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

Percentage Accurate: 65.4% → 99.2%

Time: 8.5s

Alternatives: 11

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: 65.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-9} \lor \neg \left(t_0 \leq 1.005\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + \frac{1 - x}{y}\right)\right) + \frac{x + -1}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (or (<= t_0 5e-9) (not (<= t_0 1.005)))
     (fma (/ y (+ 1.0 y)) (+ x -1.0) 1.0)
     (+
      (+ x (+ (/ (- 1.0 x) (pow y 3.0)) (/ (- 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 <= 5e-9) || !(t_0 <= 1.005)) {
		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))) + ((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 <= 5e-9) || !(t_0 <= 1.005))
		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(Float64(x + -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, 5e-9], N[Not[LessEqual[t$95$0, 1.005]], $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[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 5.0000000000000001e-9 or 1.0049999999999999 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 84.2%

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

      1. sub-neg 84.2%

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

      2. +-commutative 84.2%

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

      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* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 5.0000000000000001e-9 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.0049999999999999

   1. Initial program 8.3%

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

      1. sub-neg 8.3%

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

      2. +-commutative 8.3%

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

      3. associate-/l* 8.3%

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

      4. distribute-neg-frac 8.3%

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

      5. *-lft-identity 8.3%

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

      6. associate-*l/ 8.3%

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

      7. fma-def 8.3%

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

      8. associate-/l* 8.3%

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

      9. *-lft-identity 8.3%

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

      10. +-commutative 8.3%

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

      11. neg-sub0 8.3%

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

      12. associate--r- 8.3%

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

      13. metadata-eval 8.3%

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

      14. +-commutative 8.3%

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

   3. Simplified 8.3%

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

   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%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 5 \cdot 10^{-9} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.005\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(\frac{1 - x}{{y}^{3}} + \frac{1 - x}{y}\right)\right) + \frac{x + -1}{y \cdot y}\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-9} \lor \neg \left(t_0 \leq 1.005\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y} + \left(x + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))))
   (if (or (<= t_0 5e-9) (not (<= t_0 1.005)))
     (fma (/ y (+ 1.0 y)) (+ x -1.0) 1.0)
     (+ (/ (- 1.0 x) y) (+ x (/ (+ 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 <= 5e-9) || !(t_0 <= 1.005)) {
		tmp = fma((y / (1.0 + y)), (x + -1.0), 1.0);
	} else {
		tmp = ((1.0 - x) / y) + (x + ((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 <= 5e-9) || !(t_0 <= 1.005))
		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(Float64(x + -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, 5e-9], N[Not[LessEqual[t$95$0, 1.005]], $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[(N[(x + -1.0), $MachinePrecision] / 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)) < 5.0000000000000001e-9 or 1.0049999999999999 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 84.2%

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

      1. sub-neg 84.2%

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

      2. +-commutative 84.2%

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

      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* 99.9%

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

      9. *-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 5.0000000000000001e-9 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.0049999999999999

   1. Initial program 8.3%

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

      1. sub-neg 8.3%

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

      2. distribute-neg-frac 8.3%

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

      3. neg-mul-1 8.3%

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

      4. associate-*l/ 8.0%

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

      5. metadata-eval 8.0%

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

      6. associate-*l/ 8.0%

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

      7. associate-/r/ 8.0%

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

      8. metadata-eval 8.0%

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

      9. distribute-neg-frac 8.0%

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

      10. cancel-sign-sub-inv 8.0%

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

      11. associate-/r/ 8.2%

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

      12. associate-/r* 8.2%

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

      13. neg-mul-1 8.2%

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

      14. associate-/r/ 8.0%

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

      15. distribute-rgt-neg-in 8.0%

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

      16. associate-/r/ 8.2%

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

      17. distribute-neg-frac 8.2%

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

      18. metadata-eval 8.2%

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

      19. associate-/r/ 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. neg-mul-1 99.9%

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

      3. sub-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. mul-1-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. unpow2 99.9%

