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

Percentage Accurate: 65.9% → 99.9%

Time: 9.1s

Alternatives: 13

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -130000:\\ \;\;\;\;\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \left(\left(x - \frac{1 - x}{y \cdot y}\right) - \left(\frac{x}{y} + \frac{x}{{y}^{3}}\right)\right)\right)\\ \mathbf{elif}\;y \leq 12500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + t_0\right)\right) - \frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -130000.0)
     (+
      (/ 1.0 y)
      (+
       (/ 1.0 (pow y 3.0))
       (- (- x (/ (- 1.0 x) (* y y))) (+ (/ x y) (/ x (pow y 3.0))))))
     (if (<= y 12500.0)
       (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)
       (- (+ x (+ (/ (- 1.0 x) (pow y 3.0)) t_0)) (/ t_0 y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -130000.0) {
		tmp = (1.0 / y) + ((1.0 / pow(y, 3.0)) + ((x - ((1.0 - x) / (y * y))) - ((x / y) + (x / pow(y, 3.0)))));
	} else if (y <= 12500.0) {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	} else {
		tmp = (x + (((1.0 - x) / pow(y, 3.0)) + t_0)) - (t_0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -130000.0)
		tmp = Float64(Float64(1.0 / y) + Float64(Float64(1.0 / (y ^ 3.0)) + Float64(Float64(x - Float64(Float64(1.0 - x) / Float64(y * y))) - Float64(Float64(x / y) + Float64(x / (y ^ 3.0))))));
	elseif (y <= 12500.0)
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	else
		tmp = Float64(Float64(x + Float64(Float64(Float64(1.0 - x) / (y ^ 3.0)) + t_0)) - Float64(t_0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -130000.0], N[(N[(1.0 / y), $MachinePrecision] + N[(N[(1.0 / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x - N[(N[(1.0 - x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] + N[(x / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 12500.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e5

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. associate-*l/ 59.7%

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

      3. distribute-lft-neg-in 59.7%

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

      4. distribute-frac-neg 59.7%

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

      5. neg-sub0 59.7%

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

      6. associate--r- 59.7%

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

      7. metadata-eval 59.7%

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

      8. +-commutative 59.7%

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

      9. +-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-mul-1 100.0%

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

      4. sub-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. associate--l+ 100.0%

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

   6. Simplified 100.0%

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

   if -1.3e5 < y < 12500

   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. associate-*l/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. distribute-frac-neg 99.9%

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

      9. neg-sub0 99.9%

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

      10. associate--r- 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

      13. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 12500 < y

   1. Initial program 24.4%

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

      1. sub-neg 24.4%

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

      2. metadata-eval 24.4%

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

      3. associate--r- 24.4%

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

      4. neg-sub0 24.4%

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

      5. remove-double-neg 24.4%

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

      6. associate-/l* 52.6%

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

      7. +-commutative 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130000:\\ \;\;\;\;\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \left(\left(x - \frac{1 - x}{y \cdot y}\right) - \left(\frac{x}{y} + \frac{x}{{y}^{3}}\right)\right)\right)\\ \mathbf{elif}\;y \leq 12500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + \frac{1 - x}{y}\right)\right) - \frac{\frac{1 - x}{y}}{y}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -130000 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + t_0\right)\right) - \frac{t_0}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (or (<= y -130000.0) (not (<= y 14000.0)))
     (- (+ x (+ (/ (- 1.0 x) (pow y 3.0)) t_0)) (/ t_0 y))
     (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if ((y <= -130000.0) || !(y <= 14000.0)) {
		tmp = (x + (((1.0 - x) / pow(y, 3.0)) + t_0)) - (t_0 / y);
	} else {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if ((y <= -130000.0) || !(y <= 14000.0))
		tmp = Float64(Float64(x + Float64(Float64(Float64(1.0 - x) / (y ^ 3.0)) + t_0)) - Float64(t_0 / y));
	else
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[y, -130000.0], N[Not[LessEqual[y, 14000.0]], $MachinePrecision]], N[(N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e5 or 14000 < y

