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

Percentage Accurate: 100.0% → 100.0%

Time: 9.7s

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

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

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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;z - x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.5)
   (+ z (* x y))
   (if (<= y 2.5) (- z (* x -0.5)) (* y (+ x (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.5) {
		tmp = z + (x * y);
	} else if (y <= 2.5) {
		tmp = z - (x * -0.5);
	} else {
		tmp = y * (x + (z / 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 <= (-0.5d0)) then
        tmp = z + (x * y)
    else if (y <= 2.5d0) then
        tmp = z - (x * (-0.5d0))
    else
        tmp = y * (x + (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.5) {
		tmp = z + (x * y);
	} else if (y <= 2.5) {
		tmp = z - (x * -0.5);
	} else {
		tmp = y * (x + (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.5:
		tmp = z + (x * y)
	elif y <= 2.5:
		tmp = z - (x * -0.5)
	else:
		tmp = y * (x + (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.5)
		tmp = Float64(z + Float64(x * y));
	elseif (y <= 2.5)
		tmp = Float64(z - Float64(x * -0.5));
	else
		tmp = Float64(y * Float64(x + Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.5)
		tmp = z + (x * y);
	elseif (y <= 2.5)
		tmp = z - (x * -0.5);
	else
		tmp = y * (x + (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.5], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5], N[(z - N[(x * -0.5), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if 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 inf 97.5%

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

      1. mul-1-neg 97.5%

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

      2. *-commutative 97.5%

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

      3. distribute-rgt-neg-in 97.5%

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

   7. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. cancel-sign-sub 97.5%

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

      3. *-commutative 97.5%

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

      4. +-commutative 97.5%

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

      5. *-commutative 97.5%

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

   9. Applied egg-rr 97.5%

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

   if -0.5 < y < 2.5

   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%

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

   if 2.5 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      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 98.1%

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

      1. mul-1-neg 98.1%

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

      2. *-commutative 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in y around inf 98.1%

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

      1. cancel-sign-sub-inv 98.1%

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

      2. metadata-eval 98.1%

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

      3. *-lft-identity 98.1%

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

   10. Simplified 98.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;z - x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 0.5)) {
		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 <= (-0.5d0)) .or. (.not. (y <= 0.5d0))) 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 <= -0.5) || !(y <= 0.5)) {
		tmp = z + (x * y);
	} 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 = z + (x * y)
	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(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 <= -0.5) || ~((y <= 0.5)))
		tmp = z + (x * y);
	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[(z + N[(x * y), $MachinePrecision]), $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%

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

      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 97.8%

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

      1. mul-1-neg 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-rgt-neg-in 97.8%

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

   7. Simplified 97.8%

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

      1. *-commutative 97.8%

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

      2. cancel-sign-sub 97.8%

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

      3. *-commutative 97.8%

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

      4. +-commutative 97.8%

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

      5. *-commutative 97.8%

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

   9. Applied egg-rr 97.8%

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

   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 98.0%

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

Alternative 4: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{-126} \lor \neg \left(z \leq 0.0045\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.6e-126) (not (<= z 0.0045))) (+ z (* x y)) (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.6e-126) || !(z <= 0.0045)) {
		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 <= (-8.6d-126)) .or. (.not. (z <= 0.0045d0))) 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 <= -8.6e-126) || !(z <= 0.0045)) {
		tmp = z + (x * y);
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.6e-126) or not (z <= 0.0045):
		tmp = z + (x * y)
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.6e-126) || !(z <= 0.0045))
		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 <= -8.6e-126) || ~((z <= 0.0045)))
		tmp = z + (x * y);
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.6e-126], N[Not[LessEqual[z, 0.0045]], $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 < -8.60000000000000065e-126 or 0.00449999999999999966 < 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 84.9%

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

      1. mul-1-neg 84.9%

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

      2. *-commutative 84.9%

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

      3. distribute-rgt-neg-in 84.9%

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

   7. Simplified 84.9%

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

      1. *-commutative 84.9%

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

      2. cancel-sign-sub 84.9%

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

      3. *-commutative 84.9%

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

      4. +-commutative 84.9%

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

      5. *-commutative 84.9%

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

   9. Applied egg-rr 84.9%

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

   if -8.60000000000000065e-126 < z < 0.00449999999999999966

   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 86.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{-126} \lor \neg \left(z \leq 0.0045\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.8e+31) z (if (<= z 2.7e+67) (* x (+ y 0.5)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e+31) {
		tmp = z;
	} else if (z <= 2.7e+67) {
		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 <= (-3.8d+31)) then
        tmp = z
    else if (z <= 2.7d+67) 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 <= -3.8e+31) {
		tmp = z;
	} else if (z <= 2.7e+67) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.8e+31:
		tmp = z
	elif z <= 2.7e+67:
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.8e+31)
		tmp = z;
	elseif (z <= 2.7e+67)
		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 <= -3.8e+31)
		tmp = z;
	elseif (z <= 2.7e+67)
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.8e+31], z, If[LessEqual[z, 2.7e+67], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000001e31 or 2.6999999999999999e67 < 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 69.4%

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

   if -3.8000000000000001e31 < z < 2.6999999999999999e67

   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 79.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+67}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+31} \lor \neg \left(y \leq 1.8 \cdot 10^{+103}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.2e+31) (not (<= y 1.8e+103))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.2e+31) || !(y <= 1.8e+103)) {
		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 ((y <= (-7.2d+31)) .or. (.not. (y <= 1.8d+103))) 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 ((y <= -7.2e+31) || !(y <= 1.8e+103)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.2e+31) or not (y <= 1.8e+103):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.2e+31) || !(y <= 1.8e+103))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7.2e+31) || ~((y <= 1.8e+103)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.2e+31], N[Not[LessEqual[y, 1.8e+103]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999992e31 or 1.80000000000000008e103 < 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 81.3%

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

   if -7.19999999999999992e31 < y < 1.80000000000000008e103

   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 54.7%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+31} \lor \neg \left(y \leq 1.8 \cdot 10^{+103}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-125}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 0.0036:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.8e-125) z (if (<= z 0.0036) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.8e-125) {
		tmp = z;
	} else if (z <= 0.0036) {
		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 <= (-6.8d-125)) then
        tmp = z
    else if (z <= 0.0036d0) 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 <= -6.8e-125) {
		tmp = z;
	} else if (z <= 0.0036) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.8e-125:
		tmp = z
	elif z <= 0.0036:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.8e-125)
		tmp = z;
	elseif (z <= 0.0036)
		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 <= -6.8e-125)
		tmp = z;
	elseif (z <= 0.0036)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.8e-125], z, If[LessEqual[z, 0.0036], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -6.7999999999999995e-125 or 0.0035999999999999999 < 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 60.0%

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

   if -6.7999999999999995e-125 < z < 0.0035999999999999999

   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 86.9%

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

   6. Taylor expanded in y around 0 41.7%

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

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

   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 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. 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 42.7%

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

Reproduce

herbie shell --seed 2024165 
(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))