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

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 9

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

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

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(x / 2.0) + Float64(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[(x / 2.0), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-+l+ 100.0%

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

   2. *-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 59.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+38}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.45 \cdot 10^{-242}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-211}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.45e+38)
   (* x y)
   (if (<= y -3.45e-242)
     z
     (if (<= y 2.5e-211)
       (* x 0.5)
       (if (<= y 1.15e-62)
         z
         (if (<= y 3.55e-6) (* x 0.5) (if (<= y 1.25e+20) z (* x y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.45e+38) {
		tmp = x * y;
	} else if (y <= -3.45e-242) {
		tmp = z;
	} else if (y <= 2.5e-211) {
		tmp = x * 0.5;
	} else if (y <= 1.15e-62) {
		tmp = z;
	} else if (y <= 3.55e-6) {
		tmp = x * 0.5;
	} else if (y <= 1.25e+20) {
		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 <= (-1.45d+38)) then
        tmp = x * y
    else if (y <= (-3.45d-242)) then
        tmp = z
    else if (y <= 2.5d-211) then
        tmp = x * 0.5d0
    else if (y <= 1.15d-62) then
        tmp = z
    else if (y <= 3.55d-6) then
        tmp = x * 0.5d0
    else if (y <= 1.25d+20) 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 <= -1.45e+38) {
		tmp = x * y;
	} else if (y <= -3.45e-242) {
		tmp = z;
	} else if (y <= 2.5e-211) {
		tmp = x * 0.5;
	} else if (y <= 1.15e-62) {
		tmp = z;
	} else if (y <= 3.55e-6) {
		tmp = x * 0.5;
	} else if (y <= 1.25e+20) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.45e+38:
		tmp = x * y
	elif y <= -3.45e-242:
		tmp = z
	elif y <= 2.5e-211:
		tmp = x * 0.5
	elif y <= 1.15e-62:
		tmp = z
	elif y <= 3.55e-6:
		tmp = x * 0.5
	elif y <= 1.25e+20:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.45e+38)
		tmp = Float64(x * y);
	elseif (y <= -3.45e-242)
		tmp = z;
	elseif (y <= 2.5e-211)
		tmp = Float64(x * 0.5);
	elseif (y <= 1.15e-62)
		tmp = z;
	elseif (y <= 3.55e-6)
		tmp = Float64(x * 0.5);
	elseif (y <= 1.25e+20)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.45e+38)
		tmp = x * y;
	elseif (y <= -3.45e-242)
		tmp = z;
	elseif (y <= 2.5e-211)
		tmp = x * 0.5;
	elseif (y <= 1.15e-62)
		tmp = z;
	elseif (y <= 3.55e-6)
		tmp = x * 0.5;
	elseif (y <= 1.25e+20)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.45e+38], N[(x * y), $MachinePrecision], If[LessEqual[y, -3.45e-242], z, If[LessEqual[y, 2.5e-211], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 1.15e-62], z, If[LessEqual[y, 3.55e-6], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 1.25e+20], z, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.45000000000000003e38 or 1.25e20 < y

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 90.7%

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

      1. +-commutative 90.7%

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

      2. +-commutative 90.7%

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

      3. associate-+l+ 90.7%

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

      4. +-commutative 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in y around inf 77.4%

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

   if -1.45000000000000003e38 < y < -3.44999999999999998e-242 or 2.5000000000000001e-211 < y < 1.15e-62 or 3.5499999999999999e-6 < y < 1.25e20

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.4%

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

   if -3.44999999999999998e-242 < y < 2.5000000000000001e-211 or 1.15e-62 < y < 3.5499999999999999e-6

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 93.0%

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

      1. +-commutative 93.0%

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

      2. +-commutative 93.0%

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

      3. associate-+l+ 93.0%

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

      4. +-commutative 93.0%

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

   7. Simplified 93.0%

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

   8. Taylor expanded in x around inf 70.5%

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

   9. Taylor expanded in y around 0 69.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+38}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -3.45 \cdot 10^{-242}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-211}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-99} \lor \neg \left(x \leq 9 \cdot 10^{-95}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.7e-99) (not (<= x 9e-95))) (* x (+ y 0.5)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.7e-99) || !(x <= 9e-95)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	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 ((x <= (-2.7d-99)) .or. (.not. (x <= 9d-95))) then
        tmp = x * (y + 0.5d0)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.7e-99) || !(x <= 9e-95)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.7e-99) or not (x <= 9e-95):
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.7e-99) || !(x <= 9e-95))
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.7e-99) || ~((x <= 9e-95)))
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.7e-99], N[Not[LessEqual[x, 9e-95]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -2.7e-99 or 9e-95 < x