   1. Initial program 24.5%

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

      1. sub-neg 24.5%

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

      2. metadata-eval 24.5%

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

      3. associate--r- 24.5%

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

      4. neg-sub0 24.5%

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

      5. remove-double-neg 24.5%

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

      6. associate-/l* 56.1%

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

      7. +-commutative 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

   6. Simplified 100.0%

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

   if -1.3e5 < 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. associate-*l/ 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-def 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. distribute-frac-neg 99.9%

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

      9. neg-sub0 99.9%

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

      10. associate--r- 99.9%

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

      11. metadata-eval 99.9%

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

      12. +-commutative 99.9%

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

      13. +-commutative 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130000 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + \frac{1 - x}{y}\right)\right) - \frac{\frac{1 - x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + t_0\right) - \frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -4.1e+37)
     (+ (/ 1.0 y) x)
     (if (<= y 280000.0)
       (fma (/ (+ x -1.0) (+ y 1.0)) y 1.0)
       (- (+ x t_0) (/ t_0 y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -4.1e+37) {
		tmp = (1.0 / y) + x;
	} else if (y <= 280000.0) {
		tmp = fma(((x + -1.0) / (y + 1.0)), y, 1.0);
	} else {
		tmp = (x + t_0) - (t_0 / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -4.1e+37)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 280000.0)
		tmp = fma(Float64(Float64(x + -1.0) / Float64(y + 1.0)), y, 1.0);
	else
		tmp = Float64(Float64(x + t_0) - Float64(t_0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -4.1e+37], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 280000.0], N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(x + t$95$0), $MachinePrecision] - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0999999999999998e37

   1. Initial program 21.0%

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

      1. sub-neg 21.0%

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

      2. metadata-eval 21.0%

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

      3. associate--r- 21.0%

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

      4. neg-sub0 21.0%

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

      5. remove-double-neg 21.0%

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

      6. associate-/l* 58.2%

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

      7. +-commutative 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -4.0999999999999998e37 < y < 2.8e5

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-*l/ 99.7%

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

      4. distribute-lft-neg-in 99.7%

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

      5. fma-def 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. distribute-frac-neg 99.7%

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

      9. neg-sub0 99.7%

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

      10. associate--r- 99.7%

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

      11. metadata-eval 99.7%

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

      12. +-commutative 99.7%

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

      13. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 2.8e5 < y

   1. Initial program 23.1%

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

      1. sub-neg 23.1%

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

      2. metadata-eval 23.1%

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

      3. associate--r- 23.1%

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

      4. neg-sub0 23.1%

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

      5. remove-double-neg 23.1%

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

      6. associate-/l* 51.8%

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

      7. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in y around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. mul-1-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. metadata-eval 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. metadata-eval 99.7%

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

      11. sub-neg 99.7%

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

      12. div-sub 99.7%

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

      13. sub-neg 99.7%

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

      14. metadata-eval 99.7%

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

      15. unpow2 99.7%

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

      16. associate-/r* 99.7%

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

      17. +-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 280000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x + -1}{y + 1}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) - \frac{\frac{1 - x}{y}}{y}\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{y}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 230000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + t_0\right) - \frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) y)))
   (if (<= y -4.1e+37)
     (+ (/ 1.0 y) x)
     (if (<= y 230000.0)
       (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
       (- (+ x t_0) (/ t_0 y))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -4.1e+37) {
		tmp = (1.0 / y) + x;
	} else if (y <= 230000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (x + t_0) - (t_0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / y
    if (y <= (-4.1d+37)) then
        tmp = (1.0d0 / y) + x
    else if (y <= 230000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = (x + t_0) - (t_0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 - x) / y;
	double tmp;
	if (y <= -4.1e+37) {
		tmp = (1.0 / y) + x;
	} else if (y <= 230000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = (x + t_0) - (t_0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 - x) / y
	tmp = 0
	if y <= -4.1e+37:
		tmp = (1.0 / y) + x
	elif y <= 230000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = (x + t_0) - (t_0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - x) / y)
	tmp = 0.0
	if (y <= -4.1e+37)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 230000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x + t_0) - Float64(t_0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 - x) / y;
	tmp = 0.0;
	if (y <= -4.1e+37)
		tmp = (1.0 / y) + x;
	elseif (y <= 230000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = (x + t_0) - (t_0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -4.1e+37], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 230000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + t$95$0), $MachinePrecision] - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0999999999999998e37

