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

Percentage Accurate: 65.5% → 99.1%

Time: 7.7s

Alternatives: 12

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: 65.5% 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: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t_0 \leq 10^{-10} \lor \neg \left(t_0 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + \frac{1 - x}{y}\right)\right) + \frac{-1}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (or (<= t_0 1e-10) (not (<= t_0 2.0)))
     (fma (/ y (+ 1.0 y)) (+ x -1.0) 1.0)
     (+
      (+ x (+ (/ (- 1.0 x) (pow y 3.0)) (/ (- 1.0 x) y)))
      (/ -1.0 (* y y))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 1e-10) || !(t_0 <= 2.0)) {
		tmp = fma((y / (1.0 + y)), (x + -1.0), 1.0);
	} else {
		tmp = (x + (((1.0 - x) / pow(y, 3.0)) + ((1.0 - x) / y))) + (-1.0 / (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if ((t_0 <= 1e-10) || !(t_0 <= 2.0))
		tmp = fma(Float64(y / Float64(1.0 + y)), Float64(x + -1.0), 1.0);
	else
		tmp = Float64(Float64(x + Float64(Float64(Float64(1.0 - x) / (y ^ 3.0)) + Float64(Float64(1.0 - x) / y))) + Float64(-1.0 / Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 1e-10], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x + N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.00000000000000004e-10 or 2 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 85.3%

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

      1. sub-neg 85.3%

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

      2. +-commutative 85.3%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 2

   1. Initial program 12.8%

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

      1. sub-neg 12.8%

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

      2. +-commutative 12.8%

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

      3. associate-/l* 12.7%

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

      4. distribute-neg-frac 12.7%

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

      5. *-lft-identity 12.7%

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

      6. associate-*l/ 12.7%

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

      7. fma-def 12.7%

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

      8. associate-/l* 12.8%

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

      9. *-lft-identity 12.8%

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

      10. +-commutative 12.8%

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

      11. neg-sub0 12.8%

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

      12. associate--r- 12.8%

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

      13. metadata-eval 12.8%

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

      14. +-commutative 12.8%

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

   3. Simplified 12.8%

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

   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%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 10^{-10} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + \frac{1 - x}{y}\right)\right) + \frac{-1}{y \cdot y}\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (or (<= t_0 1e-10) (not (<= t_0 2.0)))
     (fma (/ y (+ 1.0 y)) (+ x -1.0) 1.0)
     (+ (/ (- 1.0 x) y) (+ x (/ -1.0 (* y y)))))))

C

double code(double x, double y) {
	double t_0 = ((1.0 - x) * y) / (1.0 + y);
	double tmp;
	if ((t_0 <= 1e-10) || !(t_0 <= 2.0)) {
		tmp = fma((y / (1.0 + y)), (x + -1.0), 1.0);
	} else {
		tmp = ((1.0 - x) / y) + (x + (-1.0 / (y * y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y))
	tmp = 0.0
	if ((t_0 <= 1e-10) || !(t_0 <= 2.0))
		tmp = fma(Float64(y / Float64(1.0 + y)), Float64(x + -1.0), 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 - x) / y) + Float64(x + Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 1e-10], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.00000000000000004e-10 or 2 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 85.3%

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

      1. sub-neg 85.3%

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

      2. +-commutative 85.3%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 2

   1. Initial program 12.8%

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

      1. sub-neg 12.8%

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

      2. distribute-neg-frac 12.8%

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

      3. neg-mul-1 12.8%

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

      4. associate-*l/ 12.5%

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

      5. metadata-eval 12.5%

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

      6. associate-*l/ 12.5%

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

      7. associate-/r/ 12.5%

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

      8. metadata-eval 12.5%

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

      9. distribute-neg-frac 12.5%

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

      10. cancel-sign-sub-inv 12.5%

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

      11. associate-/r/ 12.4%

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

      12. associate-/r* 12.4%

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

      13. neg-mul-1 12.4%

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

      14. associate-/r/ 12.5%

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

      15. distribute-rgt-neg-in 12.5%

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

      16. associate-/r/ 12.4%

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

      17. distribute-neg-frac 12.4%

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

      18. metadata-eval 12.4%

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

      19. associate-/r/ 12.5%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. +-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. sub-neg 99.5%

