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

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 6

Speedup: 1.3×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11500 \lor \neg \left(y \leq 8.2 \cdot 10^{-6}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -11500.0) (not (<= y 8.2e-6))) (+ z (* x y)) (+ z (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -11500.0) || !(y <= 8.2e-6)) {
		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 <= (-11500.0d0)) .or. (.not. (y <= 8.2d-6))) 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 <= -11500.0) || !(y <= 8.2e-6)) {
		tmp = z + (x * y);
	} else {
		tmp = z + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -11500.0) or not (y <= 8.2e-6):
		tmp = z + (x * y)
	else:
		tmp = z + (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -11500.0) || !(y <= 8.2e-6))
		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 <= -11500.0) || ~((y <= 8.2e-6)))
		tmp = z + (x * y);
	else
		tmp = z + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -11500.0], N[Not[LessEqual[y, 8.2e-6]], $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 < -11500 or 8.1999999999999994e-6 < 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 98.6%

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

   if -11500 < y < 8.1999999999999994e-6

   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* 99.8%

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

      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 98.5%

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

      1. *-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11500 \lor \neg \left(y \leq 8.2 \cdot 10^{-6}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / 2.0d0) + (x * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 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 5: 65.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ z + x \cdot 0.5 \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return z + (x * 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 * 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z + N[(x * 0.5), $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. Taylor expanded in y around 0 69.5%

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

   1. *-commutative 69.5%

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

6. Simplified 69.5%

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

7. Final simplification 69.5%

   \[\leadsto z + x \cdot 0.5 \]

Alternative 6: 40.0% 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 44.0%

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

5. Final simplification 44.0%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023320 
(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))