= 135000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((y / (y + 1.0)) * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -135000000.0], N[Not[LessEqual[y, 135000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35e8 or 1.35e8 < y

   1. Initial program 28.3%

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

      1. associate-/l* 50.3%

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

      2. remove-double-neg 50.3%

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

      3. remove-double-neg 50.3%

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

      4. +-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y around inf 99.4%

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

      1. associate--l+ 99.4%

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

      2. div-sub 99.4%

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

      3. remove-double-neg 99.4%

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

      4. neg-sub0 99.4%

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

      5. associate--r- 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. metadata-eval 99.4%

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

      9. sub-neg 99.4%

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

      10. distribute-neg-frac 99.4%

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

      11. unsub-neg 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if -1.35e8 < y < 1.35e8

   1. Initial program 99.4%

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

      1. associate-/l* 99.4%

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

      2. remove-double-neg 99.4%

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

      3. remove-double-neg 99.4%

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

      4. +-commutative 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -135000000 \lor \neg \left(y \leq 135000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{y + 1} \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -52000000 \lor \neg \left(y \leq 116000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -52000000.0) (not (<= y 116000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (/ (* y (- 1.0 x)) (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -52000000.0) || !(y <= 116000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-52000000.0d0)) .or. (.not. (y <= 116000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((y * (1.0d0 - x)) / ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -52000000.0) || !(y <= 116000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -52000000.0) or not (y <= 116000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -52000000.0) || !(y <= 116000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(y * Float64(1.0 - x)) / Float64(-1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -52000000.0) || ~((y <= 116000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + ((y * (1.0 - x)) / (-1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -52000000.0], N[Not[LessEqual[y, 116000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e7 or 1.16e8 < y

   1. Initial program 28.3%

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

      1. associate-/l* 50.3%

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

      2. remove-double-neg 50.3%

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

      3. remove-double-neg 50.3%

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

      4. +-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y around inf 99.4%

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

      1. associate--l+ 99.4%

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

      2. div-sub 99.4%

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

      3. remove-double-neg 99.4%

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

      4. neg-sub0 99.4%

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

      5. associate--r- 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. metadata-eval 99.4%

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

      9. sub-neg 99.4%

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

      10. distribute-neg-frac 99.4%

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

      11. unsub-neg 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if -5.2e7 < y < 1.16e8

   1. Initial program 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -52000000 \lor \neg \left(y \leq 116000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot \left(1 - x\right)}{-1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	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)) .or. (.not. (y <= 1.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 32.5%

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

      1. associate-/l* 53.0%

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

      2. remove-double-neg 53.0%

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

      3. remove-double-neg 53.0%

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

      4. +-commutative 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in y around inf 96.6%

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

      1. associate--l+ 96.6%

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

      2. div-sub 96.6%

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

      3. remove-double-neg 96.6%

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

      4. neg-sub0 96.6%

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

      5. associate--r- 96.6%

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

      6. metadata-eval 96.6%

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

      7. +-commutative 96.6%

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

      8. metadata-eval 96.6%

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

      9. sub-neg 96.6%

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

      10. distribute-neg-frac 96.6%

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

      11. unsub-neg 96.6%

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

      12. sub-neg 96.6%

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

      13. metadata-eval 96.6%

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

   7. Simplified 96.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.3%

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

4. Final simplification 97.3%

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

Alternative 12: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 5e-16) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 5e-16) {
		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 <= 5d-16) 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 <= 5e-16) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 5e-16:
		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 <= 5e-16)
		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 <= 5e-16)
		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, 5e-16], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.0000000000000004e-16 < y

   1. Initial program 33.9%

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

      1. associate-/l* 54.0%

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

      2. remove-double-neg 54.0%

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

      3. remove-double-neg 54.0%

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

      4. +-commutative 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in y around inf 64.8%

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

   if -1 < y < 5.0000000000000004e-16

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. metadata-eval 83.9%

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

      2. distribute-neg-frac 83.9%

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

      3. distribute-neg-frac2 83.9%

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

      4. distribute-neg-in 83.9%

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

      5. metadata-eval 83.9%

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

      6. sub-neg 83.9%

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

   7. Simplified 83.9%

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

   8. Taylor expanded in y around 0 83.3%

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

      1. neg-mul-1 83.3%

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

      2. unsub-neg 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 5e-16) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 5e-16) {
		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 <= 5d-16) 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 <= 5e-16) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 5e-16:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 5e-16)
		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 <= 5e-16)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 5e-16], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 5.0000000000000004e-16 < y

   1. Initial program 33.9%

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

      1. associate-/l* 54.0%

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

      2. remove-double-neg 54.0%

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

      3. remove-double-neg 54.0%

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

      4. +-commutative 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in y around inf 64.8%

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

   if -1 < y < 5.0000000000000004e-16

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 82.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.0% accurate, 11.0× 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 63.1%

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

   1. associate-/l* 74.3%

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

   2. remove-double-neg 74.3%

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

   3. remove-double-neg 74.3%

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

   4. +-commutative 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in y around 0 38.7%

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

6. Final simplification 38.7%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 99.6% accurate, 0.5× 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 2024044 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))