Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + y \cdot x\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (y * x)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{2} + x \cdot y\right) + z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* x y)) z))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (x * y)) + z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (x * y)) + z
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (x * y)) + z;
}

Python

def code(x, y, z):
	return ((x / 2.0) + (x * y)) + z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(x * y)) + z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (x * y)) + z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]

2. Final simplification 100.0%

   \[\leadsto \left(\frac{x}{2} + x \cdot y\right) + z \]

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -360000000000 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -360000000000.0) (not (<= y 0.5)))
   (+ z (* x y))
   (+ z (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -360000000000.0) || !(y <= 0.5)) {
		tmp = z + (x * y);
	} else {
		tmp = z + (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-360000000000.0d0)) .or. (.not. (y <= 0.5d0))) then
        tmp = z + (x * y)
    else
        tmp = z + (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -360000000000.0) || !(y <= 0.5)) {
		tmp = z + (x * y);
	} else {
		tmp = z + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -360000000000.0) or not (y <= 0.5):
		tmp = z + (x * y)
	else:
		tmp = z + (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -360000000000.0) || !(y <= 0.5))
		tmp = Float64(z + Float64(x * y));
	else
		tmp = Float64(z + Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -360000000000.0) || ~((y <= 0.5)))
		tmp = z + (x * y);
	else
		tmp = z + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -360000000000.0], N[Not[LessEqual[y, 0.5]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6e11 or 0.5 < y

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r/ 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. sub-neg 100.0%

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

      11. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

      13. distribute-neg-in 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

      14. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

      15. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]

   4. Taylor expanded in y around inf 99.4%

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

   if -3.6e11 < y < 0.5

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 99.9%

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

      5. associate-/r/ 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. sub-neg 100.0%

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

      11. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

      13. distribute-neg-in 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

      14. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

      15. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]

   4. Taylor expanded in y around 0 98.3%

      \[\leadsto \color{blue}{0.5 \cdot x} + z \]
   5. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \color{blue}{x \cdot 0.5} + z \]

   6. Simplified 98.3%

      \[\leadsto \color{blue}{x \cdot 0.5} + z \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -360000000000 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \]

Alternative 3: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+37}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+83) (* x y) (if (<= y 2.9e+37) (+ z (* x 0.5)) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+83) {
		tmp = x * y;
	} else if (y <= 2.9e+37) {
		tmp = z + (x * 0.5);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.4d+83)) then
        tmp = x * y
    else if (y <= 2.9d+37) then
        tmp = z + (x * 0.5d0)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+83) {
		tmp = x * y;
	} else if (y <= 2.9e+37) {
		tmp = z + (x * 0.5);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e+83:
		tmp = x * y
	elif y <= 2.9e+37:
		tmp = z + (x * 0.5)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+83)
		tmp = Float64(x * y);
	elseif (y <= 2.9e+37)
		tmp = Float64(z + Float64(x * 0.5));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e+83)
		tmp = x * y;
	elseif (y <= 2.9e+37)
		tmp = z + (x * 0.5);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+83], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.9e+37], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999991e83 or 2.89999999999999978e37 < y

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r/ 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. sub-neg 100.0%

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

      11. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

      13. distribute-neg-in 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

      14. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

      15. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]
   4. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + x \cdot 0.5\right)} + z \]

      2. flip-+ 49.1%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5}} + z \]

      3. pow2 49.1%

         \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

   5. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{2} - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5}} + z \]
   6. Step-by-step derivation

      1. unpow2 49.1%

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)} - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

      2. difference-of-squares 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot y + x \cdot 0.5\right) \cdot \left(x \cdot y - x \cdot 0.5\right)}}{x \cdot y - x \cdot 0.5} + z \]

      3. +-commutative 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot 0.5 + x \cdot y\right)} \cdot \left(x \cdot y - x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

      4. distribute-lft-in 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(0.5 + y\right)\right)} \cdot \left(x \cdot y - x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

      5. distribute-lft-out-- 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \color{blue}{\left(x \cdot \left(y - 0.5\right)\right)}}{x \cdot y - x \cdot 0.5} + z \]

      6. sub-neg 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \color{blue}{\left(y + \left(-0.5\right)\right)}\right)}{x \cdot y - x \cdot 0.5} + z \]