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

      9. div-sub 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 5 \cdot 10^{-9} \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.005\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{y} + \left(x + \frac{x + -1}{y \cdot y}\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -260000 \lor \neg \left(y \leq 320000\right):\\ \;\;\;\;\frac{1 - x}{y} + \left(x + \frac{x + -1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -260000.0) (not (<= y 320000.0)))
   (+ (/ (- 1.0 x) y) (+ x (/ (+ x -1.0) (* y y))))
   (- 1.0 (/ (* (- 1.0 x) y) (+ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -260000.0) || !(y <= 320000.0)) {
		tmp = ((1.0 - x) / y) + (x + ((x + -1.0) / (y * y)));
	} else {
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-260000.0d0)) .or. (.not. (y <= 320000.0d0))) then
        tmp = ((1.0d0 - x) / y) + (x + ((x + (-1.0d0)) / (y * y)))
    else
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -260000.0) || !(y <= 320000.0)) {
		tmp = ((1.0 - x) / y) + (x + ((x + -1.0) / (y * y)));
	} else {
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -260000.0) or not (y <= 320000.0):
		tmp = ((1.0 - x) / y) + (x + ((x + -1.0) / (y * y)))
	else:
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -260000.0) || !(y <= 320000.0))
		tmp = Float64(Float64(Float64(1.0 - x) / y) + Float64(x + Float64(Float64(x + -1.0) / Float64(y * y))));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -260000.0) || ~((y <= 320000.0)))
		tmp = ((1.0 - x) / y) + (x + ((x + -1.0) / (y * y)));
	else
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -260000.0], N[Not[LessEqual[y, 320000.0]], $MachinePrecision]], N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(x + N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e5 or 3.2e5 < y

   1. Initial program 32.2%

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

      1. sub-neg 32.2%

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

      2. distribute-neg-frac 32.2%

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

      3. neg-mul-1 32.2%

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

      4. associate-*l/ 32.0%

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

      5. metadata-eval 32.0%

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

      6. associate-*l/ 32.0%

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

      7. associate-/r/ 32.0%

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

      8. metadata-eval 32.0%

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

      9. distribute-neg-frac 32.0%

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

      10. cancel-sign-sub-inv 32.0%

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

      11. associate-/r/ 32.2%

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

      12. associate-/r* 32.2%

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

      13. neg-mul-1 32.2%

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

      14. associate-/r/ 32.0%

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

      15. distribute-rgt-neg-in 32.0%

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

      16. associate-/r/ 32.2%

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

      17. distribute-neg-frac 32.2%

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

      18. metadata-eval 32.2%

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

      19. associate-/r/ 32.0%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. neg-mul-1 99.9%

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

      3. sub-neg 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. mul-1-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. unpow2 99.9%

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

      9. div-sub 99.9%

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

   6. Simplified 99.9%

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

   if -2.6e5 < y < 3.2e5

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -260000 \lor \neg \left(y \leq 320000\right):\\ \;\;\;\;\frac{1 - x}{y} + \left(x + \frac{x + -1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -150000000 \lor \neg \left(y \leq 128000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -150000000.0) (not (<= y 128000000.0)))
   (+ x (/ (- 1.0 x) y))
   (+ 1.0 (* y (/ (+ x -1.0) (+ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -150000000.0) || !(y <= 128000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-150000000.0d0)) .or. (.not. (y <= 128000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 + (y * ((x + (-1.0d0)) / (1.0d0 + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -150000000.0) || !(y <= 128000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -150000000.0) or not (y <= 128000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -150000000.0) || !(y <= 128000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x + -1.0) / Float64(1.0 + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -150000000.0) || ~((y <= 128000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 + (y * ((x + -1.0) / (1.0 + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -150000000.0], N[Not[LessEqual[y, 128000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e8 or 1.28e8 < y

