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

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 7

Speedup: 0.1×

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: 60.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+15}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.06 \cdot 10^{-99}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-225}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-179}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-111}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 26000000:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.55e+15)
   (* x y)
   (if (<= y -1.06e-99)
     z
     (if (<= y -3.2e-225)
       (* x 0.5)
       (if (<= y 5.9e-179)
         z
         (if (<= y 5.6e-111) (* x 0.5) (if (<= y 26000000.0) z (* x y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e+15) {
		tmp = x * y;
	} else if (y <= -1.06e-99) {
		tmp = z;
	} else if (y <= -3.2e-225) {
		tmp = x * 0.5;
	} else if (y <= 5.9e-179) {
		tmp = z;
	} else if (y <= 5.6e-111) {
		tmp = x * 0.5;
	} else if (y <= 26000000.0) {
		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 <= (-1.55d+15)) then
        tmp = x * y
    else if (y <= (-1.06d-99)) then
        tmp = z
    else if (y <= (-3.2d-225)) then
        tmp = x * 0.5d0
    else if (y <= 5.9d-179) then
        tmp = z
    else if (y <= 5.6d-111) then
        tmp = x * 0.5d0
    else if (y <= 26000000.0d0) 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 <= -1.55e+15) {
		tmp = x * y;
	} else if (y <= -1.06e-99) {
		tmp = z;
	} else if (y <= -3.2e-225) {
		tmp = x * 0.5;
	} else if (y <= 5.9e-179) {
		tmp = z;
	} else if (y <= 5.6e-111) {
		tmp = x * 0.5;
	} else if (y <= 26000000.0) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.55e+15:
		tmp = x * y
	elif y <= -1.06e-99:
		tmp = z
	elif y <= -3.2e-225:
		tmp = x * 0.5
	elif y <= 5.9e-179:
		tmp = z
	elif y <= 5.6e-111:
		tmp = x * 0.5
	elif y <= 26000000.0:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.55e+15)
		tmp = Float64(x * y);
	elseif (y <= -1.06e-99)
		tmp = z;
	elseif (y <= -3.2e-225)
		tmp = Float64(x * 0.5);
	elseif (y <= 5.9e-179)
		tmp = z;
	elseif (y <= 5.6e-111)
		tmp = Float64(x * 0.5);
	elseif (y <= 26000000.0)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.55e+15)
		tmp = x * y;
	elseif (y <= -1.06e-99)
		tmp = z;
	elseif (y <= -3.2e-225)
		tmp = x * 0.5;
	elseif (y <= 5.9e-179)
		tmp = z;
	elseif (y <= 5.6e-111)
		tmp = x * 0.5;
	elseif (y <= 26000000.0)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.55e+15], N[(x * y), $MachinePrecision], If[LessEqual[y, -1.06e-99], z, If[LessEqual[y, -3.2e-225], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 5.9e-179], z, If[LessEqual[y, 5.6e-111], N[(x * 0.5), $MachinePrecision], If[LessEqual[y, 26000000.0], z, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e15 or 2.6e7 < 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 76.3%

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

   if -1.55e15 < y < -1.0599999999999999e-99 or -3.19999999999999975e-225 < y < 5.90000000000000029e-179 or 5.5999999999999999e-111 < y < 2.6e7

   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 64.5%

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

   if -1.0599999999999999e-99 < y < -3.19999999999999975e-225 or 5.90000000000000029e-179 < y < 5.5999999999999999e-111

   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 71.0%

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

   6. Taylor expanded in y around 0 71.0%

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

      1. *-commutative 71.0%

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

   8. Simplified 71.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+15}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.06 \cdot 10^{-99}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-225}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-179}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-111}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;y \leq 26000000:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -235 \lor \neg \left(x \leq 2.8 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -235.0) (not (<= x 2.8e-30))) (* x (+ y 0.5)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -235.0) || !(x <= 2.8e-30)) {
		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 <= (-235.0d0)) .or. (.not. (x <= 2.8d-30))) 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 <= -235.0) || !(x <= 2.8e-30)) {
		tmp = x * (y + 0.5);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -235.0) or not (x <= 2.8e-30):
		tmp = x * (y + 0.5)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -235.0) || !(x <= 2.8e-30))
		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 <= -235.0) || ~((x <= 2.8e-30)))
		tmp = x * (y + 0.5);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -235.0], N[Not[LessEqual[x, 2.8e-30]], $MachinePrecision]], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -235 or 2.79999999999999988e-30 < x

   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 83.6%

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

   if -235 < x < 2.79999999999999988e-30

   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%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -235 \lor \neg \left(x \leq 2.8 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 29000000:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.4e+19)
   (* x y)
   (if (<= y 29000000.0) (+ z (* x 0.5)) (* x (+ y 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.4e+19) {
		tmp = x * y;
	} else if (y <= 29000000.0) {
		tmp = z + (x * 0.5);
	} 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 (y <= (-4.4d+19)) then
        tmp = x * y
    else if (y <= 29000000.0d0) then
        tmp = z + (x * 0.5d0)
    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 (y <= -4.4e+19) {
		tmp = x * y;
	} else if (y <= 29000000.0) {
		tmp = z + (x * 0.5);
	} else {
		tmp = x * (y + 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.4e+19:
		tmp = x * y
	elif y <= 29000000.0:
		tmp = z + (x * 0.5)
	else:
		tmp = x * (y + 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.4e+19)
		tmp = Float64(x * y);
	elseif (y <= 29000000.0)
		tmp = Float64(z + Float64(x * 0.5));
	else
		tmp = Float64(x * Float64(y + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.4e+19)
		tmp = x * y;
	elseif (y <= 29000000.0)
		tmp = z + (x * 0.5);
	else
		tmp = x * (y + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.4e+19], N[(x * y), $MachinePrecision], If[LessEqual[y, 29000000.0], N[(z + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4e19

   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 83.6%

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

   if -4.4e19 < y < 2.9e7

   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 98.0%

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

      1. *-commutative 98.0%

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

   7. Simplified 98.0%

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

   if 2.9e7 < 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 x around inf 69.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 29000000:\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+129} \lor \neg \left(x \leq 1.36 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.02e+129) (not (<= x 1.36e+60))) (* x 0.5) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e+129) || !(x <= 1.36e+60)) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.02d+129)) .or. (.not. (x <= 1.36d+60))) then
        tmp = x * 0.5d0
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e+129) || !(x <= 1.36e+60)) {
		tmp = x * 0.5;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.02e+129) or not (x <= 1.36e+60):
		tmp = x * 0.5
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.02e+129) || !(x <= 1.36e+60))
		tmp = Float64(x * 0.5);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.02e+129) || ~((x <= 1.36e+60)))
		tmp = x * 0.5;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.02e+129], N[Not[LessEqual[x, 1.36e+60]], $MachinePrecision]], N[(x * 0.5), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -1.01999999999999996e129 or 1.36000000000000002e60 < x

   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 88.8%

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

   6. Taylor expanded in y around 0 40.5%

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

      1. *-commutative 40.5%

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

   8. Simplified 40.5%

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

   if -1.01999999999999996e129 < x < 1.36000000000000002e60

   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 55.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+129} \lor \neg \left(x \leq 1.36 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
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: 40.0% 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 38.8%

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

6. Final simplification 38.8%

   \[\leadsto z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))