   1. Initial program 21.0%

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

      1. sub-neg 21.0%

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

      2. metadata-eval 21.0%

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

      3. associate--r- 21.0%

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

      4. neg-sub0 21.0%

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

      5. remove-double-neg 21.0%

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

      6. associate-/l* 58.2%

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

      7. +-commutative 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -4.0999999999999998e37 < y < 2.3e5

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l/ 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. neg-sub0 99.7%

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

      6. associate--r- 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 2.3e5 < y

   1. Initial program 23.1%

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

      1. sub-neg 23.1%

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

      2. metadata-eval 23.1%

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

      3. associate--r- 23.1%

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

      4. neg-sub0 23.1%

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

      5. remove-double-neg 23.1%

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

      6. associate-/l* 51.8%

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

      7. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in y around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. mul-1-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. metadata-eval 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. metadata-eval 99.7%

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

      11. sub-neg 99.7%

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

      12. div-sub 99.7%

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

      13. sub-neg 99.7%

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

      14. metadata-eval 99.7%

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

      15. unpow2 99.7%

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

      16. associate-/r* 99.7%

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

      17. +-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 230000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{1 - x}{y}\right) - \frac{\frac{1 - x}{y}}{y}\\ \end{array} \]

Alternative 5: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 420000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.1e+37)
   (+ (/ 1.0 y) x)
   (if (<= y 420000000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ y 1.0))))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+37) {
		tmp = (1.0 / y) + x;
	} else if (y <= 420000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 - x) / 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 <= (-4.1d+37)) then
        tmp = (1.0d0 / y) + x
    else if (y <= 420000000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (y + 1.0d0)))
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+37) {
		tmp = (1.0 / y) + x;
	} else if (y <= 420000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.1e+37:
		tmp = (1.0 / y) + x
	elif y <= 420000000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.1e+37)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 420000000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.1e+37)
		tmp = (1.0 / y) + x;
	elseif (y <= 420000000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (y + 1.0)));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.1e+37], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 420000000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0999999999999998e37

   1. Initial program 21.0%

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

      1. sub-neg 21.0%

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

      2. metadata-eval 21.0%

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

      3. associate--r- 21.0%

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

      4. neg-sub0 21.0%

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

      5. remove-double-neg 21.0%

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

      6. associate-/l* 58.2%

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

      7. +-commutative 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -4.0999999999999998e37 < y < 4.2e8

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-*l/ 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

      4. distribute-frac-neg 99.5%

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

      5. neg-sub0 99.5%

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

      6. associate--r- 99.5%

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

      7. metadata-eval 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   if 4.2e8 < y

   1. Initial program 22.3%

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

      1. sub-neg 22.3%

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

      2. metadata-eval 22.3%

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

      3. associate--r- 22.3%

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

      4. neg-sub0 22.3%

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

      5. remove-double-neg 22.3%

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

      6. associate-/l* 51.5%

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

      7. +-commutative 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in y around -inf 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. sub-neg 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 420000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 6: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -180000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 36000:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -180000.0)
   (+ (/ 1.0 y) x)
   (if (<= y 36000.0) (+ 1.0 (* y (/ x (+ y 1.0)))) (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -180000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 36000.0) {
		tmp = 1.0 + (y * (x / (y + 1.0)));
	} else {
		tmp = x + ((1.0 - x) / 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 <= (-180000.0d0)) then
        tmp = (1.0d0 / y) + x
    else if (y <= 36000.0d0) then
        tmp = 1.0d0 + (y * (x / (y + 1.0d0)))
    else
        tmp = x + ((1.0d0 - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -180000.0) {
		tmp = (1.0 / y) + x;
	} else if (y <= 36000.0) {
		tmp = 1.0 + (y * (x / (y + 1.0)));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -180000.0:
		tmp = (1.0 / y) + x
	elif y <= 36000.0:
		tmp = 1.0 + (y * (x / (y + 1.0)))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -180000.0)
		tmp = Float64(Float64(1.0 / y) + x);
	elseif (y <= 36000.0)
		tmp = Float64(1.0 + Float64(y * Float64(x / Float64(y + 1.0))));
	else
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -180000.0)
		tmp = (1.0 / y) + x;
	elseif (y <= 36000.0)
		tmp = 1.0 + (y * (x / (y + 1.0)));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -180000.0], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 36000.0], N[(1.0 + N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e5

