Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3

Percentage Accurate: 93.9% → 99.9%

Time: 6.9s

Alternatives: 11

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) \cdot \frac{\frac{x + -3}{-3}}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (- 1.0 x) (/ (/ (+ x -3.0) -3.0) y)))

C

double code(double x, double y) {
	return (1.0 - x) * (((x + -3.0) / -3.0) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) * (((x + (-3.0d0)) / (-3.0d0)) / y)
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) * (((x + -3.0) / -3.0) / y);
}

Python

def code(x, y):
	return (1.0 - x) * (((x + -3.0) / -3.0) / y)

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) * Float64(Float64(Float64(x + -3.0) / -3.0) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) * (((x + -3.0) / -3.0) / y);
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] * N[(N[(N[(x + -3.0), $MachinePrecision] / -3.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-*l/ 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-/l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 92.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{x \cdot \left(1 - x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.8) (not (<= x 3.0)))
   (* -0.3333333333333333 (/ (* x (- 1.0 x)) y))
   (/ (+ 1.0 (* x -1.3333333333333333)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = -0.3333333333333333 * ((x * (1.0 - x)) / y);
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.8d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = (-0.3333333333333333d0) * ((x * (1.0d0 - x)) / y)
    else
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = -0.3333333333333333 * ((x * (1.0 - x)) / y);
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.8) or not (x <= 3.0):
		tmp = -0.3333333333333333 * ((x * (1.0 - x)) / y)
	else:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.8) || !(x <= 3.0))
		tmp = Float64(-0.3333333333333333 * Float64(Float64(x * Float64(1.0 - x)) / y));
	else
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.8) || ~((x <= 3.0)))
		tmp = -0.3333333333333333 * ((x * (1.0 - x)) / y);
	else
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.8], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(-0.3333333333333333 * N[(N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998 or 3 < x

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-lft-identity 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. times-frac 99.7%

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

      9. *-commutative 99.7%

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

      10. neg-mul-1 99.7%

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

      11. distribute-rgt-neg-in 99.7%

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

      12. times-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.7%

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

      16. distribute-lft-neg-in 99.7%

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

      17. distribute-frac-neg 99.7%

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

      18. sub-neg 99.7%

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

      19. +-commutative 99.7%

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

      20. distribute-neg-in 99.7%

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

      21. remove-double-neg 99.7%

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

      22. metadata-eval 99.7%

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

      23. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around inf 99.0%

      \[\leadsto \left(1 - x\right) \cdot \left(\color{blue}{\frac{x}{y}} \cdot -0.3333333333333333\right) \]

   5. Taylor expanded in y around 0 88.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{x \cdot \left(1 - x\right)}{y}} \]

   if -3.7999999999999998 < x < 3

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

      4. *-lft-identity 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. times-frac 99.5%

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

      9. *-commutative 99.5%

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

      10. neg-mul-1 99.5%

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

      11. distribute-rgt-neg-in 99.5%

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

      12. times-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.4%

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

      16. distribute-lft-neg-in 99.4%

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

      17. distribute-frac-neg 99.4%

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

      18. sub-neg 99.4%

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

      19. +-commutative 99.4%

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

      20. distribute-neg-in 99.4%

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

      21. remove-double-neg 99.4%

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

      22. metadata-eval 99.4%

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

      23. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{1 + \color{blue}{x \cdot -1.3333333333333333}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1 + x \cdot -1.3333333333333333}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{x \cdot \left(1 - x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\left(1 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.8) (not (<= x 3.0)))
   (* (- 1.0 x) (* -0.3333333333333333 (/ x y)))
   (/ (+ 1.0 (* x -1.3333333333333333)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = (1.0 - x) * (-0.3333333333333333 * (x / y));
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.8d0)) .or. (.not. (x <= 3.0d0))) then
        tmp = (1.0d0 - x) * ((-0.3333333333333333d0) * (x / y))
    else
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.8) || !(x <= 3.0)) {
		tmp = (1.0 - x) * (-0.3333333333333333 * (x / y));
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.8) or not (x <= 3.0):
		tmp = (1.0 - x) * (-0.3333333333333333 * (x / y))
	else:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.8) || !(x <= 3.0))
		tmp = Float64(Float64(1.0 - x) * Float64(-0.3333333333333333 * Float64(x / y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.8) || ~((x <= 3.0)))
		tmp = (1.0 - x) * (-0.3333333333333333 * (x / y));
	else
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.8], N[Not[LessEqual[x, 3.0]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] * N[(-0.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998 or 3 < x

