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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 8

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-/l* 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-lft-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. Final simplification 100.0%

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

Alternative 2: 60.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+54}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-286}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-103}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+86}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.25e+54)
   (* x y)
   (if (<= y 5.8e-286)
     z
     (if (<= y 1.06e-103) (* x 0.5) (if (<= y 1.4e+86) z (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25e+54) {
		tmp = x * y;
	} else if (y <= 5.8e-286) {
		tmp = z;
	} else if (y <= 1.06e-103) {
		tmp = x * 0.5;
	} else if (y <= 1.4e+86) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.25d+54)) then
        tmp = x * y
    else if (y <= 5.8d-286) then
        tmp = z
    else if (y <= 1.06d-103) then
        tmp = x * 0.5d0
    else if (y <= 1.4d+86) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.25e+54) {
		tmp = x * y;
	} else if (y <= 5.8e-286) {
		tmp = z;
	} else if (y <= 1.06e-103) {
		tmp = x * 0.5;
	} else if (y <= 1.4e+86) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.25e+54:
		tmp = x * y
	elif y <= 5.8e-286:
		tmp = z
	elif y <= 1.06e-103:
		tmp = x * 0.5
	elif y <= 1.4e+86:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.25e+54)
		tmp = Float64(x * y);
	elseif (y <= 5.8e-286)
		tmp = z;
	elseif (y <= 1.06e-103)
		tmp = Float64(x * 0.5);
	elseif (y <= 1.4e+86)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.25e+54)
		tmp = x * y;
	elseif (y <= 5.8e-286)
		tmp = z;
	elseif (y <= 1.06e-103)
		tmp = x * 0.5;
	elseif (y <= 1.4e+86)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.25e+54], N[(x * y), $MachinePrecision], If[LessEqual[y, 5.8e-286], z, If[LessEqual[y, 1.06e-103], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 1.4e+86], z, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25000000000000001e54 or 1.40000000000000002e86 < 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 x around inf 75.3%

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

      1. +-commutative 75.3%

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

   7. Simplified 75.3%

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

   8. Taylor expanded in y around inf 75.3%

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

   if -1.25000000000000001e54 < y < 5.7999999999999996e-286 or 1.06000000000000004e-103 < y < 1.40000000000000002e86

   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. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out-- 100.0%

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

      11. cancel-sign-sub 100.0%

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

      12. distribute-lft-neg-in 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. neg-sub0 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

      15. associate-+l- 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. 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 65.4%

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

      1. associate-*r* 65.4%

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

      2. neg-mul-1 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in z around inf 57.7%

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

   if 5.7999999999999996e-286 < y < 1.06000000000000004e-103

   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 66.8%

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

      1. +-commutative 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in y around 0 66.8%

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

      1. *-commutative 66.8%

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

   10. Simplified 66.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+54}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-286}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-103}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+86}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-103} \lor \neg \left(x \leq 4.8 \cdot 10^{-135}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.6e-103) (not (<= x 4.8e-135))) (* x (+ y 0.5)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.6e-103) || !(x <= 4.8e-135)) {
		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 <= (-2.6d-103)) .or. (.not. (x <= 4.8d-135))) 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 <= -2.6e-103) || !(x <= 4.8e-135)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.6e-103) or not (x <= 4.8e-135):
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.6e-103) || !(x <= 4.8e-135))
		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 <= -2.6e-103) || ~((x <= 4.8e-135)))
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.6e-103], N[Not[LessEqual[x, 4.8e-135]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999996e-103 or 4.7999999999999997e-135 < 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 76.7%

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

      1. +-commutative 76.7%

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

   7. Simplified 76.7%

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

   if -2.59999999999999996e-103 < x < 4.7999999999999997e-135

   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. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out-- 100.0%

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

      11. cancel-sign-sub 100.0%

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

      12. distribute-lft-neg-in 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. neg-sub0 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

      15. associate-+l- 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. 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 96.0%

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

      1. associate-*r* 96.0%

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

      2. neg-mul-1 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in z around inf 75.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-103} \lor \neg \left(x \leq 4.8 \cdot 10^{-135}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-64} \lor \neg \left(z \leq 2.15 \cdot 10^{+17}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9e-64) (not (<= z 2.15e+17))) (+ z (* x y)) (* x (+ y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e-64) || !(z <= 2.15e+17)) {
		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 <= (-9d-64)) .or. (.not. (z <= 2.15d+17))) 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 <= -9e-64) || !(z <= 2.15e+17)) {
		tmp = z + (x * y);
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9e-64) or not (z <= 2.15e+17):
		tmp = z + (x * y)
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9e-64) || !(z <= 2.15e+17))
		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 <= -9e-64) || ~((z <= 2.15e+17)))
		tmp = z + (x * y);
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9e-64], N[Not[LessEqual[z, 2.15e+17]], $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 < -9.00000000000000019e-64 or 2.15e17 < 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. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out-- 100.0%

