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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 8

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. Add Preprocessing

Alternative 2: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-189}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-48}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.5e-5)
   z
   (if (<= z -1.7e-189)
     (* x y)
     (if (<= z 1.75e-158) (* x 0.5) (if (<= z 4.5e-48) (* x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e-5) {
		tmp = z;
	} else if (z <= -1.7e-189) {
		tmp = x * y;
	} else if (z <= 1.75e-158) {
		tmp = x * 0.5;
	} else if (z <= 4.5e-48) {
		tmp = x * y;
	} 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 <= (-2.5d-5)) then
        tmp = z
    else if (z <= (-1.7d-189)) then
        tmp = x * y
    else if (z <= 1.75d-158) then
        tmp = x * 0.5d0
    else if (z <= 4.5d-48) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.5e-5) {
		tmp = z;
	} else if (z <= -1.7e-189) {
		tmp = x * y;
	} else if (z <= 1.75e-158) {
		tmp = x * 0.5;
	} else if (z <= 4.5e-48) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.5e-5:
		tmp = z
	elif z <= -1.7e-189:
		tmp = x * y
	elif z <= 1.75e-158:
		tmp = x * 0.5
	elif z <= 4.5e-48:
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.5e-5)
		tmp = z;
	elseif (z <= -1.7e-189)
		tmp = Float64(x * y);
	elseif (z <= 1.75e-158)
		tmp = Float64(x * 0.5);
	elseif (z <= 4.5e-48)
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.5e-5)
		tmp = z;
	elseif (z <= -1.7e-189)
		tmp = x * y;
	elseif (z <= 1.75e-158)
		tmp = x * 0.5;
	elseif (z <= 4.5e-48)
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.5e-5], z, If[LessEqual[z, -1.7e-189], N[(x * y), $MachinePrecision], If[LessEqual[z, 1.75e-158], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 4.5e-48], N[(x * y), $MachinePrecision], z]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.50000000000000012e-5 or 4.49999999999999988e-48 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 63.7%

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

   if -2.50000000000000012e-5 < z < -1.7000000000000001e-189 or 1.75000000000000006e-158 < z < 4.49999999999999988e-48

   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. Taylor expanded in y around inf 60.6%

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

   if -1.7000000000000001e-189 < z < 1.75000000000000006e-158

   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. Taylor expanded in x around inf 98.1%

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

   6. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   8. Simplified 63.3%

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

Alternative 3: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 0.5)) {
		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 <= 0.5d0))) 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 <= 0.5)) {
		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 <= 0.5):
		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 <= 0.5))
		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 <= 0.5)))
		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, 0.5]], $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 0.5 < 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. fma-define 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 99.3%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y}, z\right) \]
   6. Step-by-step derivation

      1. fma-undefine 99.3%

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

   7. Applied egg-rr 99.3%

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

   if -0.5 < y < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. distribute-rgt-neg-out 100.0%

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

      6. unsub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

      9. *-commutative 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. neg-mul-1 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-/l* 100.0%

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

      14. distribute-lft-out-- 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 98.7%

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

Alternative 4: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+117} \lor \neg \left(x \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.8e+117) (not (<= x 1.9e+32))) (* x (+ y 0.5)) (+ (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.8e+117) || !(x <= 1.9e+32)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = (x * y) + 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 <= (-9.8d+117)) .or. (.not. (x <= 1.9d+32))) then
        tmp = x * (y + 0.5d0)
    else
        tmp = (x * y) + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.8e+117) || !(x <= 1.9e+32)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = (x * y) + z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.8e+117) or not (x <= 1.9e+32):
		tmp = x * (y + 0.5)
	else:
		tmp = (x * y) + z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.8e+117) || !(x <= 1.9e+32))
		tmp = Float64(x * Float64(y + 0.5));
	else
		tmp = Float64(Float64(x * y) + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.8e+117) || ~((x <= 1.9e+32)))
		tmp = x * (y + 0.5);
	else
		tmp = (x * y) + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.8e+117], N[Not[LessEqual[x, 1.9e+32]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.8000000000000002e117 or 1.9000000000000002e32 < x

   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. Taylor expanded in x around inf 88.5%

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

   if -9.8000000000000002e117 < x < 1.9000000000000002e32

   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. fma-define 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 86.5%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y}, z\right) \]
   6. Step-by-step derivation

      1. fma-undefine 86.5%

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

   7. Applied egg-rr 86.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+117} \lor \neg \left(x \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-36} \lor \neg \left(x \leq 6.6 \cdot 10^{-66}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.5e-36) (not (<= x 6.6e-66))) (* x (+ y 0.5)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.5e-36) || !(x <= 6.6e-66)) {
		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 <= (-2.5d-36)) .or. (.not. (x <= 6.6d-66))) 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 <= -2.5e-36) || !(x <= 6.6e-66)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.5e-36) or not (x <= 6.6e-66):
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.5e-36) || !(x <= 6.6e-66))
		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 <= -2.5e-36) || ~((x <= 6.6e-66)))
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.5e-36], N[Not[LessEqual[x, 6.6e-66]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000002e-36 or 6.5999999999999998e-66 < x

   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. Taylor expanded in x around inf 80.8%

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

   if -2.50000000000000002e-36 < x < 6.5999999999999998e-66

   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. Taylor expanded in x around 0 73.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-36} \lor \neg \left(x \leq 6.6 \cdot 10^{-66}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -740000 \lor \neg \left(x \leq 4.7 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -740000.0) (not (<= x 4.7e+32))) (* x 0.5) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -740000.0) || !(x <= 4.7e+32)) {
		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 ((x <= (-740000.0d0)) .or. (.not. (x <= 4.7d+32))) 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 ((x <= -740000.0) || !(x <= 4.7e+32)) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -740000.0) or not (x <= 4.7e+32):
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -740000.0) || !(x <= 4.7e+32))
		tmp = Float64(x * 0.5);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -740000.0) || ~((x <= 4.7e+32)))
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -740000.0], N[Not[LessEqual[x, 4.7e+32]], $MachinePrecision]], N[(x * 0.5), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -7.4e5 or 4.70000000000000023e32 < x

   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. Taylor expanded in x around inf 84.6%

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

   6. Taylor expanded in y around 0 46.9%

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

      1. *-commutative 46.9%

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

   8. Simplified 46.9%

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

   if -7.4e5 < x < 4.70000000000000023e32

   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. Taylor expanded in x around 0 64.4%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -740000 \lor \neg \left(x \leq 4.7 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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%

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

   2. remove-double-neg 100.0%

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

   3. distribute-neg-in 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. distribute-rgt-neg-out 100.0%

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

   6. unsub-neg 100.0%

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

   7. +-commutative 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. cancel-sign-sub-inv 100.0%

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

   11. neg-mul-1 100.0%

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

   12. *-commutative 100.0%

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

   13. associate-/l* 100.0%

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

   14. distribute-lft-out-- 100.0%

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

   15. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 41.3% 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. Add Preprocessing

5. Taylor expanded in x around 0 42.0%

   \[\leadsto \color{blue}{z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))