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

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 8

Speedup: 0.1×

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(0.5 + y), $MachinePrecision] * x + z), $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. Taylor expanded in x around 0 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(0.5 + y, x, z\right) \]

Alternative 2: 61.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-61}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-300}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-157}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-88}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e+54)
   (* y x)
   (if (<= y -2.9e-61)
     z
     (if (<= y -1.6e-203)
       (* 0.5 x)
       (if (<= y 2.85e-300)
         z
         (if (<= y 6.2e-157)
           (* 0.5 x)
           (if (<= y 5.2e-88)
             z
             (if (<= y 2e-31) (* 0.5 x) (if (<= y 1.25e+19) z (* y x))))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e+54) {
		tmp = y * x;
	} else if (y <= -2.9e-61) {
		tmp = z;
	} else if (y <= -1.6e-203) {
		tmp = 0.5 * x;
	} else if (y <= 2.85e-300) {
		tmp = z;
	} else if (y <= 6.2e-157) {
		tmp = 0.5 * x;
	} else if (y <= 5.2e-88) {
		tmp = z;
	} else if (y <= 2e-31) {
		tmp = 0.5 * x;
	} else if (y <= 1.25e+19) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	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 <= (-7.5d+54)) then
        tmp = y * x
    else if (y <= (-2.9d-61)) then
        tmp = z
    else if (y <= (-1.6d-203)) then
        tmp = 0.5d0 * x
    else if (y <= 2.85d-300) then
        tmp = z
    else if (y <= 6.2d-157) then
        tmp = 0.5d0 * x
    else if (y <= 5.2d-88) then
        tmp = z
    else if (y <= 2d-31) then
        tmp = 0.5d0 * x
    else if (y <= 1.25d+19) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e+54) {
		tmp = y * x;
	} else if (y <= -2.9e-61) {
		tmp = z;
	} else if (y <= -1.6e-203) {
		tmp = 0.5 * x;
	} else if (y <= 2.85e-300) {
		tmp = z;
	} else if (y <= 6.2e-157) {
		tmp = 0.5 * x;
	} else if (y <= 5.2e-88) {
		tmp = z;
	} else if (y <= 2e-31) {
		tmp = 0.5 * x;
	} else if (y <= 1.25e+19) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5e+54:
		tmp = y * x
	elif y <= -2.9e-61:
		tmp = z
	elif y <= -1.6e-203:
		tmp = 0.5 * x
	elif y <= 2.85e-300:
		tmp = z
	elif y <= 6.2e-157:
		tmp = 0.5 * x
	elif y <= 5.2e-88:
		tmp = z
	elif y <= 2e-31:
		tmp = 0.5 * x
	elif y <= 1.25e+19:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e+54)
		tmp = Float64(y * x);
	elseif (y <= -2.9e-61)
		tmp = z;
	elseif (y <= -1.6e-203)
		tmp = Float64(0.5 * x);
	elseif (y <= 2.85e-300)
		tmp = z;
	elseif (y <= 6.2e-157)
		tmp = Float64(0.5 * x);
	elseif (y <= 5.2e-88)
		tmp = z;
	elseif (y <= 2e-31)
		tmp = Float64(0.5 * x);
	elseif (y <= 1.25e+19)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.5e+54)
		tmp = y * x;
	elseif (y <= -2.9e-61)
		tmp = z;
	elseif (y <= -1.6e-203)
		tmp = 0.5 * x;
	elseif (y <= 2.85e-300)
		tmp = z;
	elseif (y <= 6.2e-157)
		tmp = 0.5 * x;
	elseif (y <= 5.2e-88)
		tmp = z;
	elseif (y <= 2e-31)
		tmp = 0.5 * x;
	elseif (y <= 1.25e+19)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e+54], N[(y * x), $MachinePrecision], If[LessEqual[y, -2.9e-61], z, If[LessEqual[y, -1.6e-203], N[(0.5 * x), $MachinePrecision], If[LessEqual[y, 2.85e-300], z, If[LessEqual[y, 6.2e-157], N[(0.5 * x), $MachinePrecision], If[LessEqual[y, 5.2e-88], z, If[LessEqual[y, 2e-31], N[(0.5 * x), $MachinePrecision], If[LessEqual[y, 1.25e+19], z, N[(y * x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.50000000000000042e54 or 1.25e19 < y