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-lft-identity 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. times-frac 99.7%

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

      9. *-commutative 99.7%

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

      10. neg-mul-1 99.7%

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

      11. distribute-rgt-neg-in 99.7%

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

      12. times-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.7%

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

      16. distribute-lft-neg-in 99.7%

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

      17. distribute-frac-neg 99.7%

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

      18. sub-neg 99.7%

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

      19. +-commutative 99.7%

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

      20. distribute-neg-in 99.7%

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

      21. remove-double-neg 99.7%

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

      22. metadata-eval 99.7%

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

      23. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around inf 99.0%

      \[\leadsto \left(1 - x\right) \cdot \left(\color{blue}{\frac{x}{y}} \cdot -0.3333333333333333\right) \]

   if -3.7999999999999998 < x < 3

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

      4. *-lft-identity 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. times-frac 99.5%

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

      9. *-commutative 99.5%

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

      10. neg-mul-1 99.5%

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

      11. distribute-rgt-neg-in 99.5%

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

      12. times-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.4%

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

      16. distribute-lft-neg-in 99.4%

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

      17. distribute-frac-neg 99.4%

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

      18. sub-neg 99.4%

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

      19. +-commutative 99.4%

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

      20. distribute-neg-in 99.4%

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

      21. remove-double-neg 99.4%

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

      22. metadata-eval 99.4%

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

      23. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{1 + \color{blue}{x \cdot -1.3333333333333333}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1 + x \cdot -1.3333333333333333}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \lor \neg \left(x \leq 3\right):\\ \;\;\;\;\left(1 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]

Alternative 4: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(\frac{x}{3} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.3) (not (<= x 1.3)))
   (* (/ x y) (+ (/ x 3.0) -1.0))
   (/ (+ 1.0 (* x -1.3333333333333333)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.3) || !(x <= 1.3)) {
		tmp = (x / y) * ((x / 3.0) + -1.0);
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.3d0)) .or. (.not. (x <= 1.3d0))) then
        tmp = (x / y) * ((x / 3.0d0) + (-1.0d0))
    else
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.3) || !(x <= 1.3)) {
		tmp = (x / y) * ((x / 3.0) + -1.0);
	} else {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.3) or not (x <= 1.3):
		tmp = (x / y) * ((x / 3.0) + -1.0)
	else:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.3) || !(x <= 1.3))
		tmp = Float64(Float64(x / y) * Float64(Float64(x / 3.0) + -1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.3) || ~((x <= 1.3)))
		tmp = (x / y) * ((x / 3.0) + -1.0);
	else
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.3], N[Not[LessEqual[x, 1.3]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * N[(N[(x / 3.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999998 or 1.30000000000000004 < x

   1. Initial program 89.0%

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

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{1 - x}{y} \cdot \frac{3 - x}{3}} \]

      2. div-sub 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 99.1%

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

      1. neg-mul-1 99.1%

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

      2. distribute-neg-frac 99.1%

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

   6. Simplified 99.1%

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

   if -2.2999999999999998 < x < 1.30000000000000004

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

      4. *-lft-identity 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. times-frac 99.5%

