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

Percentage Accurate: 100.0% → 100.0%

Time: 2.7s

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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.0078\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.5) (not (<= y 0.0078))) (+ (* x y) z) (+ z (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 0.0078)) {
		tmp = (x * y) + z;
	} 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 <= (-0.5d0)) .or. (.not. (y <= 0.0078d0))) then
        tmp = (x * y) + z
    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 <= -0.5) || !(y <= 0.0078)) {
		tmp = (x * y) + z;
	} else {
		tmp = z + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.5) or not (y <= 0.0078):
		tmp = (x * y) + z
	else:
		tmp = z + (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.5) || !(y <= 0.0078))
		tmp = Float64(Float64(x * y) + z);
	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 <= -0.5) || ~((y <= 0.0078)))
		tmp = (x * y) + z;
	else
		tmp = z + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.5], N[Not[LessEqual[y, 0.0078]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 0.0077999999999999996 < 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%

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

   if -0.5 < y < 0.0077999999999999996

   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 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.0078\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \]

Alternative 3: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+56}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+23}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e+56) (* x y) (if (<= y 1.42e+23) (+ z (* x 0.5)) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+56) {
		tmp = x * y;
	} else if (y <= 1.42e+23) {
		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 <= (-3.4d+56)) then
        tmp = x * y
    else if (y <= 1.42d+23) 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 <= -3.4e+56) {
		tmp = x * y;
	} else if (y <= 1.42e+23) {
		tmp = z + (x * 0.5);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.4e+56:
		tmp = x * y
	elif y <= 1.42e+23:
		tmp = z + (x * 0.5)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e+56)
		tmp = Float64(x * y);
	elseif (y <= 1.42e+23)
		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 <= -3.4e+56)
		tmp = x * y;
	elseif (y <= 1.42e+23)
		tmp = z + (x * 0.5);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.4e+56], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.42e+23], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.40000000000000001e56 or 1.42000000000000004e23 < 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 100.0%

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

   5. Taylor expanded in y around inf 81.0%

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

   if -3.40000000000000001e56 < y < 1.42000000000000004e23

   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 95.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+56}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+23}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+59}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.8e+59) (* x y) (if (<= y 4.6e+17) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.8e+59) {
		tmp = x * y;
	} else if (y <= 4.6e+17) {
		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 <= (-7.8d+59)) then
        tmp = x * y
    else if (y <= 4.6d+17) 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 <= -7.8e+59) {
		tmp = x * y;
	} else if (y <= 4.6e+17) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.8e+59:
		tmp = x * y
	elif y <= 4.6e+17:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.8e+59)
		tmp = Float64(x * y);
	elseif (y <= 4.6e+17)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.8e+59)
		tmp = x * y;
	elseif (y <= 4.6e+17)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.8e+59], N[(x * y), $MachinePrecision], If[LessEqual[y, 4.6e+17], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.80000000000000043e59 or 4.6e17 < 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 100.0%

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

   5. Taylor expanded in y around inf 81.0%

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

   if -7.80000000000000043e59 < y < 4.6e17

   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 55.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+59}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+17}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

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: 40.4% 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 41.9%

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

5. Final simplification 41.9%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023268 
(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))