      7. metadata-eval 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + \color{blue}{-0.5}\right)\right)}{x \cdot y - x \cdot 0.5} + z \]

      8. distribute-lft-out-- 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{\color{blue}{x \cdot \left(y - 0.5\right)}} + z \]

      9. sub-neg 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{x \cdot \color{blue}{\left(y + \left(-0.5\right)\right)}} + z \]

      10. metadata-eval 49.4%

          \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{x \cdot \left(y + \color{blue}{-0.5}\right)} + z \]

   7. Simplified 49.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{x \cdot \left(y + -0.5\right)}} + z \]

   8. Taylor expanded in y around inf 26.4%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot {y}^{2}}}{x \cdot \left(y + -0.5\right)} + z \]
   9. Step-by-step derivation

      1. unpow2 26.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot {y}^{2}}{x \cdot \left(y + -0.5\right)} + z \]

      2. unpow2 26.4%

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)}}{x \cdot \left(y + -0.5\right)} + z \]

      3. swap-sqr 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}}{x \cdot \left(y + -0.5\right)} + z \]

   10. Simplified 49.4%

       \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}}{x \cdot \left(y + -0.5\right)} + z \]
   11. Step-by-step derivation

       1. frac-2neg 49.4%

          \[\leadsto \color{blue}{\frac{-\left(x \cdot y\right) \cdot \left(x \cdot y\right)}{-x \cdot \left(y + -0.5\right)}} + z \]

       2. div-inv 49.3%

          \[\leadsto \color{blue}{\left(-\left(x \cdot y\right) \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{-x \cdot \left(y + -0.5\right)}} + z \]

       3. fma-def 49.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(x \cdot y\right) \cdot \left(x \cdot y\right), \frac{1}{-x \cdot \left(y + -0.5\right)}, z\right)} \]

       4. pow2 49.3%

          \[\leadsto \mathsf{fma}\left(-\color{blue}{{\left(x \cdot y\right)}^{2}}, \frac{1}{-x \cdot \left(y + -0.5\right)}, z\right) \]

       5. distribute-rgt-neg-in 49.3%

          \[\leadsto \mathsf{fma}\left(-{\left(x \cdot y\right)}^{2}, \frac{1}{\color{blue}{x \cdot \left(-\left(y + -0.5\right)\right)}}, z\right) \]

   12. Applied egg-rr 49.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-{\left(x \cdot y\right)}^{2}, \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)}, z\right)} \]
   13. Step-by-step derivation

       1. fma-udef 49.3%

          \[\leadsto \color{blue}{\left(-{\left(x \cdot y\right)}^{2}\right) \cdot \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)} + z} \]

       2. +-commutative 49.3%

          \[\leadsto \color{blue}{z + \left(-{\left(x \cdot y\right)}^{2}\right) \cdot \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)}} \]

       3. distribute-lft-neg-out 49.3%

          \[\leadsto z + \color{blue}{\left(-{\left(x \cdot y\right)}^{2} \cdot \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)}\right)} \]

       4. associate-*r/ 49.4%

          \[\leadsto z + \left(-\color{blue}{\frac{{\left(x \cdot y\right)}^{2} \cdot 1}{x \cdot \left(-\left(y + -0.5\right)\right)}}\right) \]

       5. *-rgt-identity 49.4%

          \[\leadsto z + \left(-\frac{\color{blue}{{\left(x \cdot y\right)}^{2}}}{x \cdot \left(-\left(y + -0.5\right)\right)}\right) \]

       6. unsub-neg 49.4%

          \[\leadsto \color{blue}{z - \frac{{\left(x \cdot y\right)}^{2}}{x \cdot \left(-\left(y + -0.5\right)\right)}} \]

       7. associate-/r* 69.4%

          \[\leadsto z - \color{blue}{\frac{\frac{{\left(x \cdot y\right)}^{2}}{x}}{-\left(y + -0.5\right)}} \]

       8. unpow2 69.4%

          \[\leadsto z - \frac{\frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}}{x}}{-\left(y + -0.5\right)} \]

       9. swap-sqr 49.5%

          \[\leadsto z - \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(y \cdot y\right)}}{x}}{-\left(y + -0.5\right)} \]