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

      9. *-commutative 99.5%

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

      10. neg-mul-1 99.5%

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

      11. distribute-rgt-neg-in 99.5%

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

      12. times-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.4%

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

      16. distribute-lft-neg-in 99.4%

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

      17. distribute-frac-neg 99.4%

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

      18. sub-neg 99.4%

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

      19. +-commutative 99.4%

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

      20. distribute-neg-in 99.4%

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

      21. remove-double-neg 99.4%

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

      22. metadata-eval 99.4%

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

      23. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{1 + \color{blue}{x \cdot -1.3333333333333333}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1 + x \cdot -1.3333333333333333}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(\frac{x}{3} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \end{array} \]

Alternative 5: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot -0.3333333333333333}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.8)
   (/ (* (- 1.0 x) -0.3333333333333333) (/ y x))
   (if (<= x 3.0)
     (/ (+ 1.0 (* x -1.3333333333333333)) y)
     (* (- 1.0 x) (* -0.3333333333333333 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.8) {
		tmp = ((1.0 - x) * -0.3333333333333333) / (y / x);
	} else if (x <= 3.0) {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	} else {
		tmp = (1.0 - x) * (-0.3333333333333333 * (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 (x <= (-3.8d0)) then
        tmp = ((1.0d0 - x) * (-0.3333333333333333d0)) / (y / x)
    else if (x <= 3.0d0) then
        tmp = (1.0d0 + (x * (-1.3333333333333333d0))) / y
    else
        tmp = (1.0d0 - x) * ((-0.3333333333333333d0) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.8) {
		tmp = ((1.0 - x) * -0.3333333333333333) / (y / x);
	} else if (x <= 3.0) {
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	} else {
		tmp = (1.0 - x) * (-0.3333333333333333 * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.8:
		tmp = ((1.0 - x) * -0.3333333333333333) / (y / x)
	elif x <= 3.0:
		tmp = (1.0 + (x * -1.3333333333333333)) / y
	else:
		tmp = (1.0 - x) * (-0.3333333333333333 * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(Float64(Float64(1.0 - x) * -0.3333333333333333) / Float64(y / x));
	elseif (x <= 3.0)
		tmp = Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y);
	else
		tmp = Float64(Float64(1.0 - x) * Float64(-0.3333333333333333 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.8)
		tmp = ((1.0 - x) * -0.3333333333333333) / (y / x);
	elseif (x <= 3.0)
		tmp = (1.0 + (x * -1.3333333333333333)) / y;
	else
		tmp = (1.0 - x) * (-0.3333333333333333 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.8], N[(N[(N[(1.0 - x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.0], N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * N[(-0.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999998

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-lft-identity 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. times-frac 99.7%

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

      9. *-commutative 99.7%

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

      10. neg-mul-1 99.7%

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

      11. distribute-rgt-neg-in 99.7%

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

      12. times-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.7%

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

      16. distribute-lft-neg-in 99.7%

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

      17. distribute-frac-neg 99.7%

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

      18. sub-neg 99.7%

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

      19. +-commutative 99.7%

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

      20. distribute-neg-in 99.7%

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

      21. remove-double-neg 99.7%

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

      22. metadata-eval 99.7%

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

      23. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around inf 99.5%

      \[\leadsto \left(1 - x\right) \cdot \left(\color{blue}{\frac{x}{y}} \cdot -0.3333333333333333\right) \]
   5. Step-by-step derivation

      1. associate-*l/ 99.5%

         \[\leadsto \left(1 - x\right) \cdot \color{blue}{\frac{x \cdot -0.3333333333333333}{y}} \]

      2. *-commutative 99.5%

         \[\leadsto \left(1 - x\right) \cdot \frac{\color{blue}{-0.3333333333333333 \cdot x}}{y} \]

      3. associate-/l* 99.4%

         \[\leadsto \left(1 - x\right) \cdot \color{blue}{\frac{-0.3333333333333333}{\frac{y}{x}}} \]

      4. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot -0.3333333333333333}{\frac{y}{x}}} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\left(1 - x\right) \cdot -0.3333333333333333}{\frac{y}{x}}} \]

   if -3.7999999999999998 < x < 3

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

      4. *-lft-identity 99.5%

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

      5. metadata-eval 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. times-frac 99.5%

