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

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

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

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / 2.0d0) + (x * y)) + z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 60.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+40}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{-19}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.65e+40)
   (* x y)
   (if (<= y -1.08e-19) z (if (<= y 0.5) (* x 0.5) (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.65e+40) {
		tmp = x * y;
	} else if (y <= -1.08e-19) {
		tmp = z;
	} else if (y <= 0.5) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.65d+40)) then
        tmp = x * y
    else if (y <= (-1.08d-19)) then
        tmp = z
    else if (y <= 0.5d0) then
        tmp = x * 0.5d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.65e+40) {
		tmp = x * y;
	} else if (y <= -1.08e-19) {
		tmp = z;
	} else if (y <= 0.5) {
		tmp = x * 0.5;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.65e+40:
		tmp = x * y
	elif y <= -1.08e-19:
		tmp = z
	elif y <= 0.5:
		tmp = x * 0.5
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.65e+40)
		tmp = Float64(x * y);
	elseif (y <= -1.08e-19)
		tmp = z;
	elseif (y <= 0.5)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.65e+40)
		tmp = x * y;
	elseif (y <= -1.08e-19)
		tmp = z;
	elseif (y <= 0.5)
		tmp = x * 0.5;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.65e+40], N[(x * y), $MachinePrecision], If[LessEqual[y, -1.08e-19], z, If[LessEqual[y, 0.5], N[(x * 0.5), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6499999999999999e40 or 0.5 < 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. Add Preprocessing

   5. Taylor expanded in y around inf 79.6%

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

   if -1.6499999999999999e40 < y < -1.08e-19

   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 65.7%

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

   if -1.08e-19 < 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. Add Preprocessing

   5. Taylor expanded in x around inf 53.7%

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

   6. Taylor expanded in y around 0 53.7%

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

Alternative 3: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -130000 \lor \neg \left(y \leq 1.4 \cdot 10^{-12}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -130000.0) (not (<= y 1.4e-12)))
   (+ z (* x y))
   (- z (* x -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -130000.0) || !(y <= 1.4e-12)) {
		tmp = z + (x * y);
	} else {
		tmp = z - (x * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-130000.0d0)) .or. (.not. (y <= 1.4d-12))) then
        tmp = z + (x * y)
    else
        tmp = z - (x * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -130000.0) || !(y <= 1.4e-12)) {
		tmp = z + (x * y);
	} else {
		tmp = z - (x * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -130000.0) or not (y <= 1.4e-12):
		tmp = z + (x * y)
	else:
		tmp = z - (x * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -130000.0) || !(y <= 1.4e-12))
		tmp = Float64(z + Float64(x * y));
	else
		tmp = Float64(z - Float64(x * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -130000.0) || ~((y <= 1.4e-12)))
		tmp = z + (x * y);
	else
		tmp = z - (x * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -130000.0], N[Not[LessEqual[y, 1.4e-12]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e5 or 1.4000000000000001e-12 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      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 inf 99.5%

      \[\leadsto z - \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   7. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. cancel-sign-sub 99.5%

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

      3. *-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

   if -1.3e5 < y < 1.4000000000000001e-12

   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 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130000 \lor \neg \left(y \leq 1.4 \cdot 10^{-12}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-56} \lor \neg \left(z \leq 7.8 \cdot 10^{+71}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.4e-56) (not (<= z 7.8e+71))) (+ z (* x y)) (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.4e-56) || !(z <= 7.8e+71)) {
		tmp = z + (x * y);
	} else {
		tmp = x * (y + 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 ((z <= (-6.4d-56)) .or. (.not. (z <= 7.8d+71))) then
        tmp = z + (x * y)
    else
        tmp = x * (y + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.4e-56) || !(z <= 7.8e+71)) {
		tmp = z + (x * y);
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.4e-56) or not (z <= 7.8e+71):
		tmp = z + (x * y)
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.4e-56) || !(z <= 7.8e+71))
		tmp = Float64(z + Float64(x * y));
	else
		tmp = Float64(x * Float64(y + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.4e-56) || ~((z <= 7.8e+71)))
		tmp = z + (x * y);
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.4e-56], N[Not[LessEqual[z, 7.8e+71]], $MachinePrecision]], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.39999999999999971e-56 or 7.8000000000000002e71 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      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 inf 86.8%

      \[\leadsto z - \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 86.8%

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

      2. *-commutative 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

   7. Simplified 86.8%

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

      1. *-commutative 86.8%

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

      2. cancel-sign-sub 86.8%

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

      3. *-commutative 86.8%

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

      4. +-commutative 86.8%

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

      5. *-commutative 86.8%

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

   9. Applied egg-rr 86.8%

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

   if -6.39999999999999971e-56 < z < 7.8000000000000002e71

   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 88.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-56} \lor \neg \left(z \leq 7.8 \cdot 10^{+71}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e+72) z (if (<= z 8.2e+71) (* x (+ y 0.5)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+72) {
		tmp = z;
	} else if (z <= 8.2e+71) {
		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 (z <= (-1.3d+72)) then
        tmp = z
    else if (z <= 8.2d+71) 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 (z <= -1.3e+72) {
		tmp = z;
	} else if (z <= 8.2e+71) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.3e+72:
		tmp = z
	elif z <= 8.2e+71:
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e+72)
		tmp = z;
	elseif (z <= 8.2e+71)
		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 (z <= -1.3e+72)
		tmp = z;
	elseif (z <= 8.2e+71)
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e+72], z, If[LessEqual[z, 8.2e+71], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999991e72 or 8.2000000000000004e71 < 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 73.2%

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

   if -1.29999999999999991e72 < z < 8.2000000000000004e71

   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 83.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+72}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-57}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+71}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e-57) z (if (<= z 7.8e+71) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e-57) {
		tmp = z;
	} else if (z <= 7.8e+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 <= (-5.1d-57)) then
        tmp = z
    else if (z <= 7.8d+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 <= -5.1e-57) {
		tmp = z;
	} else if (z <= 7.8e+71) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.1e-57:
		tmp = z
	elif z <= 7.8e+71:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e-57)
		tmp = z;
	elseif (z <= 7.8e+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 <= -5.1e-57)
		tmp = z;
	elseif (z <= 7.8e+71)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e-57], z, If[LessEqual[z, 7.8e+71], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1e-57 or 7.8000000000000002e71 < 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 63.1%

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

   if -5.1e-57 < z < 7.8000000000000002e71

   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 88.3%

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

   6. Taylor expanded in y around 0 41.5%

      \[\leadsto x \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. 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: 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. 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 38.2%

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

Reproduce

herbie shell --seed 2024172 
(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))