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. metadata-eval 24.7%

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

      3. associate--r- 24.7%

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

      4. neg-sub0 24.7%

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

      5. remove-double-neg 24.7%

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

      6. associate-/l* 60.0%

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

      7. +-commutative 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in y around -inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-neg-frac 99.1%

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

      6. +-commutative 99.1%

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

      7. distribute-neg-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. sub-neg 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in x around 0 99.1%

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

   if -1.8e5 < y < 36000

   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. associate-*l/ 99.9%

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

      3. distribute-lft-neg-in 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. neg-sub0 99.9%

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

      6. associate--r- 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 97.2%

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

   if 36000 < y

   1. Initial program 23.1%

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

      1. sub-neg 23.1%

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

      2. metadata-eval 23.1%

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

      3. associate--r- 23.1%

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

      4. neg-sub0 23.1%

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

      5. remove-double-neg 23.1%

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

      6. associate-/l* 51.8%

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

      7. +-commutative 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in y around -inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. distribute-neg-frac 99.1%

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

      6. +-commutative 99.1%

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

      7. distribute-neg-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. sub-neg 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -180000:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;y \leq 36000:\\ \;\;\;\;1 + y \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 7: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.15))) (+ x (/ (- 1.0 x) y)) (+ 1.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.15)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * 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)) .or. (.not. (y <= 1.15d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.15)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.15):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.15))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.15)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.15]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.1499999999999999 < y

   1. Initial program 28.0%

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

      1. sub-neg 28.0%

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

      2. metadata-eval 28.0%

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

      3. associate--r- 28.0%

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

      4. neg-sub0 28.0%

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

      5. remove-double-neg 28.0%

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

      6. associate-/l* 58.1%

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

      7. +-commutative 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in y around -inf 95.9%

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

      1. +-commutative 95.9%

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

      2. mul-1-neg 95.9%

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

      3. sub-neg 95.9%

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

      4. metadata-eval 95.9%

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

      5. distribute-neg-frac 95.9%

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

      6. +-commutative 95.9%

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

      7. distribute-neg-in 95.9%

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

      8. metadata-eval 95.9%

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

      9. sub-neg 95.9%

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

   6. Simplified 95.9%

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

   if -1 < y < 1.1499999999999999

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.6%

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

   5. Taylor expanded in y around 0 97.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.15\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot x\\ \end{array} \]

Alternative 8: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

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) - 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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + ((y * x) - y)
    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) - y);
	}
	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) - y)
	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(Float64(y * x) - y));
	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) - y);
	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[(N[(y * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 28.0%

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

      1. sub-neg 28.0%

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

      2. metadata-eval 28.0%

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

      3. associate--r- 28.0%

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

      4. neg-sub0 28.0%

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

      5. remove-double-neg 28.0%

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

      6. associate-/l* 58.1%

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

      7. +-commutative 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in y around -inf 95.9%

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

      1. +-commutative 95.9%

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

      2. mul-1-neg 95.9%

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

      3. sub-neg 95.9%

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

      4. metadata-eval 95.9%

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

      5. distribute-neg-frac 95.9%

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

      6. +-commutative 95.9%

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

      7. distribute-neg-in 95.9%

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

      8. metadata-eval 95.9%

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

      9. sub-neg 95.9%

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

   6. Simplified 95.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

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

      3. associate--r- 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. /-rgt-identity 98.4%

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

      3. associate-/r/ 98.3%

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

      4. div-sub 98.3%

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

      5. remove-double-div 98.3%

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

      6. associate-/r/ 98.4%

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

      7. /-rgt-identity 98.4%

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

      8. *-commutative 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 97.1%

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

Alternative 9: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 28.0%

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

      1. sub-neg 28.0%

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

      2. metadata-eval 28.0%

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

      3. associate--r- 28.0%

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

      4. neg-sub0 28.0%

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

      5. remove-double-neg 28.0%

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

      6. associate-/l* 58.1%

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

      7. +-commutative 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in y around -inf 95.9%