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

      9. *-commutative 99.5%

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

      10. neg-mul-1 99.5%

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

      11. distribute-rgt-neg-in 99.5%

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

      12. times-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.4%

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

      16. distribute-lft-neg-in 99.4%

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

      17. distribute-frac-neg 99.4%

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

      18. sub-neg 99.4%

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

      19. +-commutative 99.4%

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

      20. distribute-neg-in 99.4%

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

      21. remove-double-neg 99.4%

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

      22. metadata-eval 99.4%

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

      23. metadata-eval 99.4%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{1 + \color{blue}{x \cdot -1.3333333333333333}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1 + x \cdot -1.3333333333333333}{y}} \]

   if 3 < x

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

      4. *-lft-identity 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-lft-neg-in 99.6%

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

      7. metadata-eval 99.6%

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

      8. times-frac 99.6%

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

      9. *-commutative 99.6%

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

      10. neg-mul-1 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. times-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.7%

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

      16. distribute-lft-neg-in 99.7%

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

      17. distribute-frac-neg 99.7%

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

      18. sub-neg 99.7%

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

      19. +-commutative 99.7%

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

      20. distribute-neg-in 99.7%

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

      21. remove-double-neg 99.7%

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

      22. metadata-eval 99.7%

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

      23. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around inf 98.6%

      \[\leadsto \left(1 - x\right) \cdot \left(\color{blue}{\frac{x}{y}} \cdot -0.3333333333333333\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{\left(1 - x\right) \cdot -0.3333333333333333}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3:\\ \;\;\;\;\frac{1 + x \cdot -1.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot \left(-0.3333333333333333 \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (- 1.0 x) (* (/ (+ x -3.0) y) -0.3333333333333333)))

C

double code(double x, double y) {
	return (1.0 - x) * (((x + -3.0) / y) * -0.3333333333333333);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) * (((x + (-3.0d0)) / y) * (-0.3333333333333333d0))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) * (((x + -3.0) / y) * -0.3333333333333333);
}

Python

def code(x, y):
	return (1.0 - x) * (((x + -3.0) / y) * -0.3333333333333333)

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) * Float64(Float64(Float64(x + -3.0) / y) * -0.3333333333333333))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) * (((x + -3.0) / y) * -0.3333333333333333);
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] * N[(N[(N[(x + -3.0), $MachinePrecision] / y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-*l/ 99.6%

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

   3. *-commutative 99.6%

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

   4. *-lft-identity 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-lft-neg-in 99.6%

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

   7. metadata-eval 99.6%

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

   8. times-frac 99.6%

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

   9. *-commutative 99.6%

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

   10. neg-mul-1 99.6%

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

   11. distribute-rgt-neg-in 99.6%

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

   12. times-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   15. metadata-eval 99.6%

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

   16. distribute-lft-neg-in 99.6%

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

   17. distribute-frac-neg 99.6%

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

   18. sub-neg 99.6%

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

   19. +-commutative 99.6%

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

   20. distribute-neg-in 99.6%

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

   21. remove-double-neg 99.6%

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

   22. metadata-eval 99.6%

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

   23. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

4. Final simplification 99.6%

   \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right) \]

Alternative 7: 58.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.75) (* -1.3333333333333333 (/ x y)) (/ 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.75) {
		tmp = -1.3333333333333333 * (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 (x <= (-0.75d0)) then
        tmp = (-1.3333333333333333d0) * (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 (x <= -0.75) {
		tmp = -1.3333333333333333 * (x / y);
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.75:
		tmp = -1.3333333333333333 * (x / y)
	else:
		tmp = 1.0 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.75)
		tmp = Float64(-1.3333333333333333 * Float64(x / y));
	else
		tmp = Float64(1.0 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.75)
		tmp = -1.3333333333333333 * (x / y);
	else
		tmp = 1.0 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.75], N[(-1.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.75

