Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3

Percentage Accurate: 97.7% → 100.0%

Time: 4.2s

Alternatives: 8

Speedup: 1.3×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - 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 * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Step-by-step derivation

   1. sub-neg 96.9%

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

   2. +-commutative 96.9%

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

   3. distribute-lft1-in 96.9%

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

   4. associate-+r+ 96.9%

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

   5. +-commutative 96.9%

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

   6. *-commutative 96.9%

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

   7. neg-mul-1 96.9%

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

   8. associate-*r* 96.9%

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

   9. *-commutative 96.9%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+246}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+107}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -1.9e+290)
     t_0
     (if (<= x -1.6e+246)
       (* x y)
       (if (<= x -6e+144)
         t_0
         (if (<= x -2.6e+107)
           (* x y)
           (if (<= x -2.4e+59)
             t_0
             (if (<= x -1.4e-79)
               (* x y)
               (if (<= x 1.15e-8)
                 z
                 (if (<= x 3.8e+105)
                   (* x y)
                   (if (<= x 1.8e+216) t_0 (* x y))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -1.9e+290) {
		tmp = t_0;
	} else if (x <= -1.6e+246) {
		tmp = x * y;
	} else if (x <= -6e+144) {
		tmp = t_0;
	} else if (x <= -2.6e+107) {
		tmp = x * y;
	} else if (x <= -2.4e+59) {
		tmp = t_0;
	} else if (x <= -1.4e-79) {
		tmp = x * y;
	} else if (x <= 1.15e-8) {
		tmp = z;
	} else if (x <= 3.8e+105) {
		tmp = x * y;
	} else if (x <= 1.8e+216) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = z * -x
    if (x <= (-1.9d+290)) then
        tmp = t_0
    else if (x <= (-1.6d+246)) then
        tmp = x * y
    else if (x <= (-6d+144)) then
        tmp = t_0
    else if (x <= (-2.6d+107)) then
        tmp = x * y
    else if (x <= (-2.4d+59)) then
        tmp = t_0
    else if (x <= (-1.4d-79)) then
        tmp = x * y
    else if (x <= 1.15d-8) then
        tmp = z
    else if (x <= 3.8d+105) then
        tmp = x * y
    else if (x <= 1.8d+216) then
        tmp = t_0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -1.9e+290) {
		tmp = t_0;
	} else if (x <= -1.6e+246) {
		tmp = x * y;
	} else if (x <= -6e+144) {
		tmp = t_0;
	} else if (x <= -2.6e+107) {
		tmp = x * y;
	} else if (x <= -2.4e+59) {
		tmp = t_0;
	} else if (x <= -1.4e-79) {
		tmp = x * y;
	} else if (x <= 1.15e-8) {
		tmp = z;
	} else if (x <= 3.8e+105) {
		tmp = x * y;
	} else if (x <= 1.8e+216) {
		tmp = t_0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -1.9e+290:
		tmp = t_0
	elif x <= -1.6e+246:
		tmp = x * y
	elif x <= -6e+144:
		tmp = t_0
	elif x <= -2.6e+107:
		tmp = x * y
	elif x <= -2.4e+59:
		tmp = t_0
	elif x <= -1.4e-79:
		tmp = x * y
	elif x <= 1.15e-8:
		tmp = z
	elif x <= 3.8e+105:
		tmp = x * y
	elif x <= 1.8e+216:
		tmp = t_0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -1.9e+290)
		tmp = t_0;
	elseif (x <= -1.6e+246)
		tmp = Float64(x * y);
	elseif (x <= -6e+144)
		tmp = t_0;
	elseif (x <= -2.6e+107)
		tmp = Float64(x * y);
	elseif (x <= -2.4e+59)
		tmp = t_0;
	elseif (x <= -1.4e-79)
		tmp = Float64(x * y);
	elseif (x <= 1.15e-8)
		tmp = z;
	elseif (x <= 3.8e+105)
		tmp = Float64(x * y);
	elseif (x <= 1.8e+216)
		tmp = t_0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (x <= -1.9e+290)
		tmp = t_0;
	elseif (x <= -1.6e+246)
		tmp = x * y;
	elseif (x <= -6e+144)
		tmp = t_0;
	elseif (x <= -2.6e+107)
		tmp = x * y;
	elseif (x <= -2.4e+59)
		tmp = t_0;
	elseif (x <= -1.4e-79)
		tmp = x * y;
	elseif (x <= 1.15e-8)
		tmp = z;
	elseif (x <= 3.8e+105)
		tmp = x * y;
	elseif (x <= 1.8e+216)
		tmp = t_0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[x, -1.9e+290], t$95$0, If[LessEqual[x, -1.6e+246], N[(x * y), $MachinePrecision], If[LessEqual[x, -6e+144], t$95$0, If[LessEqual[x, -2.6e+107], N[(x * y), $MachinePrecision], If[LessEqual[x, -2.4e+59], t$95$0, If[LessEqual[x, -1.4e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.15e-8], z, If[LessEqual[x, 3.8e+105], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.8e+216], t$95$0, N[(x * y), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9e290 or -1.60000000000000007e246 < x < -5.9999999999999998e144 or -2.6000000000000001e107 < x < -2.4000000000000002e59 or 3.8e105 < x < 1.8000000000000001e216