       10. associate-*l/ 51.6%

           \[\leadsto z - \frac{\color{blue}{\frac{x \cdot x}{x} \cdot \left(y \cdot y\right)}}{-\left(y + -0.5\right)} \]

       11. associate-/l* 56.8%

           \[\leadsto z - \frac{\color{blue}{\frac{x}{\frac{x}{x}}} \cdot \left(y \cdot y\right)}{-\left(y + -0.5\right)} \]

       12. *-inverses 56.8%

           \[\leadsto z - \frac{\frac{x}{\color{blue}{1}} \cdot \left(y \cdot y\right)}{-\left(y + -0.5\right)} \]

       13. /-rgt-identity 56.8%

           \[\leadsto z - \frac{\color{blue}{x} \cdot \left(y \cdot y\right)}{-\left(y + -0.5\right)} \]

       14. neg-sub0 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{\color{blue}{0 - \left(y + -0.5\right)}} \]

       15. +-commutative 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{0 - \color{blue}{\left(-0.5 + y\right)}} \]

       16. associate--r+ 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{\color{blue}{\left(0 - -0.5\right) - y}} \]

       17. metadata-eval 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{\color{blue}{0.5} - y} \]

   14. Simplified 56.8%

       \[\leadsto \color{blue}{z - \frac{x \cdot \left(y \cdot y\right)}{0.5 - y}} \]

   15. Taylor expanded in y around inf 72.4%

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

   if -2.39999999999999991e83 < y < 2.89999999999999978e37

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 99.9%

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

      5. associate-/r/ 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. sub-neg 100.0%

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

      11. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

      13. distribute-neg-in 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

      14. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

      15. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]

   4. Taylor expanded in y around 0 92.9%

      \[\leadsto \color{blue}{0.5 \cdot x} + z \]
   5. Step-by-step derivation

      1. *-commutative 92.9%

         \[\leadsto \color{blue}{x \cdot 0.5} + z \]

   6. Simplified 92.9%

      \[\leadsto \color{blue}{x \cdot 0.5} + z \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+83}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+37}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 59.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+83}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+36}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.3e+83) (* x y) (if (<= y 6.2e+36) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e+83) {
		tmp = x * y;
	} else if (y <= 6.2e+36) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.3d+83)) then
        tmp = x * y
    else if (y <= 6.2d+36) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e+83) {
		tmp = x * y;
	} else if (y <= 6.2e+36) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.3e+83:
		tmp = x * y
	elif y <= 6.2e+36:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.3e+83)
		tmp = Float64(x * y);
	elseif (y <= 6.2e+36)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.3e+83)
		tmp = x * y;
	elseif (y <= 6.2e+36)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.3e+83], N[(x * y), $MachinePrecision], If[LessEqual[y, 6.2e+36], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.29999999999999985e83 or 6.1999999999999999e36 < y

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r/ 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. sub-neg 100.0%

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

      11. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

      13. distribute-neg-in 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

      14. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

      15. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]
   4. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{\left(x \cdot y + x \cdot 0.5\right)} + z \]

      2. flip-+ 49.1%

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5}} + z \]

      3. pow2 49.1%

         \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{2}} - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

   5. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{2} - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5}} + z \]
   6. Step-by-step derivation

      1. unpow2 49.1%

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)} - \left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

      2. difference-of-squares 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot y + x \cdot 0.5\right) \cdot \left(x \cdot y - x \cdot 0.5\right)}}{x \cdot y - x \cdot 0.5} + z \]

      3. +-commutative 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot 0.5 + x \cdot y\right)} \cdot \left(x \cdot y - x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

      4. distribute-lft-in 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(0.5 + y\right)\right)} \cdot \left(x \cdot y - x \cdot 0.5\right)}{x \cdot y - x \cdot 0.5} + z \]

      5. distribute-lft-out-- 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \color{blue}{\left(x \cdot \left(y - 0.5\right)\right)}}{x \cdot y - x \cdot 0.5} + z \]

      6. sub-neg 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \color{blue}{\left(y + \left(-0.5\right)\right)}\right)}{x \cdot y - x \cdot 0.5} + z \]

      7. metadata-eval 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + \color{blue}{-0.5}\right)\right)}{x \cdot y - x \cdot 0.5} + z \]

      8. distribute-lft-out-- 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{\color{blue}{x \cdot \left(y - 0.5\right)}} + z \]