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-lft-identity 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. times-frac 99.7%

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

      9. *-commutative 99.7%

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

      10. neg-mul-1 99.7%

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

      11. distribute-rgt-neg-in 99.7%

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

      12. times-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.7%

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

      16. distribute-lft-neg-in 99.7%

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

      17. distribute-frac-neg 99.7%

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

      18. sub-neg 99.7%

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

      19. +-commutative 99.7%

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

      20. distribute-neg-in 99.7%

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

      21. remove-double-neg 99.7%

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

      22. metadata-eval 99.7%

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

      23. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around 0 24.2%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

   5. Taylor expanded in x around inf 24.2%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y}} \]

   if -0.75 < x

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

      4. *-lft-identity 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-lft-neg-in 99.6%

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

      7. metadata-eval 99.6%

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

      8. times-frac 99.6%

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

      9. *-commutative 99.6%

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

      10. neg-mul-1 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. times-frac 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.5%

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

      16. distribute-lft-neg-in 99.5%

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

      17. distribute-frac-neg 99.5%

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

      18. sub-neg 99.5%

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

      19. +-commutative 99.5%

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

      20. distribute-neg-in 99.5%

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

      21. remove-double-neg 99.5%

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

      22. metadata-eval 99.5%

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

      23. metadata-eval 99.5%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \color{blue}{\left(\frac{x + -3}{y} \cdot -0.3333333333333333\right) \cdot \left(1 - x\right)} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{\left(x + -3\right) \cdot -0.3333333333333333}{y}} \cdot \left(1 - x\right) \]

      3. metadata-eval 99.9%

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

      4. div-inv 99.9%

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

      5. div-inv 99.9%

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

      6. associate-*l* 99.9%

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

      7. frac-2neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. metadata-eval 99.9%

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

      13. div-sub 99.9%

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

      14. metadata-eval 99.9%

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

      15. associate-/r/ 99.8%

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

      16. un-div-inv 99.9%

          \[\leadsto \color{blue}{\frac{1 - \frac{x}{3}}{\frac{y}{1 - x}}} \]

      17. sub-neg 99.9%

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

      18. div-inv 99.9%

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

      19. distribute-rgt-neg-in 99.9%

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

      20. metadata-eval 99.9%

          \[\leadsto \frac{1 + x \cdot \left(-\color{blue}{0.3333333333333333}\right)}{\frac{y}{1 - x}} \]

      21. metadata-eval 99.9%

          \[\leadsto \frac{1 + x \cdot \color{blue}{-0.3333333333333333}}{\frac{y}{1 - x}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1 + x \cdot -0.3333333333333333}{\frac{y}{1 - x}}} \]

   6. Taylor expanded in x around 0 68.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 8: 57.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 + x \cdot -1.3333333333333333}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ 1.0 (* x -1.3333333333333333)) y))

C

double code(double x, double y) {
	return (1.0 + (x * -1.3333333333333333)) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (x * (-1.3333333333333333d0))) / y
end function

Java

public static double code(double x, double y) {
	return (1.0 + (x * -1.3333333333333333)) / y;
}

Python

def code(x, y):
	return (1.0 + (x * -1.3333333333333333)) / y

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(x * -1.3333333333333333)) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (x * -1.3333333333333333)) / y;
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-*l/ 99.6%

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

   3. *-commutative 99.6%

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

   4. *-lft-identity 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-lft-neg-in 99.6%

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

   7. metadata-eval 99.6%

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

   8. times-frac 99.6%

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

   9. *-commutative 99.6%

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

   10. neg-mul-1 99.6%

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

   11. distribute-rgt-neg-in 99.6%

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

   12. times-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   15. metadata-eval 99.6%

