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

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 7

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%

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

   9. distribute-rgt-out-- 100.0%

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

   10. fma-define 100.0%

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

   11. sub-neg 100.0%

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

   12. metadata-eval 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 59.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -2.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-194}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-74}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-23}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e+31)
   (* x y)
   (if (<= y -2.06e-159)
     z
     (if (<= y 2.65e-194)
       (* x 0.5)
       (if (<= y 2.2e-74)
         z
         (if (<= y 3.1e-23) (* x 0.5) (if (<= y 2.7e+107) z (* x y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e+31) {
		tmp = x * y;
	} else if (y <= -2.06e-159) {
		tmp = z;
	} else if (y <= 2.65e-194) {
		tmp = x * 0.5;
	} else if (y <= 2.2e-74) {
		tmp = z;
	} else if (y <= 3.1e-23) {
		tmp = x * 0.5;
	} else if (y <= 2.7e+107) {
		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 <= (-3.8d+31)) then
        tmp = x * y
    else if (y <= (-2.06d-159)) then
        tmp = z
    else if (y <= 2.65d-194) then
        tmp = x * 0.5d0
    else if (y <= 2.2d-74) then
        tmp = z
    else if (y <= 3.1d-23) then
        tmp = x * 0.5d0
    else if (y <= 2.7d+107) 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 <= -3.8e+31) {
		tmp = x * y;
	} else if (y <= -2.06e-159) {
		tmp = z;
	} else if (y <= 2.65e-194) {
		tmp = x * 0.5;
	} else if (y <= 2.2e-74) {
		tmp = z;
	} else if (y <= 3.1e-23) {
		tmp = x * 0.5;
	} else if (y <= 2.7e+107) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.8e+31:
		tmp = x * y
	elif y <= -2.06e-159:
		tmp = z
	elif y <= 2.65e-194:
		tmp = x * 0.5
	elif y <= 2.2e-74:
		tmp = z
	elif y <= 3.1e-23:
		tmp = x * 0.5
	elif y <= 2.7e+107:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e+31)
		tmp = Float64(x * y);
	elseif (y <= -2.06e-159)
		tmp = z;
	elseif (y <= 2.65e-194)
		tmp = Float64(x * 0.5);
	elseif (y <= 2.2e-74)
		tmp = z;
	elseif (y <= 3.1e-23)
		tmp = Float64(x * 0.5);
	elseif (y <= 2.7e+107)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.8e+31)
		tmp = x * y;
	elseif (y <= -2.06e-159)
		tmp = z;
	elseif (y <= 2.65e-194)
		tmp = x * 0.5;
	elseif (y <= 2.2e-74)
		tmp = z;
	elseif (y <= 3.1e-23)
		tmp = x * 0.5;
	elseif (y <= 2.7e+107)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e+31], N[(x * y), $MachinePrecision], If[LessEqual[y, -2.06e-159], z, If[LessEqual[y, 2.65e-194], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 2.2e-74], z, If[LessEqual[y, 3.1e-23], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 2.7e+107], z, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000001e31 or 2.7000000000000001e107 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 67.6%

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

   if -3.8000000000000001e31 < y < -2.05999999999999989e-159 or 2.6500000000000001e-194 < y < 2.2000000000000001e-74 or 3.0999999999999999e-23 < y < 2.7000000000000001e107

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 63.3%

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

   if -2.05999999999999989e-159 < y < 2.6500000000000001e-194 or 2.2000000000000001e-74 < y < 3.0999999999999999e-23

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 63.3%

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

   6. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   8. Simplified 63.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -2.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-194}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-74}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-23}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+23}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.5e+23) z (if (<= z 6e+49) (* x (+ y 0.5)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5e+23) {
		tmp = z;
	} else if (z <= 6e+49) {
		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 <= (-6.5d+23)) then
        tmp = z
    else if (z <= 6d+49) 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 <= -6.5e+23) {
		tmp = z;
	} else if (z <= 6e+49) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.5e+23:
		tmp = z
	elif z <= 6e+49:
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.5e+23)
		tmp = z;
	elseif (z <= 6e+49)
		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 <= -6.5e+23)
		tmp = z;
	elseif (z <= 6e+49)
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.5e+23], z, If[LessEqual[z, 6e+49], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999996e23 or 6.0000000000000005e49 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 72.8%

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

   if -6.4999999999999996e23 < z < 6.0000000000000005e49

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 81.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+23}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9e-5)
   (* x (+ y 0.5))
   (if (<= y 1.85e+109) (+ z (* x 0.5)) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9e-5) {
		tmp = x * (y + 0.5);
	} else if (y <= 1.85e+109) {
		tmp = z + (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 <= (-9d-5)) then
        tmp = x * (y + 0.5d0)
    else if (y <= 1.85d+109) then
        tmp = z + (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 <= -9e-5) {
		tmp = x * (y + 0.5);
	} else if (y <= 1.85e+109) {
		tmp = z + (x * 0.5);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -9e-5:
		tmp = x * (y + 0.5)
	elif y <= 1.85e+109:
		tmp = z + (x * 0.5)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9e-5)
		tmp = Float64(x * Float64(y + 0.5));
	elseif (y <= 1.85e+109)
		tmp = Float64(z + 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 <= -9e-5)
		tmp = x * (y + 0.5);
	elseif (y <= 1.85e+109)
		tmp = z + (x * 0.5);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9e-5], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+109], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.00000000000000057e-5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. neg-mul-1 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. distribute-rgt-out-- 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 66.5%

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

   if -9.00000000000000057e-5 < y < 1.8500000000000001e109

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 94.3%

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

      1. *-commutative 94.3%

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

   7. Simplified 94.3%

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

   if 1.8500000000000001e109 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 66.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+109}:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -780000000:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -780000000.0) z (if (<= z 3.6e-11) (* x 0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -780000000.0) {
		tmp = z;
	} else if (z <= 3.6e-11) {
		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 <= (-780000000.0d0)) then
        tmp = z
    else if (z <= 3.6d-11) 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 <= -780000000.0) {
		tmp = z;
	} else if (z <= 3.6e-11) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -780000000.0:
		tmp = z
	elif z <= 3.6e-11:
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -780000000.0)
		tmp = z;
	elseif (z <= 3.6e-11)
		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 <= -780000000.0)
		tmp = z;
	elseif (z <= 3.6e-11)
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -780000000.0], z, If[LessEqual[z, 3.6e-11], N[(x * 0.5), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8e8 or 3.59999999999999985e-11 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 68.4%

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

   if -7.8e8 < z < 3.59999999999999985e-11

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 83.4%

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

   6. Taylor expanded in y around 0 43.9%

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

      1. *-commutative 43.9%

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

   8. Simplified 43.9%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -780000000:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-/l* 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-rgt-out-- 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 7: 40.4% 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%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-/l* 100.0%

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

   8. *-commutative 100.0%

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

   9. distribute-rgt-out-- 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 45.3%

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

6. Final simplification 45.3%

   \[\leadsto z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))