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

      4. associate--l+ 99.5%

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

      5. +-commutative 99.5%

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

      6. mul-1-neg 99.5%

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

      7. unsub-neg 99.5%

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

      8. unpow2 99.5%

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

      9. div-sub 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.5%

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

      1. unpow2 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -200000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 1450000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y} + \left(x + \frac{-1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -200000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 1450000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ 1.0 y)))
     (+ (/ (- 1.0 x) y) (+ x (/ -1.0 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -200000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 1450000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y));
	} else {
		tmp = ((1.0 - x) / y) + (x + (-1.0 / (y * 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 <= (-200000000000.0d0)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 1450000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (1.0d0 + y))
    else
        tmp = ((1.0d0 - x) / y) + (x + ((-1.0d0) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -200000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 1450000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y));
	} else {
		tmp = ((1.0 - x) / y) + (x + (-1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -200000000000.0:
		tmp = x - (-1.0 / y)
	elif y <= 1450000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y))
	else:
		tmp = ((1.0 - x) / y) + (x + (-1.0 / (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -200000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 1450000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(1.0 + y)));
	else
		tmp = Float64(Float64(Float64(1.0 - x) / y) + Float64(x + Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -200000000000.0)
		tmp = x - (-1.0 / y);
	elseif (y <= 1450000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y));
	else
		tmp = ((1.0 - x) / y) + (x + (-1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -200000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1450000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e11

   1. Initial program 36.0%

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

      1. sub-neg 36.0%

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

      2. +-commutative 36.0%

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

      3. associate-/l* 58.6%

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

      4. distribute-neg-frac 58.6%

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

      5. *-lft-identity 58.6%

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

      6. associate-*l/ 58.6%

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

      7. fma-def 58.6%

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

      8. associate-/l* 58.6%

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

      9. *-lft-identity 58.6%

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

      10. +-commutative 58.6%

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

      11. neg-sub0 58.6%

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

      12. associate--r- 58.6%

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

      13. metadata-eval 58.6%

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

      14. +-commutative 58.6%

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

   3. Simplified 58.6%

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

   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. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -2e11 < y < 1.45e6

   1. Initial program 100.0%

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

   if 1.45e6 < y

   1. Initial program 38.7%

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

      1. sub-neg 38.7%

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

      2. distribute-neg-frac 38.7%

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

      3. neg-mul-1 38.7%

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

      4. associate-*l/ 38.4%

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

      5. metadata-eval 38.4%

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

      6. associate-*l/ 38.4%

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

      7. associate-/r/ 38.4%

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

      8. metadata-eval 38.4%

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

      9. distribute-neg-frac 38.4%

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

      10. cancel-sign-sub-inv 38.4%

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

      11. associate-/r/ 38.2%

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

      12. associate-/r* 38.2%

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

      13. neg-mul-1 38.2%

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

      14. associate-/r/ 38.4%

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

      15. distribute-rgt-neg-in 38.4%

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

      16. associate-/r/ 38.2%

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

      17. distribute-neg-frac 38.2%

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

      18. metadata-eval 38.2%

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

      19. associate-/r/ 38.4%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. +-commutative 99.6%

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

      2. neg-mul-1 99.6%

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

      3. sub-neg 99.6%

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

      4. associate--l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. mul-1-neg 99.6%