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

   16. distribute-lft-neg-in 99.6%

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

   17. distribute-frac-neg 99.6%

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

   18. sub-neg 99.6%

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

   19. +-commutative 99.6%

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

   20. distribute-neg-in 99.6%

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

   21. remove-double-neg 99.6%

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

   22. metadata-eval 99.6%

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

   23. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

4. Taylor expanded in x around 0 58.0%

   \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

5. Taylor expanded in y around 0 58.0%

   \[\leadsto \color{blue}{\frac{1 + -1.3333333333333333 \cdot x}{y}} \]
6. Step-by-step derivation

   1. *-commutative 58.0%

      \[\leadsto \frac{1 + \color{blue}{x \cdot -1.3333333333333333}}{y} \]

7. Simplified 58.0%

   \[\leadsto \color{blue}{\frac{1 + x \cdot -1.3333333333333333}{y}} \]

8. Final simplification 58.0%

   \[\leadsto \frac{1 + x \cdot -1.3333333333333333}{y} \]

Alternative 9: 58.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. *-lft-identity 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. times-frac 99.7%

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

      9. *-commutative 99.7%

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

      10. neg-mul-1 99.7%

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

      11. distribute-rgt-neg-in 99.7%

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

      12. times-frac 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.7%

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

      16. distribute-lft-neg-in 99.7%

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

      17. distribute-frac-neg 99.7%

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

      18. sub-neg 99.7%

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

      19. +-commutative 99.7%

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

      20. distribute-neg-in 99.7%

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

      21. remove-double-neg 99.7%

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

      22. metadata-eval 99.7%

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

      23. metadata-eval 99.7%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

   4. Taylor expanded in x around 0 24.2%

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

   5. Taylor expanded in x around inf 24.2%

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

      1. associate-*r/ 24.2%

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

      2. neg-mul-1 24.2%

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

   7. Simplified 24.2%

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

   if -1 < x

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

      4. *-lft-identity 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-lft-neg-in 99.6%

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

      7. metadata-eval 99.6%

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

      8. times-frac 99.6%

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

      9. *-commutative 99.6%

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

      10. neg-mul-1 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. times-frac 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

          \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

      15. metadata-eval 99.5%

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

      16. distribute-lft-neg-in 99.5%

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

      17. distribute-frac-neg 99.5%

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

      18. sub-neg 99.5%

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

      19. +-commutative 99.5%

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

      20. distribute-neg-in 99.5%

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

      21. remove-double-neg 99.5%

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

      22. metadata-eval 99.5%

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

      23. metadata-eval 99.5%

          \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \color{blue}{\left(\frac{x + -3}{y} \cdot -0.3333333333333333\right) \cdot \left(1 - x\right)} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{\left(x + -3\right) \cdot -0.3333333333333333}{y}} \cdot \left(1 - x\right) \]

      3. metadata-eval 99.9%

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

      4. div-inv 99.9%

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

      5. div-inv 99.9%

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

      6. associate-*l* 99.9%

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

      7. frac-2neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. metadata-eval 99.9%

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

      13. div-sub 99.9%

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

      14. metadata-eval 99.9%

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

      15. associate-/r/ 99.8%

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

      16. un-div-inv 99.9%

          \[\leadsto \color{blue}{\frac{1 - \frac{x}{3}}{\frac{y}{1 - x}}} \]

      17. sub-neg 99.9%

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

      18. div-inv 99.9%

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

      19. distribute-rgt-neg-in 99.9%

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

      20. metadata-eval 99.9%

          \[\leadsto \frac{1 + x \cdot \left(-\color{blue}{0.3333333333333333}\right)}{\frac{y}{1 - x}} \]

      21. metadata-eval 99.9%

          \[\leadsto \frac{1 + x \cdot \color{blue}{-0.3333333333333333}}{\frac{y}{1 - x}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1 + x \cdot -0.3333333333333333}{\frac{y}{1 - x}}} \]

   6. Taylor expanded in x around 0 68.6%

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

4. Final simplification 58.5%

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

Alternative 10: 57.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-*l/ 99.6%