   1. Initial program 31.8%

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

      1. sub-neg 31.8%

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

      2. +-commutative 31.8%

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

      3. associate-/l* 53.0%

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

      4. distribute-neg-frac 53.0%

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

      5. *-lft-identity 53.0%

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

      6. associate-*l/ 53.0%

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

      7. fma-def 53.0%

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

      8. associate-/l* 53.0%

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

      9. *-lft-identity 53.0%

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

      10. +-commutative 53.0%

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

      11. neg-sub0 53.0%

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

      12. associate--r- 53.0%

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

      13. metadata-eval 53.0%

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

      14. +-commutative 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in y around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. sub-neg 99.7%

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

   6. Simplified 99.7%

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

   if -1.5e8 < y < 1.28e8

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. distribute-neg-frac 99.3%

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

      3. neg-mul-1 99.3%

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

      4. associate-*l/ 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-*l/ 99.2%

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

      7. associate-/r/ 99.2%

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

      8. metadata-eval 99.2%

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

      9. distribute-neg-frac 99.2%

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

      10. cancel-sign-sub-inv 99.2%

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

      11. associate-/r/ 99.1%

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

      12. associate-/r* 99.1%

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

      13. neg-mul-1 99.1%

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

      14. associate-/r/ 99.2%

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

      15. distribute-rgt-neg-in 99.2%

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

      16. associate-/r/ 99.1%

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

      17. distribute-neg-frac 99.1%

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

      18. metadata-eval 99.1%

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

      19. associate-/r/ 99.2%

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

   3. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150000000 \lor \neg \left(y \leq 128000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{x + -1}{1 + y}\\ \end{array} \]

Alternative 5: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -150000000 \lor \neg \left(y \leq 170000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -150000000.0) (not (<= y 170000000.0)))
   (+ x (/ (- 1.0 x) y))
   (- 1.0 (/ (* (- 1.0 x) y) (+ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -150000000.0) || !(y <= 170000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-150000000.0d0)) .or. (.not. (y <= 170000000.0d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 - (((1.0d0 - x) * y) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -150000000.0) || !(y <= 170000000.0)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -150000000.0) or not (y <= 170000000.0):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -150000000.0) || !(y <= 170000000.0))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 - x) * y) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -150000000.0) || ~((y <= 170000000.0)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 - (((1.0 - x) * y) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -150000000.0], N[Not[LessEqual[y, 170000000.0]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(1.0 - x), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e8 or 1.7e8 < y

   1. Initial program 31.8%

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

      1. sub-neg 31.8%

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

      2. +-commutative 31.8%

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

      3. associate-/l* 53.0%

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

      4. distribute-neg-frac 53.0%

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

      5. *-lft-identity 53.0%

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

      6. associate-*l/ 53.0%

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

      7. fma-def 53.0%

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

      8. associate-/l* 53.0%

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

      9. *-lft-identity 53.0%

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

      10. +-commutative 53.0%

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

      11. neg-sub0 53.0%

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

      12. associate--r- 53.0%

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

      13. metadata-eval 53.0%

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

      14. +-commutative 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in y around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. sub-neg 99.7%

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

   6. Simplified 99.7%

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

   if -1.5e8 < y < 1.7e8

   1. Initial program 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150000000 \lor \neg \left(y \leq 170000000\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{1 + y}\\ \end{array} \]

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.92\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.92))) (+ x (/ (- 1.0 x) y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.92)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.92d0))) then
        tmp = x + ((1.0d0 - x) / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.92)) {
		tmp = x + ((1.0 - x) / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.92):
		tmp = x + ((1.0 - x) / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.92))
		tmp = Float64(x + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.92)))
		tmp = x + ((1.0 - x) / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.92]], $MachinePrecision]], N[(x + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.92000000000000004 < y