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

      7. unsub-neg 99.6%

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

      8. unpow2 99.6%

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

      9. div-sub 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -200000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 1450000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y} + \left(x + \frac{-1}{y \cdot y}\right)\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -48000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 220000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -48000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 220000000.0)
     (+ 1.0 (* y (/ (+ x -1.0) (+ 1.0 y))))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -48000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 220000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)));
	} 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 <= (-48000000000.0d0)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 220000000.0d0) then
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (1.0d0 + y)))
    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 <= -48000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 220000000.0) {
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -48000000000.0:
		tmp = x - (-1.0 / y)
	elif y <= 220000000.0:
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -48000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 220000000.0)
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(1.0 + y))));
	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 <= -48000000000.0)
		tmp = x - (-1.0 / y);
	elseif (y <= 220000000.0)
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -48000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 220000000.0], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + y), $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 < -4.8e10

   1. Initial program 36.0%

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

      1. sub-neg 36.0%

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

      2. +-commutative 36.0%

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

      3. associate-/l* 58.6%

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

      4. distribute-neg-frac 58.6%

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

      5. *-lft-identity 58.6%

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

      6. associate-*l/ 58.6%

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

      7. fma-def 58.6%

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

      8. associate-/l* 58.6%

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

      9. *-lft-identity 58.6%

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

      10. +-commutative 58.6%

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

      11. neg-sub0 58.6%

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

      12. associate--r- 58.6%

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

      13. metadata-eval 58.6%

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

      14. +-commutative 58.6%

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

   3. Simplified 58.6%

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

   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. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -4.8e10 < y < 2.2e8

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. neg-mul-1 99.6%

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

      4. associate-*l/ 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-*l/ 99.6%

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

      7. associate-/r/ 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-neg-frac 99.6%

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

      10. cancel-sign-sub-inv 99.6%

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

      11. associate-/r/ 99.4%

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

      12. associate-/r* 99.4%

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

      13. neg-mul-1 99.4%

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

      14. associate-/r/ 99.6%

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

      15. distribute-rgt-neg-in 99.6%

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

      16. associate-/r/ 99.4%

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

      17. distribute-neg-frac 99.4%

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

      18. metadata-eval 99.4%

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

      19. associate-/r/ 99.6%

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

   3. Simplified 99.5%

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

   if 2.2e8 < y

   1. Initial program 37.8%

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

      1. sub-neg 37.8%

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

      2. +-commutative 37.8%

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

      3. associate-/l* 60.1%

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

      4. distribute-neg-frac 60.1%

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

      5. *-lft-identity 60.1%

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

      6. associate-*l/ 60.1%

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

      7. fma-def 60.1%

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

      8. associate-/l* 60.2%

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

      9. *-lft-identity 60.2%

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

      10. +-commutative 60.2%

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

      11. neg-sub0 60.2%

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

      12. associate--r- 60.2%

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

      13. metadata-eval 60.2%

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

      14. +-commutative 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -48000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 220000000:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 5: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -360000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 17000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -360000000000.0)
   (- x (/ -1.0 y))
   (if (<= y 17000000.0)
     (+ 1.0 (/ (* y (+ x -1.0)) (+ 1.0 y)))
     (+ x (/ (- 1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -360000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 17000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y));
	} 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 <= (-360000000000.0d0)) then
        tmp = x - ((-1.0d0) / y)
    else if (y <= 17000000.0d0) then
        tmp = 1.0d0 + ((y * (x + (-1.0d0))) / (1.0d0 + y))
    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 <= -360000000000.0) {
		tmp = x - (-1.0 / y);
	} else if (y <= 17000000.0) {
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y));
	} else {
		tmp = x + ((1.0 - x) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -360000000000.0:
		tmp = x - (-1.0 / y)
	elif y <= 17000000.0:
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y))
	else:
		tmp = x + ((1.0 - x) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -360000000000.0)
		tmp = Float64(x - Float64(-1.0 / y));
	elseif (y <= 17000000.0)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x + -1.0)) / Float64(1.0 + y)));
	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 <= -360000000000.0)
		tmp = x - (-1.0 / y);
	elseif (y <= 17000000.0)
		tmp = 1.0 + ((y * (x + -1.0)) / (1.0 + y));
	else
		tmp = x + ((1.0 - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -360000000000.0], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 17000000.0], N[(1.0 + N[(N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6e11