      9. sub-neg 49.4%

         \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{x \cdot \color{blue}{\left(y + \left(-0.5\right)\right)}} + z \]

      10. metadata-eval 49.4%

          \[\leadsto \frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{x \cdot \left(y + \color{blue}{-0.5}\right)} + z \]

   7. Simplified 49.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(0.5 + y\right)\right) \cdot \left(x \cdot \left(y + -0.5\right)\right)}{x \cdot \left(y + -0.5\right)}} + z \]

   8. Taylor expanded in y around inf 26.4%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot {y}^{2}}}{x \cdot \left(y + -0.5\right)} + z \]
   9. Step-by-step derivation

      1. unpow2 26.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot {y}^{2}}{x \cdot \left(y + -0.5\right)} + z \]

      2. unpow2 26.4%

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)}}{x \cdot \left(y + -0.5\right)} + z \]

      3. swap-sqr 49.4%

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}}{x \cdot \left(y + -0.5\right)} + z \]

   10. Simplified 49.4%

       \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}}{x \cdot \left(y + -0.5\right)} + z \]
   11. Step-by-step derivation

       1. frac-2neg 49.4%

          \[\leadsto \color{blue}{\frac{-\left(x \cdot y\right) \cdot \left(x \cdot y\right)}{-x \cdot \left(y + -0.5\right)}} + z \]

       2. div-inv 49.3%

          \[\leadsto \color{blue}{\left(-\left(x \cdot y\right) \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{-x \cdot \left(y + -0.5\right)}} + z \]

       3. fma-def 49.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-\left(x \cdot y\right) \cdot \left(x \cdot y\right), \frac{1}{-x \cdot \left(y + -0.5\right)}, z\right)} \]

       4. pow2 49.3%

          \[\leadsto \mathsf{fma}\left(-\color{blue}{{\left(x \cdot y\right)}^{2}}, \frac{1}{-x \cdot \left(y + -0.5\right)}, z\right) \]

       5. distribute-rgt-neg-in 49.3%

          \[\leadsto \mathsf{fma}\left(-{\left(x \cdot y\right)}^{2}, \frac{1}{\color{blue}{x \cdot \left(-\left(y + -0.5\right)\right)}}, z\right) \]

   12. Applied egg-rr 49.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-{\left(x \cdot y\right)}^{2}, \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)}, z\right)} \]
   13. Step-by-step derivation

       1. fma-udef 49.3%

          \[\leadsto \color{blue}{\left(-{\left(x \cdot y\right)}^{2}\right) \cdot \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)} + z} \]

       2. +-commutative 49.3%

          \[\leadsto \color{blue}{z + \left(-{\left(x \cdot y\right)}^{2}\right) \cdot \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)}} \]

       3. distribute-lft-neg-out 49.3%

          \[\leadsto z + \color{blue}{\left(-{\left(x \cdot y\right)}^{2} \cdot \frac{1}{x \cdot \left(-\left(y + -0.5\right)\right)}\right)} \]

       4. associate-*r/ 49.4%

          \[\leadsto z + \left(-\color{blue}{\frac{{\left(x \cdot y\right)}^{2} \cdot 1}{x \cdot \left(-\left(y + -0.5\right)\right)}}\right) \]

       5. *-rgt-identity 49.4%

          \[\leadsto z + \left(-\frac{\color{blue}{{\left(x \cdot y\right)}^{2}}}{x \cdot \left(-\left(y + -0.5\right)\right)}\right) \]

       6. unsub-neg 49.4%

          \[\leadsto \color{blue}{z - \frac{{\left(x \cdot y\right)}^{2}}{x \cdot \left(-\left(y + -0.5\right)\right)}} \]

       7. associate-/r* 69.4%

          \[\leadsto z - \color{blue}{\frac{\frac{{\left(x \cdot y\right)}^{2}}{x}}{-\left(y + -0.5\right)}} \]

       8. unpow2 69.4%

          \[\leadsto z - \frac{\frac{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right)}}{x}}{-\left(y + -0.5\right)} \]

       9. swap-sqr 49.5%

          \[\leadsto z - \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(y \cdot y\right)}}{x}}{-\left(y + -0.5\right)} \]