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

      11. cancel-sign-sub 100.0%

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

      12. distribute-lft-neg-in 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. neg-sub0 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

      15. associate-+l- 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. 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 89.6%

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

      1. associate-*r* 89.6%

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

      2. neg-mul-1 89.6%

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

   7. Simplified 89.6%

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

      1. cancel-sign-sub 89.6%

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

      2. +-commutative 89.6%

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

   9. Applied egg-rr 89.6%

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

   if -9.00000000000000019e-64 < z < 2.15e17

   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. Step-by-step derivation

      1. +-commutative 86.9%

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

   7. Simplified 86.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-64} \lor \neg \left(z \leq 2.15 \cdot 10^{+17}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.0002\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.5) (not (<= y 0.0002))) (+ z (* x y)) (- z (* x -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 0.0002)) {
		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 <= 0.0002d0))) 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 <= 0.0002)) {
		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 <= 0.0002):
		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 <= 0.0002))
		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 <= 0.0002)))
		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, 0.0002]], $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 2.0000000000000001e-4 < 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. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out-- 100.0%

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

      11. cancel-sign-sub 100.0%

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

      12. distribute-lft-neg-in 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. neg-sub0 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

      15. associate-+l- 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. 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 99.0%

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

      1. associate-*r* 99.0%

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

      2. neg-mul-1 99.0%

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

   7. Simplified 99.0%

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

      1. cancel-sign-sub 99.0%

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

      2. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

   if -0.5 < y < 2.0000000000000001e-4

   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. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out-- 100.0%

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

      11. cancel-sign-sub 100.0%

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

      12. distribute-lft-neg-in 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. neg-sub0 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

      15. associate-+l- 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. 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 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.0002\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-64}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e-64) z (if (<= z 6.8e+19) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-64) {
		tmp = z;
	} else if (z <= 6.8e+19) {
		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 <= (-2.8d-64)) then
        tmp = z
    else if (z <= 6.8d+19) 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 <= -2.8e-64) {
		tmp = z;
	} else if (z <= 6.8e+19) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.8e-64:
		tmp = z
	elif z <= 6.8e+19:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e-64)
		tmp = z;
	elseif (z <= 6.8e+19)
		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 <= -2.8e-64)
		tmp = z;
	elseif (z <= 6.8e+19)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e-64], z, If[LessEqual[z, 6.8e+19], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -2.80000000000000004e-64 or 6.8e19 < 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. +-commutative 100.0%

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

      3. remove-double-neg 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out-- 100.0%

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

      11. cancel-sign-sub 100.0%

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

      12. distribute-lft-neg-in 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. neg-sub0 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

      15. associate-+l- 100.0%

          \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

      16. neg-sub0 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. 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 89.6%

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

      1. associate-*r* 89.6%

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

      2. neg-mul-1 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in z around inf 61.7%

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

   if -2.80000000000000004e-64 < z < 6.8e19

   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. Step-by-step derivation

      1. +-commutative 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in y around 0 38.7%

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

      1. *-commutative 38.7%

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

   10. Simplified 38.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-64}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+19}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
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. +-commutative 100.0%

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

   3. remove-double-neg 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. sub-neg 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-lft-out-- 100.0%

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

   11. cancel-sign-sub 100.0%

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

   12. distribute-lft-neg-in 100.0%

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

   13. distribute-rgt-neg-in 100.0%

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

   14. neg-sub0 100.0%

       \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

   15. associate-+l- 100.0%

       \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

   16. neg-sub0 100.0%

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

   17. +-commutative 100.0%

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

   18. sub-neg 100.0%

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

   19. 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: 41.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z
end function

Java

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

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. remove-double-neg 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. sub-neg 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. *-commutative 100.0%

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

   10. distribute-lft-out-- 100.0%

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

   11. cancel-sign-sub 100.0%

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

   12. distribute-lft-neg-in 100.0%

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

   13. distribute-rgt-neg-in 100.0%

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

   14. neg-sub0 100.0%

       \[\leadsto z - x \cdot \color{blue}{\left(0 - \left(y - \frac{-1}{2}\right)\right)} \]

   15. associate-+l- 100.0%

       \[\leadsto z - x \cdot \color{blue}{\left(\left(0 - y\right) + \frac{-1}{2}\right)} \]

   16. neg-sub0 100.0%

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

   17. +-commutative 100.0%

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

   18. sub-neg 100.0%

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

   19. 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 77.3%

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

   1. associate-*r* 77.3%

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

   2. neg-mul-1 77.3%

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

7. Simplified 77.3%

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

8. Taylor expanded in z around inf 40.8%

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

9. Final simplification 40.8%

   \[\leadsto z \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))