   1. Initial program 36.0%

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

      1. sub-neg 36.0%

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

      2. +-commutative 36.0%

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

      3. associate-/l* 58.6%

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

      4. distribute-neg-frac 58.6%

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

      5. *-lft-identity 58.6%

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

      6. associate-*l/ 58.6%

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

      7. fma-def 58.6%

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

      8. associate-/l* 58.6%

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

      9. *-lft-identity 58.6%

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

      10. +-commutative 58.6%

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

      11. neg-sub0 58.6%

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

      12. associate--r- 58.6%

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

      13. metadata-eval 58.6%

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

      14. +-commutative 58.6%

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

   3. Simplified 58.6%

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

   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. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   if -3.6e11 < y < 1.7e7

   1. Initial program 99.6%

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

   if 1.7e7 < y

   1. Initial program 37.8%

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

      1. sub-neg 37.8%

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

      2. +-commutative 37.8%

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

      3. associate-/l* 60.1%

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

      4. distribute-neg-frac 60.1%

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

      5. *-lft-identity 60.1%

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

      6. associate-*l/ 60.1%

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

      7. fma-def 60.1%

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

      8. associate-/l* 60.2%

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

      9. *-lft-identity 60.2%

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

      10. +-commutative 60.2%

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

      11. neg-sub0 60.2%

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

      12. associate--r- 60.2%

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

      13. metadata-eval 60.2%

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

      14. +-commutative 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -360000000000:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 17000000:\\ \;\;\;\;1 + \frac{y \cdot \left(x + -1\right)}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1 - x}{y}\\ \end{array} \]

Alternative 6: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-77}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.21:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ -1.0 y))))
   (if (<= y -1.0)
     t_0
     (if (<= y 8e-77)
       (- 1.0 y)
       (if (<= y 3.2e-30) (* x y) (if (<= y 0.21) (- 1.0 y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 8e-77) {
		tmp = 1.0 - y;
	} else if (y <= 3.2e-30) {
		tmp = x * y;
	} else if (y <= 0.21) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - ((-1.0d0) / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 8d-77) then
        tmp = 1.0d0 - y
    else if (y <= 3.2d-30) then
        tmp = x * y
    else if (y <= 0.21d0) then
        tmp = 1.0d0 - y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x - (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 8e-77) {
		tmp = 1.0 - y;
	} else if (y <= 3.2e-30) {
		tmp = x * y;
	} else if (y <= 0.21) {
		tmp = 1.0 - y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x - (-1.0 / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 8e-77:
		tmp = 1.0 - y
	elif y <= 3.2e-30:
		tmp = x * y
	elif y <= 0.21:
		tmp = 1.0 - y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x - Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 8e-77)
		tmp = Float64(1.0 - y);
	elseif (y <= 3.2e-30)
		tmp = Float64(x * y);
	elseif (y <= 0.21)
		tmp = Float64(1.0 - y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x - (-1.0 / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 8e-77)
		tmp = 1.0 - y;
	elseif (y <= 3.2e-30)
		tmp = x * y;
	elseif (y <= 0.21)
		tmp = 1.0 - y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 8e-77], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 3.2e-30], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.21], N[(1.0 - y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.209999999999999992 < y

   1. Initial program 37.6%

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

      1. sub-neg 37.6%

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

      2. +-commutative 37.6%

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

      3. associate-/l* 59.7%

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

      4. distribute-neg-frac 59.7%

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

      5. *-lft-identity 59.7%

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

      6. associate-*l/ 59.7%

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

      7. fma-def 59.7%

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

      8. associate-/l* 59.8%

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

      9. *-lft-identity 59.8%

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

      10. +-commutative 59.8%

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

      11. neg-sub0 59.8%

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

      12. associate--r- 59.8%

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

      13. metadata-eval 59.8%

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

      14. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in x around 0 98.8%

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

   if -1 < y < 7.9999999999999994e-77 or 3.2e-30 < y < 0.209999999999999992

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.8%

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

      1. mul-1-neg 77.8%

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

      2. sub-neg 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in y around 0 77.8%

