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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 8

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. Final simplification 100.0%

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

Alternative 2: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.84 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.84) (not (<= y 1.0)))
   (- 1.0 (/ x y))
   (+ x (* y (- -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.84) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = 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 <= (-0.84d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x + (y * ((-1.0d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -0.84) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x + (y * (-1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.84) or not (y <= 1.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x + (y * (-1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.84) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x + Float64(y * Float64(-1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.84) || ~((y <= 1.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x + (y * (-1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.84], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.839999999999999969 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

      4. div-sub 99.1%

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

      5. unsub-neg 99.1%

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

      6. mul-1-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. metadata-eval 99.1%

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

      9. distribute-lft-in 99.1%

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

      10. metadata-eval 99.1%

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

      11. sub-neg 99.1%

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

      12. associate-*r/ 99.1%

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

      13. mul-1-neg 99.1%

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

      14. unsub-neg 99.1%

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

      15. sub-neg 99.1%

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

      16. metadata-eval 99.1%

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

      17. +-commutative 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around inf 98.6%

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

   if -0.839999999999999969 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. unpow2 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. mul-1-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. associate--r+ 100.0%

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

      15. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. distribute-rgt1-in 99.3%

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

      5. *-commutative 99.3%

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

      6. neg-mul-1 99.3%

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

      7. sub-neg 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.84 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(-1 - y\right)\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.6× speedup

Math

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

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 y)))))

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 - 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 + (-1.0d0)) / y)
    else
        tmp = x + (y * ((-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 <= 1.0)) {
		tmp = 1.0 - ((x + -1.0) / y);
	} else {
		tmp = x + (y * (-1.0 - y));
	}
	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 - y))
	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 - 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 + -1.0) / y);
	else
		tmp = x + (y * (-1.0 - 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[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(-1.0 - y), $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. Taylor expanded in y around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

      4. div-sub 99.1%

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

      5. unsub-neg 99.1%

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

      6. mul-1-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. metadata-eval 99.1%

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

      9. distribute-lft-in 99.1%

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

      10. metadata-eval 99.1%

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

      11. sub-neg 99.1%

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

      12. associate-*r/ 99.1%

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

      13. mul-1-neg 99.1%

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

      14. unsub-neg 99.1%

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

      15. sub-neg 99.1%

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

      16. metadata-eval 99.1%

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

      17. +-commutative 99.1%

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

   4. Simplified 99.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. unpow2 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. mul-1-neg 100.0%

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

      10. unsub-neg 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. associate--r+ 100.0%

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

      15. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. distribute-rgt1-in 99.3%

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

      5. *-commutative 99.3%

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

      6. neg-mul-1 99.3%

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

      7. sub-neg 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.2%

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

Alternative 4: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -340:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e+56)
   1.0
   (if (<= y -340.0) (/ (- x) y) (if (<= y 1.0) (- x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+56) {
		tmp = 1.0;
	} else if (y <= -340.0) {
		tmp = -x / y;
	} 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 <= (-3.1d+56)) then
        tmp = 1.0d0
    else if (y <= (-340.0d0)) then
        tmp = -x / y
    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 <= -3.1e+56) {
		tmp = 1.0;
	} else if (y <= -340.0) {
		tmp = -x / y;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e+56:
		tmp = 1.0
	elif y <= -340.0:
		tmp = -x / y
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e+56)
		tmp = 1.0;
	elseif (y <= -340.0)
		tmp = Float64(Float64(-x) / y);
	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 <= -3.1e+56)
		tmp = 1.0;
	elseif (y <= -340.0)
		tmp = -x / y;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e+56], 1.0, If[LessEqual[y, -340.0], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.10000000000000005e56 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around inf 78.3%

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

   if -3.10000000000000005e56 < y < -340

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in x around inf 76.7%

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

   3. Taylor expanded in y around inf 76.7%

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

      1. associate-*r/ 76.7%

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

      2. neg-mul-1 76.7%

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

   5. Simplified 76.7%

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

   if -340 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. distribute-lft-neg-in 98.5%

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

      4. cancel-sign-sub 98.5%

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

      5. +-commutative 98.5%

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

      6. metadata-eval 98.5%

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

      7. distribute-lft-in 98.5%

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

      8. metadata-eval 98.5%

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

      9. sub-neg 98.5%

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

      10. mul-1-neg 98.5%

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

      11. remove-double-neg 98.5%

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

      12. sub-neg 98.5%

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

      13. metadata-eval 98.5%

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

      14. +-commutative 98.5%

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

      15. distribute-neg-in 98.5%

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

      16. metadata-eval 98.5%

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

      17. mul-1-neg 98.5%

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

      18. *-commutative 98.5%

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

      19. mul-1-neg 98.5%

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

      20. unsub-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in x around 0 98.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -340:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 97.9% 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. Taylor expanded in y around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

      4. div-sub 99.1%

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

      5. unsub-neg 99.1%

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

      6. mul-1-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. metadata-eval 99.1%

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

      9. distribute-lft-in 99.1%

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

      10. metadata-eval 99.1%

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

      11. sub-neg 99.1%

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

      12. associate-*r/ 99.1%

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

      13. mul-1-neg 99.1%

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

      14. unsub-neg 99.1%

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

      15. sub-neg 99.1%

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

      16. metadata-eval 99.1%

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

      17. +-commutative 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around inf 98.6%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. distribute-lft-neg-in 99.2%

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

      4. cancel-sign-sub 99.2%

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

      5. +-commutative 99.2%

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

      6. metadata-eval 99.2%

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

      7. distribute-lft-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. sub-neg 99.2%

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

      10. mul-1-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

      15. distribute-neg-in 99.2%

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

      16. metadata-eval 99.2%

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

      17. mul-1-neg 99.2%

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

      18. *-commutative 99.2%

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

      19. mul-1-neg 99.2%

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

      20. unsub-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around 0 98.6%

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

4. Final simplification 98.6%

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

Alternative 6: 85.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -16.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -16.0) {
		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 <= (-16.0d0)) 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 <= -16.0) {
		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 <= -16.0:
		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 <= -16.0)
		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 <= -16.0)
		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, -16.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -16 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around inf 74.3%

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

   if -16 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. *-commutative 99.2%

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

      3. distribute-lft-neg-in 99.2%

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

      4. cancel-sign-sub 99.2%

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

      5. +-commutative 99.2%

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

      6. metadata-eval 99.2%

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

      7. distribute-lft-in 99.2%

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

      8. metadata-eval 99.2%

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

      9. sub-neg 99.2%

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

      10. mul-1-neg 99.2%

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

      11. remove-double-neg 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

      15. distribute-neg-in 99.2%

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

      16. metadata-eval 99.2%

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

      17. mul-1-neg 99.2%

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

      18. *-commutative 99.2%

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

      19. mul-1-neg 99.2%

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

      20. unsub-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around 0 98.6%

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

4. Final simplification 87.0%

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

Alternative 7: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e-12) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e-12) {
		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 <= (-1.65d-12)) 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 <= -1.65e-12) {
		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 <= -1.65e-12:
		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 <= -1.65e-12)
		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 <= -1.65e-12)
		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, -1.65e-12], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e-12 or 1 < y

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around inf 72.8%

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

   if -1.65e-12 < y < 1

   1. Initial program 100.0%

      \[\frac{x - y}{1 - y} \]

   2. Taylor expanded in y around 0 71.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 39.1% 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. Taylor expanded in y around inf 37.4%

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

3. Final simplification 37.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023332 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))