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around inf 83.1%

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

   if -2.7e-99 < x < 9e-95

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-99} \lor \neg \left(x \leq 9 \cdot 10^{-95}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+35} \lor \neg \left(y \leq 6.7 \cdot 10^{+28}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.1e+35) (not (<= y 6.7e+28))) (* x y) (+ z (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.1e+35) || !(y <= 6.7e+28)) {
		tmp = 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 <= (-5.1d+35)) .or. (.not. (y <= 6.7d+28))) then
        tmp = 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 <= -5.1e+35) || !(y <= 6.7e+28)) {
		tmp = x * y;
	} else {
		tmp = z + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.1e+35) or not (y <= 6.7e+28):
		tmp = x * y
	else:
		tmp = z + (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.1e+35) || !(y <= 6.7e+28))
		tmp = 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 <= -5.1e+35) || ~((y <= 6.7e+28)))
		tmp = x * y;
	else
		tmp = z + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.1e+35], N[Not[LessEqual[y, 6.7e+28]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.10000000000000017e35 or 6.7e28 < y

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 90.5%

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

      1. +-commutative 90.5%

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

      2. +-commutative 90.5%

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

      3. associate-+l+ 90.5%

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

      4. +-commutative 90.5%

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

   7. Simplified 90.5%

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

   8. Taylor expanded in y around inf 77.8%

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

   if -5.10000000000000017e35 < y < 6.7e28

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 96.2%

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

      1. *-commutative 96.2%

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

   7. Simplified 96.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+35} \lor \neg \left(y \leq 6.7 \cdot 10^{+28}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -82 \lor \neg \left(y \leq 0.00112\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -82.0) (not (<= y 0.00112))) (+ (* x y) z) (+ z (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -82.0) || !(y <= 0.00112)) {
		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 <= (-82.0d0)) .or. (.not. (y <= 0.00112d0))) 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 <= -82.0) || !(y <= 0.00112)) {
		tmp = (x * y) + z;
	} else {
		tmp = z + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -82.0) or not (y <= 0.00112):
		tmp = (x * y) + z
	else:
		tmp = z + (x * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -82.0) || !(y <= 0.00112))
		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 <= -82.0) || ~((y <= 0.00112)))
		tmp = (x * y) + z;
	else
		tmp = z + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -82.0], N[Not[LessEqual[y, 0.00112]], $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 < -82 or 0.0011199999999999999 < y

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 99.4%

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

   if -82 < y < 0.0011199999999999999

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -82 \lor \neg \left(y \leq 0.00112\right):\\ \;\;\;\;x \cdot y + z\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-49}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-71}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5e-49) z (if (<= z 2.4e-71) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e-49) {
		tmp = z;
	} else if (z <= 2.4e-71) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	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 (z <= (-5.5d-49)) then
        tmp = z
    else if (z <= 2.4d-71) then
        tmp = x * 0.5d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5e-49) {
		tmp = z;
	} else if (z <= 2.4e-71) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.5e-49:
		tmp = z
	elif z <= 2.4e-71:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5e-49)
		tmp = z;
	elseif (z <= 2.4e-71)
		tmp = Float64(x * 0.5);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.5e-49)
		tmp = z;
	elseif (z <= 2.4e-71)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5e-49], z, If[LessEqual[z, 2.4e-71], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -5.50000000000000031e-49 or 2.4e-71 < z

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.8%

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

   if -5.50000000000000031e-49 < z < 2.4e-71

   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. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 89.6%

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

   9. Taylor expanded in y around 0 45.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-49}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-71}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 8: 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. +-commutative 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-/l* 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-rgt-out-- 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto z + x \cdot \left(y - -0.5\right) \]
6. Add Preprocessing

Alternative 9: 40.5% 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. +-commutative 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-/l* 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-rgt-out-- 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y - -0.5\right) + z} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 36.8%

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

6. Final simplification 36.8%

   \[\leadsto z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))