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

      1. neg-mul-1 77.8%

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

      2. sub-neg 77.8%

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

   9. Simplified 77.8%

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

   if 7.9999999999999994e-77 < y < 3.2e-30

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. *-lft-identity 99.8%

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

      6. associate-*l/ 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 79.1%

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

   5. Taylor expanded in y around 0 79.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-77}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.21:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-1}{y}\\ \end{array} \]

Alternative 7: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-77}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.43:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 8e-77)
     (- 1.0 y)
     (if (<= y 3.6e-30) (* x y) (if (<= y 0.43) (- 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 8e-77) {
		tmp = 1.0 - y;
	} else if (y <= 3.6e-30) {
		tmp = x * y;
	} else if (y <= 0.43) {
		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 <= 8d-77) then
        tmp = 1.0d0 - y
    else if (y <= 3.6d-30) then
        tmp = x * y
    else if (y <= 0.43d0) 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 <= 8e-77) {
		tmp = 1.0 - y;
	} else if (y <= 3.6e-30) {
		tmp = x * y;
	} else if (y <= 0.43) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 8e-77:
		tmp = 1.0 - y
	elif y <= 3.6e-30:
		tmp = x * y
	elif y <= 0.43:
		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 <= 8e-77)
		tmp = Float64(1.0 - y);
	elseif (y <= 3.6e-30)
		tmp = Float64(x * y);
	elseif (y <= 0.43)
		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 <= 8e-77)
		tmp = 1.0 - y;
	elseif (y <= 3.6e-30)
		tmp = x * y;
	elseif (y <= 0.43)
		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, 8e-77], N[(1.0 - y), $MachinePrecision], If[LessEqual[y, 3.6e-30], N[(x * y), $MachinePrecision], If[LessEqual[y, 0.43], N[(1.0 - y), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.429999999999999993 < y

   1. Initial program 37.6%

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

      1. sub-neg 37.6%

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

      2. +-commutative 37.6%

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

      3. associate-/l* 59.7%

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

      4. distribute-neg-frac 59.7%

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

      5. *-lft-identity 59.7%

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

      6. associate-*l/ 59.7%

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

      7. fma-def 59.7%

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

      8. associate-/l* 59.8%

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

      9. *-lft-identity 59.8%

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

      10. +-commutative 59.8%

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

      11. neg-sub0 59.8%

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

      12. associate--r- 59.8%

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

      13. metadata-eval 59.8%

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

      14. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in y around inf 74.7%

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

   if -1 < y < 7.9999999999999994e-77 or 3.6000000000000003e-30 < y < 0.429999999999999993

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. *-lft-identity 99.9%

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

      6. associate-*l/ 99.9%

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

      7. fma-def 99.9%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.8%

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

      1. mul-1-neg 77.8%

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

      2. sub-neg 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in y around 0 77.8%

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

      1. neg-mul-1 77.8%

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

      2. sub-neg 77.8%

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

   9. Simplified 77.8%

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

   if 7.9999999999999994e-77 < y < 3.6000000000000003e-30

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. *-lft-identity 99.8%

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

      6. associate-*l/ 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 79.1%

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

   5. Taylor expanded in y around 0 79.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-77}:\\ \;\;\;\;1 - y\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.43:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.78\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.78)))
   (- x (/ -1.0 y))
   (+ 1.0 (* y (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.78)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.78d0))) then
        tmp = x - ((-1.0d0) / y)
    else
        tmp = 1.0d0 + (y * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.78)) {
		tmp = x - (-1.0 / y);
	} else {
		tmp = 1.0 + (y * (x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.78):
		tmp = x - (-1.0 / y)
	else:
		tmp = 1.0 + (y * (x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.78))
		tmp = Float64(x - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.78)))
		tmp = x - (-1.0 / y);
	else
		tmp = 1.0 + (y * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.78]], $MachinePrecision]], N[(x - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.78000000000000003 < y