   1. Initial program 95.1%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Step-by-step derivation

      1. sub-neg 95.1%

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

      2. +-commutative 95.1%

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

      3. distribute-lft1-in 95.1%

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

      4. associate-+r+ 95.1%

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

      5. +-commutative 95.1%

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

      6. *-commutative 95.1%

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

      7. neg-mul-1 95.1%

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

      8. associate-*r* 95.1%

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

      9. *-commutative 95.1%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   5. Taylor expanded in y around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. distribute-rgt-neg-in 72.2%

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

   7. Simplified 72.2%

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

   if -1.9e290 < x < -1.60000000000000007e246 or -5.9999999999999998e144 < x < -2.6000000000000001e107 or -2.4000000000000002e59 < x < -1.40000000000000006e-79 or 1.15e-8 < x < 3.8e105 or 1.8000000000000001e216 < x

   1. Initial program 94.3%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around inf 69.6%

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

   if -1.40000000000000006e-79 < x < 1.15e-8

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in x around 0 81.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+290}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+246}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+144}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+107}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-79} \lor \neg \left(x \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4e-79) (not (<= x 5.5e-8))) (* x (- y z)) (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4e-79) || !(x <= 5.5e-8)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - 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 ((x <= (-1.4d-79)) .or. (.not. (x <= 5.5d-8))) then
        tmp = x * (y - z)
    else
        tmp = z * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4e-79) || !(x <= 5.5e-8)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4e-79) or not (x <= 5.5e-8):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4e-79) || !(x <= 5.5e-8))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4e-79) || ~((x <= 5.5e-8)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4e-79], N[Not[LessEqual[x, 5.5e-8]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000006e-79 or 5.5000000000000003e-8 < x

   1. Initial program 94.6%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Step-by-step derivation

      1. sub-neg 94.6%

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

      2. +-commutative 94.6%

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

      3. distribute-lft1-in 94.6%

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

      4. associate-+r+ 94.6%

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

      5. +-commutative 94.6%

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

      6. *-commutative 94.6%

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

      7. neg-mul-1 94.6%

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

      8. associate-*r* 94.6%

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

      9. *-commutative 94.6%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.2%

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

   if -1.40000000000000006e-79 < x < 5.5000000000000003e-8

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around 0 82.0%

      \[\leadsto \color{blue}{z \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-79} \lor \neg \left(x \leq 5.5 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 4: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-79} \lor \neg \left(x \leq 9.6 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e-79) (not (<= x 9.6e-6))) (* x (- y z)) (- z (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-79) || !(x <= 9.6e-6)) {
		tmp = x * (y - z);
	} else {
		tmp = z - (x * 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 <= (-2d-79)) .or. (.not. (x <= 9.6d-6))) then
        tmp = x * (y - z)
    else
        tmp = z - (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-79) || !(x <= 9.6e-6)) {
		tmp = x * (y - z);
	} else {
		tmp = z - (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e-79) or not (x <= 9.6e-6):
		tmp = x * (y - z)
	else:
		tmp = z - (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e-79) || !(x <= 9.6e-6))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z - Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e-79) || ~((x <= 9.6e-6)))
		tmp = x * (y - z);
	else
		tmp = z - (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e-79], N[Not[LessEqual[x, 9.6e-6]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z - N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-79 or 9.5999999999999996e-6 < x