   1. Initial program 32.6%

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

      1. sub-neg 32.6%

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

      2. +-commutative 32.6%

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

      3. associate-/l* 53.4%

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

      4. distribute-neg-frac 53.4%

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

      5. *-lft-identity 53.4%

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

      6. associate-*l/ 53.4%

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

      7. fma-def 53.4%

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

      8. associate-/l* 53.4%

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

      9. *-lft-identity 53.4%

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

      10. +-commutative 53.4%

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

      11. neg-sub0 53.4%

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

      12. associate--r- 53.4%

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

      13. metadata-eval 53.4%

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

      14. +-commutative 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in y around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-neg-frac 98.6%

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

      6. distribute-neg-in 98.6%

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

      7. metadata-eval 98.6%

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

      8. +-commutative 98.6%

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

      9. sub-neg 98.6%

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

   6. Simplified 98.6%

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

   if -1 < y < 0.92000000000000004

   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. associate-/r/ 100.0%

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

      6. fma-def 100.0%

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

      7. neg-sub0 100.0%

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

      8. associate--r- 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.5%

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

   5. Taylor expanded in y around 0 72.5%

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

      1. neg-mul-1 72.5%

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

      2. sub-neg 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.92\right):\\ \;\;\;\;x + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 32.6%

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

      1. sub-neg 32.6%

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

      2. +-commutative 32.6%

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

      3. associate-/l* 53.4%

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

      4. distribute-neg-frac 53.4%

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

      5. *-lft-identity 53.4%

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

      6. associate-*l/ 53.4%

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

      7. fma-def 53.4%

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

      8. associate-/l* 53.4%

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

      9. *-lft-identity 53.4%

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

      10. +-commutative 53.4%

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

      11. neg-sub0 53.4%

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

      12. associate--r- 53.4%

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

      13. metadata-eval 53.4%

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

      14. +-commutative 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in y around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-neg-frac 98.6%

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

      6. distribute-neg-in 98.6%

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

      7. metadata-eval 98.6%

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

      8. +-commutative 98.6%

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

      9. sub-neg 98.6%

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

   6. Simplified 98.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. 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 98.4%

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

4. Final simplification 98.5%

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

Alternative 8: 74.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{y}{1 + y}\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5e-8) (* x (/ y (+ 1.0 y))) (if (<= y 0.65) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-8) {
		tmp = x * (y / (1.0 + y));
	} else if (y <= 0.65) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.5d-8)) then
        tmp = x * (y / (1.0d0 + y))
    else if (y <= 0.65d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5e-8) {
		tmp = x * (y / (1.0 + y));
	} else if (y <= 0.65) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5e-8:
		tmp = x * (y / (1.0 + y))
	elif y <= 0.65:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5e-8)
		tmp = Float64(x * Float64(y / Float64(1.0 + y)));
	elseif (y <= 0.65)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.5e-8)
		tmp = x * (y / (1.0 + y));
	elseif (y <= 0.65)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5e-8], N[(x * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.65], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4999999999999999e-8

   1. Initial program 34.3%

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

      1. sub-neg 34.3%

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

      2. +-commutative 34.3%

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

      3. associate-/l* 48.9%

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

      4. distribute-neg-frac 48.9%

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

      5. *-lft-identity 48.9%

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

      6. associate-*l/ 48.9%

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

      7. fma-def 48.9%

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

      8. associate-/l* 49.0%

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

      9. *-lft-identity 49.0%

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

      10. +-commutative 49.0%

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

      11. neg-sub0 49.0%

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

      12. associate--r- 49.0%

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

      13. metadata-eval 49.0%

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

      14. +-commutative 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x around inf 56.5%

