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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{1 - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x - y}{1 - y} \]
4. Add Preprocessing

Alternative 2: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3) (not (<= y 1.0))) (- 1.0 (/ (+ x -1.0) y)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3) || !(y <= 1.0)) {
		tmp = 1.0 - ((x + -1.0) / y);
	} else {
		tmp = 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 <= (-1.3d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - ((x + (-1.0d0)) / y)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.3) || !(y <= 1.0)) {
		tmp = 1.0 - ((x + -1.0) / y);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3) or not (y <= 1.0):
		tmp = 1.0 - ((x + -1.0) / y)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.3) || ~((y <= 1.0)))
		tmp = 1.0 - ((x + -1.0) / y);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000004 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. div-sub 98.7%

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

      5. unsub-neg 98.7%

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

      6. mul-1-neg 98.7%

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

      7. +-commutative 98.7%

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

      8. metadata-eval 98.7%

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

      9. distribute-lft-in 98.7%

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

      10. metadata-eval 98.7%

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

      11. sub-neg 98.7%

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

      12. associate-*r/ 98.7%

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

      13. mul-1-neg 98.7%

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

      14. unsub-neg 98.7%

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

      15. sub-neg 98.7%

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

      16. metadata-eval 98.7%

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

      17. +-commutative 98.7%

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

   5. Simplified 98.7%

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

   if -1.30000000000000004 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-lft-neg-in 98.7%

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

      4. cancel-sign-sub 98.7%

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

      5. +-commutative 98.7%

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

      6. metadata-eval 98.7%

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

      7. distribute-lft-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. sub-neg 98.7%

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

      10. mul-1-neg 98.7%

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

      11. remove-double-neg 98.7%

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

      12. sub-neg 98.7%

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

      13. metadata-eval 98.7%

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

      14. +-commutative 98.7%

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

      15. distribute-neg-in 98.7%

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

      16. metadata-eval 98.7%

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

      17. mul-1-neg 98.7%

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

      18. *-commutative 98.7%

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

      19. mul-1-neg 98.7%

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

      20. unsub-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. div-sub 98.7%

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

      5. unsub-neg 98.7%

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

      6. mul-1-neg 98.7%

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

      7. +-commutative 98.7%

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

      8. metadata-eval 98.7%

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

      9. distribute-lft-in 98.7%

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

      10. metadata-eval 98.7%

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

      11. sub-neg 98.7%

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

      12. associate-*r/ 98.7%

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

      13. mul-1-neg 98.7%

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

      14. unsub-neg 98.7%

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

      15. sub-neg 98.7%

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

      16. metadata-eval 98.7%

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

      17. +-commutative 98.7%

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

   5. Simplified 98.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-lft-neg-in 98.7%

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

      4. cancel-sign-sub 98.7%

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

      5. +-commutative 98.7%

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

      6. metadata-eval 98.7%

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

      7. distribute-lft-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. sub-neg 98.7%

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

      10. mul-1-neg 98.7%

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

      11. remove-double-neg 98.7%

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

      12. sub-neg 98.7%

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

      13. metadata-eval 98.7%

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

      14. +-commutative 98.7%

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

      15. distribute-neg-in 98.7%

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

      16. metadata-eval 98.7%

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

      17. mul-1-neg 98.7%

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

      18. *-commutative 98.7%

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

      19. mul-1-neg 98.7%

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

      20. unsub-neg 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 4: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = 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[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. div-sub 98.7%

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

      5. unsub-neg 98.7%

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

      6. mul-1-neg 98.7%

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

      7. +-commutative 98.7%

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

      8. metadata-eval 98.7%

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

      9. distribute-lft-in 98.7%

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

      10. metadata-eval 98.7%

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

      11. sub-neg 98.7%

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

      12. associate-*r/ 98.7%

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

      13. mul-1-neg 98.7%

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

      14. unsub-neg 98.7%

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

      15. sub-neg 98.7%

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

      16. metadata-eval 98.7%

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

      17. +-commutative 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around inf 98.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-lft-neg-in 98.7%

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

      4. cancel-sign-sub 98.7%

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

      5. +-commutative 98.7%

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

      6. metadata-eval 98.7%

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

      7. distribute-lft-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. sub-neg 98.7%

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

      10. mul-1-neg 98.7%

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

      11. remove-double-neg 98.7%

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

      12. sub-neg 98.7%

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

      13. metadata-eval 98.7%

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

      14. +-commutative 98.7%

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

      15. distribute-neg-in 98.7%

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

      16. metadata-eval 98.7%

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

      17. mul-1-neg 98.7%

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

      18. *-commutative 98.7%

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

      19. mul-1-neg 98.7%

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

      20. unsub-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.2%

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

Alternative 5: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14.2:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -14.2) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -14.2) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 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 <= (-14.2d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -14.2) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -14.2:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -14.2)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -14.2)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -14.2], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -14.199999999999999 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 80.5%

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

   if -14.199999999999999 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-lft-neg-in 98.7%

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

      4. cancel-sign-sub 98.7%

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

      5. +-commutative 98.7%

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

      6. metadata-eval 98.7%

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

      7. distribute-lft-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. sub-neg 98.7%

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

      10. mul-1-neg 98.7%

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

      11. remove-double-neg 98.7%

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

      12. sub-neg 98.7%

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

      13. metadata-eval 98.7%

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

      14. +-commutative 98.7%

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

      15. distribute-neg-in 98.7%

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

      16. metadata-eval 98.7%

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

      17. mul-1-neg 98.7%

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

      18. *-commutative 98.7%

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

      19. mul-1-neg 98.7%

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

      20. unsub-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0 98.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14.2:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.195:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -0.195) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.195) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 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 <= (-0.195d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.195) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.195:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.195)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.195)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.195], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -0.19500000000000001 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 80.5%

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

   if -0.19500000000000001 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 68.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.195:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.6% accurate, 7.0× 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 100.0%

   \[\frac{x - y}{1 - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 47.9%

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

4. Final simplification 47.9%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))