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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 6

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. Taylor expanded in x around 0

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

   1. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{z + \frac{1}{2} \cdot x}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 100.0

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

10. Applied rewrites 100.0%

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

Alternative 2: 84.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0255 \lor \neg \left(y \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;\left(y - -0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0255) (not (<= y 1.3e-7))) (* (- y -0.5) x) (fma 0.5 x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0255) || !(y <= 1.3e-7)) {
		tmp = (y - -0.5) * x;
	} else {
		tmp = fma(0.5, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0255) || !(y <= 1.3e-7))
		tmp = Float64(Float64(y - -0.5) * x);
	else
		tmp = fma(0.5, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0255], N[Not[LessEqual[y, 1.3e-7]], $MachinePrecision]], N[(N[(y - -0.5), $MachinePrecision] * x), $MachinePrecision], N[(0.5 * x + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0254999999999999984 or 1.29999999999999999e-7 < y

   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. lower-fma.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot y - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sign-sub N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out-- N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sign-sub-inv N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 70.7%

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

   if -0.0254999999999999984 < y < 1.29999999999999999e-7

   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. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0255 \lor \neg \left(y \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;\left(y - -0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -120000000 \lor \neg \left(y \leq 8.4 \cdot 10^{+74}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -120000000.0) (not (<= y 8.4e+74))) (* y x) (fma 0.5 x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -120000000.0) || !(y <= 8.4e+74)) {
		tmp = y * x;
	} else {
		tmp = fma(0.5, x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -120000000.0) || !(y <= 8.4e+74))
		tmp = Float64(y * x);
	else
		tmp = fma(0.5, x, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -120000000.0], N[Not[LessEqual[y, 8.4e+74]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.5 * x + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e8 or 8.3999999999999995e74 < y

   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. lower-fma.f64 25.7

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot y - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sign-sub N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out-- N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sign-sub-inv N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 73.3

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

   11. Applied rewrites 73.3%

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

   if -1.2e8 < y < 8.3999999999999995e74

   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. lower-fma.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -120000000 \lor \neg \left(y \leq 8.4 \cdot 10^{+74}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.5) || !(y <= 0.5)) {
		tmp = y * x;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.5) || !(y <= 0.5))
		tmp = Float64(y * x);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.5) || ~((y <= 0.5)))
		tmp = y * x;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.5], N[Not[LessEqual[y, 0.5]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.5 or 0.5 < y

   1. Initial program 100.0%

      \[\left(\frac{x}{2} + y \cdot x\right) + z \]
   2. 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. lower-fma.f64 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot y - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sign-sub N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out-- N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sign-sub-inv N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 66.9

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

   11. Applied rewrites 66.9%

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

   if -0.5 < y < 0.5

   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. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot y - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. mul-1-neg N/A

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

      4. fp-cancel-sign-sub N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out-- N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sign-sub-inv N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 48.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 47.3%

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

4. Final simplification 57.2%

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

Alternative 5: 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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-+.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 6: 36.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * x), $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 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. lower-fma.f64 63.7

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

5. Applied rewrites 63.7%

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

6. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot y - \frac{1}{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

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

   2. distribute-lft-out-- N/A

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

   3. mul-1-neg N/A

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

   4. fp-cancel-sign-sub N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-out-- N/A

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

   11. metadata-eval N/A

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

   12. fp-cancel-sign-sub-inv N/A

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

   13. *-lft-identity N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

8. Applied rewrites 59.6%

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

9. Taylor expanded in y around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 35.4

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

11. Applied rewrites 35.4%

    \[\leadsto \color{blue}{y \cdot x} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))