   1. Initial program 94.6%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Step-by-step derivation

      1. sub-neg 94.6%

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

      2. +-commutative 94.6%

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

      3. distribute-lft1-in 94.6%

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

      4. associate-+r+ 94.6%

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

      5. +-commutative 94.6%

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

      6. *-commutative 94.6%

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

      7. neg-mul-1 94.6%

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

      8. associate-*r* 94.6%

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

      9. *-commutative 94.6%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.2%

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

   if -2e-79 < x < 9.5999999999999996e-6

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft1-in 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in y around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. mul-1-neg 82.0%

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

      3. unsub-neg 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-79} \lor \neg \left(x \leq 9.6 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - x \cdot z\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+68}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e+68) (* x y) (if (<= y 3.3e+44) (* z (- 1.0 x)) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e+68) {
		tmp = x * y;
	} else if (y <= 3.3e+44) {
		tmp = z * (1.0 - 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 <= (-4.5d+68)) then
        tmp = x * y
    else if (y <= 3.3d+44) then
        tmp = z * (1.0d0 - 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 <= -4.5e+68) {
		tmp = x * y;
	} else if (y <= 3.3e+44) {
		tmp = z * (1.0 - x);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.5e+68:
		tmp = x * y
	elif y <= 3.3e+44:
		tmp = z * (1.0 - x)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.5e+68)
		tmp = Float64(x * y);
	elseif (y <= 3.3e+44)
		tmp = Float64(z * Float64(1.0 - x));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.5e+68)
		tmp = x * y;
	elseif (y <= 3.3e+44)
		tmp = z * (1.0 - x);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.5e+68], N[(x * y), $MachinePrecision], If[LessEqual[y, 3.3e+44], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000003e68 or 3.30000000000000013e44 < y

   1. Initial program 91.9%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around inf 78.2%

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

   if -4.5000000000000003e68 < y < 3.30000000000000013e44

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around 0 86.9%

      \[\leadsto \color{blue}{z \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+68}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+44}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.25e-79) (* x y) (if (<= x 1.4e-8) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.25e-79) {
		tmp = x * y;
	} else if (x <= 1.4e-8) {
		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 (x <= (-2.25d-79)) then
        tmp = x * y
    else if (x <= 1.4d-8) 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 (x <= -2.25e-79) {
		tmp = x * y;
	} else if (x <= 1.4e-8) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.25e-79:
		tmp = x * y
	elif x <= 1.4e-8:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.25e-79)
		tmp = Float64(x * y);
	elseif (x <= 1.4e-8)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.25e-79)
		tmp = x * y;
	elseif (x <= 1.4e-8)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.25e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.4e-8], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2500000000000001e-79 or 1.4e-8 < x

   1. Initial program 94.6%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in y around inf 52.7%

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

   if -2.2500000000000001e-79 < x < 1.4e-8

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]

   2. Taylor expanded in x around 0 81.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return z + (x * (y - 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 + (x * (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Step-by-step derivation

   1. sub-neg 96.9%

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

   2. +-commutative 96.9%

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

   3. distribute-lft1-in 96.9%

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

   4. associate-+r+ 96.9%

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

   5. +-commutative 96.9%

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

   6. *-commutative 96.9%

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

   7. neg-mul-1 96.9%

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

   8. associate-*r* 96.9%

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

   9. *-commutative 96.9%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

   \[\leadsto z + x \cdot \left(y - z\right) \]

Alternative 8: 36.2% 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 96.9%

   \[x \cdot y + \left(1 - x\right) \cdot z \]

2. Taylor expanded in x around 0 37.1%

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

3. Final simplification 37.1%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023182 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))