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

      1. +-commutative 95.9%

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

      2. mul-1-neg 95.9%

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

      3. sub-neg 95.9%

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

      4. metadata-eval 95.9%

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

      5. distribute-neg-frac 95.9%

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

      6. +-commutative 95.9%

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

      7. distribute-neg-in 95.9%

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

      8. metadata-eval 95.9%

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

      9. sub-neg 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in x around 0 95.0%

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

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.6%

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

   5. Taylor expanded in y around 0 97.6%

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

4. Final simplification 96.3%

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

Alternative 10: 85.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.95:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.95) (+ 1.0 (* y x)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.95) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 1.95d0) then
        tmp = 1.0d0 + (y * x)
    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 <= 1.95) {
		tmp = 1.0 + (y * x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.95:
		tmp = 1.0 + (y * x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.95)
		tmp = Float64(1.0 + Float64(y * x));
	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 <= 1.95)
		tmp = 1.0 + (y * x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.95], N[(1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.94999999999999996 < y

   1. Initial program 27.4%

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

      1. sub-neg 27.4%

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

      2. associate-*l/ 57.5%

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

      3. distribute-lft-neg-in 57.5%

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

      4. distribute-frac-neg 57.5%

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

      5. neg-sub0 57.5%

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

      6. associate--r- 57.5%

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

      7. metadata-eval 57.5%

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

      8. +-commutative 57.5%

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

      9. +-commutative 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in y around inf 73.4%

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

   if -1 < y < 1.94999999999999996

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.0%

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

   5. Taylor expanded in y around 0 97.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.95:\\ \;\;\;\;1 + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0305:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.0305) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0305) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.0305d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0305) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.0305:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0305)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0305)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.0305], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.030499999999999999 < y

   1. Initial program 28.5%

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

      1. sub-neg 28.5%

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

      2. associate-*l/ 58.1%

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

      3. distribute-lft-neg-in 58.1%

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

      4. distribute-frac-neg 58.1%

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

      5. neg-sub0 58.1%

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

      6. associate--r- 58.1%

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

      7. metadata-eval 58.1%

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

      8. +-commutative 58.1%

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

      9. +-commutative 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in y around inf 72.5%

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

   if -1 < y < 0.030499999999999999

   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. metadata-eval 100.0%

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

      3. associate--r- 100.0%

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

      4. neg-sub0 100.0%

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

      5. remove-double-neg 100.0%

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

      6. associate-/l* 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. /-rgt-identity 98.9%

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

      3. associate-/r/ 98.9%

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

      4. div-sub 98.9%

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

      5. remove-double-div 98.9%

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

      6. associate-/r/ 98.9%

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

      7. /-rgt-identity 98.9%

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

      8. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in x around 0 74.6%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0305:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 73.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0305:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) x (if (<= y 0.0305) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0305) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.0305d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.0305) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.0305:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0305)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.0305)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.0305], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.030499999999999999 < y

   1. Initial program 28.5%

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

      1. sub-neg 28.5%

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

      2. associate-*l/ 58.1%

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

      3. distribute-lft-neg-in 58.1%

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

      4. distribute-frac-neg 58.1%

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

      5. neg-sub0 58.1%

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

      6. associate--r- 58.1%

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

      7. metadata-eval 58.1%

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

      8. +-commutative 58.1%

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

      9. +-commutative 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in y around inf 72.5%

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

   if -1 < y < 0.030499999999999999

   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. associate-*l/ 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate--r- 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 73.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.0305:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13: 37.8% 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 64.2%

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

   1. sub-neg 64.2%

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

   2. associate-*l/ 79.1%

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

   3. distribute-lft-neg-in 79.1%

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

   4. distribute-frac-neg 79.1%

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

   5. neg-sub0 79.1%

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

   6. associate--r- 79.1%

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

   7. metadata-eval 79.1%

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

   8. +-commutative 79.1%

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

   9. +-commutative 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in y around 0 38.7%

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

5. Final simplification 38.7%

   \[\leadsto 1 \]

Developer target: 99.6% accurate, 0.7× 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 2023268 
(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))))