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

      1. associate-/l* 60.4%

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

   6. Simplified 60.4%

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

      1. associate-/r/ 71.2%

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

   8. Applied egg-rr 71.2%

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

   if -2.4999999999999999e-8 < y < 0.650000000000000022

   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. associate-/r/ 100.0%

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

      6. fma-def 100.0%

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

      7. neg-sub0 100.0%

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

      8. associate--r- 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.4%

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

   5. Taylor expanded in y around 0 74.4%

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

      1. neg-mul-1 74.4%

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

      2. sub-neg 74.4%

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

   7. Simplified 74.4%

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

   if 0.650000000000000022 < y

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. associate-/l* 60.7%

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

      4. distribute-neg-frac 60.7%

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

      5. *-lft-identity 60.7%

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

      6. associate-*l/ 60.7%

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

      7. fma-def 60.7%

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

      8. associate-/l* 60.7%

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

      9. *-lft-identity 60.7%

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

      10. +-commutative 60.7%

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

      11. neg-sub0 60.7%

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

      12. associate--r- 60.7%

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

      13. metadata-eval 60.7%

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

      14. +-commutative 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in y around inf 84.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{y}{1 + y}\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 0.92) (- 1.0 y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.92) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = x
    else if (y <= 0.92d0) then
        tmp = 1.0d0 - y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 0.92) {
		tmp = 1.0 - y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 0.92:
		tmp = 1.0 - y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.92)
		tmp = Float64(1.0 - y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 0.92)
		tmp = 1.0 - y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 0.92], N[(1.0 - y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.92000000000000004 < y

   1. Initial program 32.6%

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

      1. sub-neg 32.6%

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

      2. +-commutative 32.6%

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

      3. associate-/l* 53.4%

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

      4. distribute-neg-frac 53.4%

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

      5. *-lft-identity 53.4%

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

      6. associate-*l/ 53.4%

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

      7. fma-def 53.4%

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

      8. associate-/l* 53.4%

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

      9. *-lft-identity 53.4%

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

      10. +-commutative 53.4%

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

      11. neg-sub0 53.4%

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

      12. associate--r- 53.4%

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

      13. metadata-eval 53.4%

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

      14. +-commutative 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in y around inf 76.7%

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

   if -1 < y < 0.92000000000000004

   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. associate-/r/ 100.0%

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

      6. fma-def 100.0%

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

      7. neg-sub0 100.0%

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

      8. associate--r- 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 72.5%

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

   5. Taylor expanded in y around 0 72.5%

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

      1. neg-mul-1 72.5%

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

      2. sub-neg 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.92:\\ \;\;\;\;1 - y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 73.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) x (if (<= y 1.65e+14) 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x;
	} else if (y <= 1.65e+14) {
		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 <= 1.65d+14) 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 <= 1.65e+14) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x
	elif y <= 1.65e+14:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = x;
	elseif (y <= 1.65e+14)
		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 <= 1.65e+14)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], x, If[LessEqual[y, 1.65e+14], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.65e14 < y

   1. Initial program 32.4%

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

      1. sub-neg 32.4%

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

      2. +-commutative 32.4%

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

      3. associate-/l* 53.5%

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

      4. distribute-neg-frac 53.5%

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

      5. *-lft-identity 53.5%

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

      6. associate-*l/ 53.5%

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

      7. fma-def 53.5%

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

      8. associate-/l* 53.5%

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

      9. *-lft-identity 53.5%

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

      10. +-commutative 53.5%

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

      11. neg-sub0 53.5%

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

      12. associate--r- 53.5%

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

      13. metadata-eval 53.5%

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

      14. +-commutative 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in y around inf 77.7%

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

   if -1 < y < 1.65e14

   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* 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* 99.1%

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

      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 70.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 38.7% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 63.1%

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

   1. sub-neg 63.1%

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

   2. +-commutative 63.1%

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

   3. associate-/l* 74.5%

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

   4. distribute-neg-frac 74.5%

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

   5. *-lft-identity 74.5%

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

   6. associate-*l/ 74.5%

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

   7. fma-def 74.5%

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

   8. associate-/l* 74.5%

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

   9. *-lft-identity 74.5%

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

   10. +-commutative 74.5%

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

   11. neg-sub0 74.5%

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

   12. associate--r- 74.5%

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

   13. metadata-eval 74.5%

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

   14. +-commutative 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in y around 0 34.3%

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

5. Final simplification 34.3%

   \[\leadsto 1 \]

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 2023196 
(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))))