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

   3. *-commutative 99.6%

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

   4. *-lft-identity 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-lft-neg-in 99.6%

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

   7. metadata-eval 99.6%

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

   8. times-frac 99.6%

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

   9. *-commutative 99.6%

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

   10. neg-mul-1 99.6%

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

   11. distribute-rgt-neg-in 99.6%

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

   12. times-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   15. metadata-eval 99.6%

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

   16. distribute-lft-neg-in 99.6%

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

   17. distribute-frac-neg 99.6%

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

   18. sub-neg 99.6%

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

   19. +-commutative 99.6%

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

   20. distribute-neg-in 99.6%

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

   21. remove-double-neg 99.6%

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

   22. metadata-eval 99.6%

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

   23. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]

4. Taylor expanded in x around 0 57.7%

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

5. Taylor expanded in x around 0 57.7%

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

   1. +-commutative 57.7%

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

   2. mul-1-neg 57.7%

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

   3. sub-neg 57.7%

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

   4. div-sub 57.7%

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

7. Simplified 57.7%

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

8. Final simplification 57.7%

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

Alternative 11: 51.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. *-commutative 94.5%

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

   2. associate-*l/ 99.6%

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

   3. *-commutative 99.6%

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

   4. *-lft-identity 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-lft-neg-in 99.6%

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

   7. metadata-eval 99.6%

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

   8. times-frac 99.6%

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

   9. *-commutative 99.6%

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

   10. neg-mul-1 99.6%

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

   11. distribute-rgt-neg-in 99.6%

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

   12. times-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(-\frac{3 - x}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

   15. metadata-eval 99.6%

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

   16. distribute-lft-neg-in 99.6%

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

   17. distribute-frac-neg 99.6%

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

   18. sub-neg 99.6%

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

   19. +-commutative 99.6%

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

   20. distribute-neg-in 99.6%

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

   21. remove-double-neg 99.6%

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

   22. metadata-eval 99.6%

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

   23. metadata-eval 99.6%

       \[\leadsto \left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot \color{blue}{-0.3333333333333333}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\frac{x + -3}{y} \cdot -0.3333333333333333\right)} \]
4. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \color{blue}{\left(\frac{x + -3}{y} \cdot -0.3333333333333333\right) \cdot \left(1 - x\right)} \]

   2. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{\left(x + -3\right) \cdot -0.3333333333333333}{y}} \cdot \left(1 - x\right) \]

   3. metadata-eval 99.8%

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

   4. div-inv 99.9%

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

   5. div-inv 99.8%

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

   6. associate-*l* 99.9%

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

   7. frac-2neg 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. metadata-eval 99.9%

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

   11. sub-neg 99.9%

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

   12. metadata-eval 99.9%

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

   13. div-sub 99.9%

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

   14. metadata-eval 99.9%

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

   15. associate-/r/ 99.8%

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

   16. un-div-inv 99.8%

       \[\leadsto \color{blue}{\frac{1 - \frac{x}{3}}{\frac{y}{1 - x}}} \]

   17. sub-neg 99.8%

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

   18. div-inv 99.8%

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

   19. distribute-rgt-neg-in 99.8%

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

   20. metadata-eval 99.8%

       \[\leadsto \frac{1 + x \cdot \left(-\color{blue}{0.3333333333333333}\right)}{\frac{y}{1 - x}} \]

   21. metadata-eval 99.8%

       \[\leadsto \frac{1 + x \cdot \color{blue}{-0.3333333333333333}}{\frac{y}{1 - x}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{1 + x \cdot -0.3333333333333333}{\frac{y}{1 - x}}} \]

6. Taylor expanded in x around 0 54.1%

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

7. Final simplification 54.1%

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

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{y} \cdot \frac{3 - x}{3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) / y) * ((3.0d0 - x) / 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) / y) * Float64(Float64(3.0 - x) / 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) / y) * ((3.0 - x) / 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(3.0 - x), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023334 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))