   1. Initial program 37.6%

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

      1. sub-neg 37.6%

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

      2. +-commutative 37.6%

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

      3. associate-/l* 59.7%

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

      4. distribute-neg-frac 59.7%

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

      5. *-lft-identity 59.7%

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

      6. associate-*l/ 59.7%

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

      7. fma-def 59.7%

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

      8. associate-/l* 59.8%

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

      9. *-lft-identity 59.8%

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

      10. +-commutative 59.8%

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

      11. neg-sub0 59.8%

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

      12. associate--r- 59.8%

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

      13. metadata-eval 59.8%

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

      14. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. unsub-neg 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in x around 0 98.8%

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

   if -1 < y < 0.78000000000000003

   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. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.78\right):\\ \;\;\;\;x - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 9: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

   1. Initial program 36.0%

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

      1. sub-neg 36.0%

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

      2. +-commutative 36.0%

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

      3. associate-/l* 58.6%

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

      4. distribute-neg-frac 58.6%

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

      5. *-lft-identity 58.6%

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

      6. associate-*l/ 58.6%

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

      7. fma-def 58.6%

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

      8. associate-/l* 58.6%

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

      9. *-lft-identity 58.6%

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

      10. +-commutative 58.6%

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

      11. neg-sub0 58.6%

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

      12. associate--r- 58.6%

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

      13. metadata-eval 58.6%

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

      14. +-commutative 58.6%

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

   3. Simplified 58.6%

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

   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. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

   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. distribute-neg-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*l/ 100.0%

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

      7. associate-/r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. associate-/r/ 99.9%

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

      12. associate-/r* 99.9%

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

      13. neg-mul-1 99.9%

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

      14. associate-/r/ 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. associate-/r/ 99.9%

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

      17. distribute-neg-frac 99.9%

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

      18. metadata-eval 99.9%

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

      19. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.2%

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

   if 1 < y

   1. Initial program 38.7%

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

      1. sub-neg 38.7%

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

      2. +-commutative 38.7%

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

      3. associate-/l* 60.4%

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

      4. distribute-neg-frac 60.4%

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

      5. *-lft-identity 60.4%

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

      6. associate-*l/ 60.4%

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

      7. fma-def 60.4%

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

      8. associate-/l* 60.5%

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

      9. *-lft-identity 60.5%

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

      10. +-commutative 60.5%

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

      11. neg-sub0 60.5%

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

      12. associate--r- 60.5%

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

      13. metadata-eval 60.5%

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

      14. +-commutative 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in y around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. mul-1-neg 98.3%

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

      3. unsub-neg 98.3%

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

      4. sub-neg 98.3%

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

      5. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 99.1%

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

Alternative 10: 73.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 86000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   x
   (if (<= y 1.62e-77)
     1.0
     (if (<= y 5.1e-30) (* x y) (if (<= y 86000000000.0) 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.62e-77) {
		tmp = 1.0;
	} else if (y <= 5.1e-30) {
		tmp = x * y;
	} else if (y <= 86000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 1.62d-77) then
        tmp = 1.0d0
    else if (y <= 5.1d-30) then
        tmp = x * y
    else if (y <= 86000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.62e-77) {
		tmp = 1.0;
	} else if (y <= 5.1e-30) {
		tmp = x * y;
	} else if (y <= 86000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.62e-77:
		tmp = 1.0
	elif y <= 5.1e-30:
		tmp = x * y
	elif y <= 86000000000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.62e-77)
		tmp = 1.0;
	elseif (y <= 5.1e-30)
		tmp = Float64(x * y);
	elseif (y <= 86000000000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.62e-77)
		tmp = 1.0;
	elseif (y <= 5.1e-30)
		tmp = x * y;
	elseif (y <= 86000000000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.62e-77], 1.0, If[LessEqual[y, 5.1e-30], N[(x * y), $MachinePrecision], If[LessEqual[y, 86000000000.0], 1.0, x]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 8.6e10 < y

