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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 7

Speedup: 2.3×

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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot y + z\\ \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* x y) z)))
   (if (<= y -0.5) t_0 (if (<= y 4e-15) (fma x 0.5 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * y) + z;
	double tmp;
	if (y <= -0.5) {
		tmp = t_0;
	} else if (y <= 4e-15) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * y) + z)
	tmp = 0.0
	if (y <= -0.5)
		tmp = t_0;
	elseif (y <= 4e-15)
		tmp = fma(x, 0.5, z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]}, If[LessEqual[y, -0.5], t$95$0, If[LessEqual[y, 4e-15], N[(x * 0.5 + z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 4.0000000000000003e-15 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -0.5 < y < 4.0000000000000003e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + z \]

      3. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;x \cdot y + z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.68:\\ \;\;\;\;\left(y - -0.5\right) \cdot x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.68) (* (- y -0.5) x) (if (<= y 4.6e+70) (fma x 0.5 z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.68) {
		tmp = (y - -0.5) * x;
	} else if (y <= 4.6e+70) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.68)
		tmp = Float64(Float64(y - -0.5) * x);
	elseif (y <= 4.6e+70)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.68], N[(N[(y - -0.5), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 4.6e+70], N[(x * 0.5 + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot x + \color{blue}{y \cdot x} \]

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -0.680000000000000049 < y < 4.59999999999999987e70

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + z \]

      3. lower-fma.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if 4.59999999999999987e70 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.68:\\ \;\;\;\;\left(y - -0.5\right) \cdot x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.68 \cdot 10^{+16}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.68e+16) (* x y) (if (<= y 4.6e+70) (fma x 0.5 z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.68e+16) {
		tmp = x * y;
	} else if (y <= 4.6e+70) {
		tmp = fma(x, 0.5, z);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.68e+16)
		tmp = Float64(x * y);
	elseif (y <= 4.6e+70)
		tmp = fma(x, 0.5, z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.68e+16], N[(x * y), $MachinePrecision], If[LessEqual[y, 4.6e+70], N[(x * 0.5 + z), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.68e16 or 4.59999999999999987e70 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if -1.68e16 < y < 4.59999999999999987e70

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{z + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + z \]

      3. lower-fma.f64 94.2

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

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.68 \cdot 10^{+16}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-17}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-17) (* x y) (if (<= y 4e-15) (* 0.5 x) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-17) {
		tmp = x * y;
	} else if (y <= 4e-15) {
		tmp = 0.5 * x;
	} 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 <= (-4d-17)) then
        tmp = x * y
    else if (y <= 4d-15) then
        tmp = 0.5d0 * x
    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 <= -4e-17) {
		tmp = x * y;
	} else if (y <= 4e-15) {
		tmp = 0.5 * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e-17:
		tmp = x * y
	elif y <= 4e-15:
		tmp = 0.5 * x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-17)
		tmp = Float64(x * y);
	elseif (y <= 4e-15)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e-17)
		tmp = x * y;
	elseif (y <= 4e-15)
		tmp = 0.5 * x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-17], N[(x * y), $MachinePrecision], If[LessEqual[y, 4e-15], N[(0.5 * x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.00000000000000029e-17 or 4.0000000000000003e-15 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.6

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

   5. Applied rewrites 68.6%

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

   if -4.00000000000000029e-17 < y < 4.0000000000000003e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot x + \color{blue}{y \cdot x} \]

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 43.0%

      \[\leadsto 0.5 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-17}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 7: 36.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

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 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 41.5

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

5. Applied rewrites 41.5%

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

6. Final simplification 41.5%

   \[\leadsto x \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))