   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. Taylor expanded in y around inf 73.6%

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

   if -7.50000000000000042e54 < y < -2.8999999999999999e-61 or -1.6e-203 < y < 2.8499999999999999e-300 or 6.1999999999999996e-157 < y < 5.20000000000000027e-88 or 2e-31 < y < 1.25e19

   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. Taylor expanded in x around 0 73.7%

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

   if -2.8999999999999999e-61 < y < -1.6e-203 or 2.8499999999999999e-300 < y < 6.1999999999999996e-157 or 5.20000000000000027e-88 < y < 2e-31

   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. Taylor expanded in x around inf 71.0%

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

   5. Taylor expanded in y around 0 71.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-61}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-300}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-157}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-88}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-66} \lor \neg \left(z \leq 5.5 \cdot 10^{-7}\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e-66) (not (<= z 5.5e-7))) (+ z (* y x)) (* (+ 0.5 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-66) || !(z <= 5.5e-7)) {
		tmp = z + (y * x);
	} else {
		tmp = (0.5 + y) * x;
	}
	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 <= (-1.1d-66)) .or. (.not. (z <= 5.5d-7))) then
        tmp = z + (y * x)
    else
        tmp = (0.5d0 + y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-66) || !(z <= 5.5e-7)) {
		tmp = z + (y * x);
	} else {
		tmp = (0.5 + y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.1e-66) or not (z <= 5.5e-7):
		tmp = z + (y * x)
	else:
		tmp = (0.5 + y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e-66) || !(z <= 5.5e-7))
		tmp = Float64(z + Float64(y * x));
	else
		tmp = Float64(Float64(0.5 + y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.1e-66) || ~((z <= 5.5e-7)))
		tmp = z + (y * x);
	else
		tmp = (0.5 + y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e-66], N[Not[LessEqual[z, 5.5e-7]], $MachinePrecision]], N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e-66 or 5.5000000000000003e-7 < z

   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. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-+ 84.5%

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

      3. associate-*r/ 81.2%

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

      4. metadata-eval 81.2%

         \[\leadsto z + \frac{x \cdot \left(\color{blue}{0.25} - y \cdot y\right)}{0.5 - y} \]

   6. Applied egg-rr 81.2%

      \[\leadsto z + \color{blue}{\frac{x \cdot \left(0.25 - y \cdot y\right)}{0.5 - y}} \]
   7. Step-by-step derivation

      1. *-commutative 81.2%

         \[\leadsto z + \frac{\color{blue}{\left(0.25 - y \cdot y\right) \cdot x}}{0.5 - y} \]

      2. associate-/l* 84.0%

         \[\leadsto z + \color{blue}{\frac{0.25 - y \cdot y}{\frac{0.5 - y}{x}}} \]

   8. Simplified 84.0%

      \[\leadsto z + \color{blue}{\frac{0.25 - y \cdot y}{\frac{0.5 - y}{x}}} \]

   9. Taylor expanded in y around inf 87.8%

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

   if -1.1000000000000001e-66 < z < 5.5000000000000003e-7

   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. Taylor expanded in x around inf 87.1%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-66} \lor \neg \left(z \leq 5.5 \cdot 10^{-7}\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -500000000000 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -500000000000.0) (not (<= y 0.5)))
   (+ z (* y x))
   (+ z (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -500000000000.0) || !(y <= 0.5)) {
		tmp = z + (y * x);
	} else {
		tmp = z + (0.5 * x);
	}
	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 <= (-500000000000.0d0)) .or. (.not. (y <= 0.5d0))) then
        tmp = z + (y * x)
    else
        tmp = z + (0.5d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -500000000000.0) || !(y <= 0.5)) {
		tmp = z + (y * x);
	} else {
		tmp = z + (0.5 * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -500000000000.0) or not (y <= 0.5):
		tmp = z + (y * x)
	else:
		tmp = z + (0.5 * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -500000000000.0) || !(y <= 0.5))
		tmp = Float64(z + Float64(y * x));
	else
		tmp = Float64(z + Float64(0.5 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -500000000000.0) || ~((y <= 0.5)))
		tmp = z + (y * x);
	else
		tmp = z + (0.5 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -500000000000.0], N[Not[LessEqual[y, 0.5]], $MachinePrecision]], N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5e11 or 0.5 < y