   1. Initial program 37.0%

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

      1. sub-neg 37.0%

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

      2. +-commutative 37.0%

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

      3. associate-/l* 59.6%

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

      4. distribute-neg-frac 59.6%

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

      5. *-lft-identity 59.6%

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

      6. associate-*l/ 59.6%

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

      7. fma-def 59.6%

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

      8. associate-/l* 59.7%

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

      9. *-lft-identity 59.7%

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

      10. +-commutative 59.7%

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

      11. neg-sub0 59.7%

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

      12. associate--r- 59.7%

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

      13. metadata-eval 59.7%

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

      14. +-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y around inf 76.3%

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

   if -1 < y < 1.62000000000000006e-77 or 5.09999999999999972e-30 < y < 8.6e10

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-/l* 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. *-lft-identity 98.9%

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

      6. associate-*l/ 99.0%

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

      7. fma-def 99.0%

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

      8. associate-/l* 99.1%

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

      9. *-lft-identity 99.1%

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

      10. +-commutative 99.1%

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

      11. neg-sub0 99.1%

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

      12. associate--r- 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 75.5%

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

   if 1.62000000000000006e-77 < y < 5.09999999999999972e-30

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. *-lft-identity 99.8%

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

      6. associate-*l/ 99.7%

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

      7. fma-def 99.7%

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

      8. associate-/l* 100.0%

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

      9. *-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. metadata-eval 100.0%

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

      14. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 79.1%

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

   5. Taylor expanded in y around 0 79.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-30}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 86000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 86000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 86000000000.0) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 86000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 86000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 86000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 86000000000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 86000000000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 86000000000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 86000000000.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 8.6e10 < y

   1. Initial program 37.0%

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

      1. sub-neg 37.0%

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

      2. +-commutative 37.0%

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

      3. associate-/l* 59.6%

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

      4. distribute-neg-frac 59.6%

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

      5. *-lft-identity 59.6%

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

      6. associate-*l/ 59.6%

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

      7. fma-def 59.6%

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

      8. associate-/l* 59.7%

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

      9. *-lft-identity 59.7%

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

      10. +-commutative 59.7%

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

      11. neg-sub0 59.7%

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

      12. associate--r- 59.7%

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

      13. metadata-eval 59.7%

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

      14. +-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y around inf 76.3%

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

   if -1 < y < 8.6e10

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-/l* 99.0%

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

      4. distribute-neg-frac 99.0%

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

      5. *-lft-identity 99.0%

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

      6. associate-*l/ 99.0%

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

      7. fma-def 99.0%

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

      8. associate-/l* 99.2%

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

      9. *-lft-identity 99.2%

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

      10. +-commutative 99.2%

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

      11. neg-sub0 99.2%

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

      12. associate--r- 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around 0 71.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 86000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 38.6% 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 68.3%

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

   1. sub-neg 68.3%

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

   2. +-commutative 68.3%

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

   3. associate-/l* 79.5%

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

   4. distribute-neg-frac 79.5%

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

   5. *-lft-identity 79.5%

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

   6. associate-*l/ 79.5%

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

   7. fma-def 79.5%

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

   8. associate-/l* 79.6%

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

   9. *-lft-identity 79.6%

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

   10. +-commutative 79.6%

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

   11. neg-sub0 79.6%

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

   12. associate--r- 79.6%

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

   13. metadata-eval 79.6%

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

   14. +-commutative 79.6%

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

3. Simplified 79.6%

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

4. Taylor expanded in y around 0 38.1%

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

5. Final simplification 38.1%

   \[\leadsto 1 \]

Developer target: 99.7% 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 2023213 
(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))))