       10. associate-*l/ 51.6%

           \[\leadsto z - \frac{\color{blue}{\frac{x \cdot x}{x} \cdot \left(y \cdot y\right)}}{-\left(y + -0.5\right)} \]

       11. associate-/l* 56.8%

           \[\leadsto z - \frac{\color{blue}{\frac{x}{\frac{x}{x}}} \cdot \left(y \cdot y\right)}{-\left(y + -0.5\right)} \]

       12. *-inverses 56.8%

           \[\leadsto z - \frac{\frac{x}{\color{blue}{1}} \cdot \left(y \cdot y\right)}{-\left(y + -0.5\right)} \]

       13. /-rgt-identity 56.8%

           \[\leadsto z - \frac{\color{blue}{x} \cdot \left(y \cdot y\right)}{-\left(y + -0.5\right)} \]

       14. neg-sub0 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{\color{blue}{0 - \left(y + -0.5\right)}} \]

       15. +-commutative 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{0 - \color{blue}{\left(-0.5 + y\right)}} \]

       16. associate--r+ 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{\color{blue}{\left(0 - -0.5\right) - y}} \]

       17. metadata-eval 56.8%

           \[\leadsto z - \frac{x \cdot \left(y \cdot y\right)}{\color{blue}{0.5} - y} \]

   14. Simplified 56.8%

       \[\leadsto \color{blue}{z - \frac{x \cdot \left(y \cdot y\right)}{0.5 - y}} \]

   15. Taylor expanded in y around inf 72.4%

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

   if -3.29999999999999985e83 < y < 6.1999999999999999e36

   1. Initial program 100.0%

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

      1. remove-double-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 99.9%

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

      5. associate-/r/ 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. sub-neg 100.0%

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

      11. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

      12. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

      13. distribute-neg-in 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

      14. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

      15. fma-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]

   4. Taylor expanded in x around 0 57.3%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+83}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+36}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ z + x \cdot \left(y + 0.5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ z (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	return z + (x * (y + 0.5));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z + (x * (y + 0.5d0))
end function

Java

public static double code(double x, double y, double z) {
	return z + (x * (y + 0.5));
}

Python

def code(x, y, z):
	return z + (x * (y + 0.5))

Julia

function code(x, y, z)
	return Float64(z + Float64(x * Float64(y + 0.5)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = z + (x * (y + 0.5));
end

Wolfram

code[x_, y_, z_] := N[(z + N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. remove-double-neg 100.0%

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

   2. distribute-frac-neg 100.0%

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

   3. neg-mul-1 100.0%

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

   4. associate-/l* 99.9%

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

   5. associate-/r/ 100.0%

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

   6. distribute-lft-neg-in 100.0%

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

   7. distribute-rgt-out 100.0%

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

   8. remove-double-neg 100.0%

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

   9. distribute-neg-in 100.0%

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

   10. sub-neg 100.0%

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

   11. fma-def 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

   12. sub-neg 100.0%

       \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

   13. distribute-neg-in 100.0%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

   14. sub-neg 100.0%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

   15. fma-def 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]

4. Final simplification 100.0%

   \[\leadsto z + x \cdot \left(y + 0.5\right) \]

Alternative 6: 40.6% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 z)

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 100.0%

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

   1. remove-double-neg 100.0%

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

   2. distribute-frac-neg 100.0%

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

   3. neg-mul-1 100.0%

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

   4. associate-/l* 99.9%

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

   5. associate-/r/ 100.0%

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

   6. distribute-lft-neg-in 100.0%

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

   7. distribute-rgt-out 100.0%

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

   8. remove-double-neg 100.0%

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

   9. distribute-neg-in 100.0%

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

   10. sub-neg 100.0%

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

   11. fma-def 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -\left(\frac{-1}{2} - y\right), z\right)} \]

   12. sub-neg 100.0%

       \[\leadsto \mathsf{fma}\left(x, -\color{blue}{\left(\frac{-1}{2} + \left(-y\right)\right)}, z\right) \]

   13. distribute-neg-in 100.0%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) + \left(-\left(-y\right)\right)}, z\right) \]

   14. sub-neg 100.0%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(-\frac{-1}{2}\right) - \left(-y\right)}, z\right) \]

   15. fma-def 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + 0.5\right) + z} \]

4. Taylor expanded in x around 0 47.0%

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

5. Final simplification 47.0%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023287 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))