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-+ 68.5%

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

      3. associate-*r/ 61.7%

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

      4. metadata-eval 61.7%

         \[\leadsto z + \frac{x \cdot \left(\color{blue}{0.25} - y \cdot y\right)}{0.5 - y} \]

   6. Applied egg-rr 61.7%

      \[\leadsto z + \color{blue}{\frac{x \cdot \left(0.25 - y \cdot y\right)}{0.5 - y}} \]
   7. Step-by-step derivation

      1. *-commutative 61.7%

         \[\leadsto z + \frac{\color{blue}{\left(0.25 - y \cdot y\right) \cdot x}}{0.5 - y} \]

      2. associate-/l* 66.1%

         \[\leadsto z + \color{blue}{\frac{0.25 - y \cdot y}{\frac{0.5 - y}{x}}} \]

   8. Simplified 66.1%

      \[\leadsto z + \color{blue}{\frac{0.25 - y \cdot y}{\frac{0.5 - y}{x}}} \]

   9. Taylor expanded in y around inf 99.7%

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

   if -5e11 < y < 0.5

   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. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -500000000000 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot x\\ \end{array} \]

Alternative 5: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+137}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 0.00045:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.95e+137) z (if (<= z 0.00045) (* (+ 0.5 y) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.95e+137) {
		tmp = z;
	} else if (z <= 0.00045) {
		tmp = (0.5 + y) * x;
	} 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 <= (-1.95d+137)) then
        tmp = z
    else if (z <= 0.00045d0) then
        tmp = (0.5d0 + y) * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.95e+137) {
		tmp = z;
	} else if (z <= 0.00045) {
		tmp = (0.5 + y) * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.95e+137:
		tmp = z
	elif z <= 0.00045:
		tmp = (0.5 + y) * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.95e+137)
		tmp = z;
	elseif (z <= 0.00045)
		tmp = Float64(Float64(0.5 + y) * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.95e+137)
		tmp = z;
	elseif (z <= 0.00045)
		tmp = (0.5 + y) * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.95e+137], z, If[LessEqual[z, 0.00045], N[(N[(0.5 + y), $MachinePrecision] * x), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95000000000000015e137 or 4.4999999999999999e-4 < z

   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. Taylor expanded in x around 0 76.5%

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

   if -1.95000000000000015e137 < z < 4.4999999999999999e-4

   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. Taylor expanded in x around inf 81.1%

      \[\leadsto \color{blue}{\left(0.5 + y\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+137}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 0.00045:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6: 51.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-67}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.6e-67) z (if (<= z 5.5e-7) (* 0.5 x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e-67) {
		tmp = z;
	} else if (z <= 5.5e-7) {
		tmp = 0.5 * x;
	} 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 <= (-1.6d-67)) then
        tmp = z
    else if (z <= 5.5d-7) then
        tmp = 0.5d0 * x
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e-67) {
		tmp = z;
	} else if (z <= 5.5e-7) {
		tmp = 0.5 * x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.6e-67:
		tmp = z
	elif z <= 5.5e-7:
		tmp = 0.5 * x
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.6e-67)
		tmp = z;
	elseif (z <= 5.5e-7)
		tmp = Float64(0.5 * x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.6e-67)
		tmp = z;
	elseif (z <= 5.5e-7)
		tmp = 0.5 * x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.6e-67], z, If[LessEqual[z, 5.5e-7], N[(0.5 * x), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000011e-67 or 5.5000000000000003e-7 < z

   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. Taylor expanded in x around 0 68.7%

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

   if -1.60000000000000011e-67 < z < 5.5000000000000003e-7

   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. Taylor expanded in x around inf 87.1%

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

   5. Taylor expanded in y around 0 43.3%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-67}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 + ((0.5d0 + y) * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z + N[(N[(0.5 + y), $MachinePrecision] * x), $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. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 41.2% 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. 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. Taylor expanded in x around 0 44.1%

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

5. Final simplification 44.1%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023195 
(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))