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

Percentage Accurate: 97.7% → 100.0%

Time: 6.7s

Alternatives: 9

Speedup: 1.3×

• 20.3% 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, 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 99.2%

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

   1. +-commutative 99.2%

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

   2. *-commutative 99.2%

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

   3. distribute-rgt-out-- 99.2%

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

   4. *-lft-identity 99.2%

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

   5. associate-+l- 99.2%

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

   6. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+232}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.08 \cdot 10^{-85}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+110} \lor \neg \left(x \leq 10^{+189}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -5.8e+232)
     (* x y)
     (if (<= x -1.0)
       t_0
       (if (<= x 5.08e-85)
         z
         (if (or (<= x 1.95e+110) (not (<= x 1e+189))) (* x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -5.8e+232) {
		tmp = x * y;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 5.08e-85) {
		tmp = z;
	} else if ((x <= 1.95e+110) || !(x <= 1e+189)) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	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 <= (-5.8d+232)) then
        tmp = x * y
    else if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 5.08d-85) then
        tmp = z
    else if ((x <= 1.95d+110) .or. (.not. (x <= 1d+189))) then
        tmp = x * y
    else
        tmp = t_0
    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 <= -5.8e+232) {
		tmp = x * y;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 5.08e-85) {
		tmp = z;
	} else if ((x <= 1.95e+110) || !(x <= 1e+189)) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -5.8e+232:
		tmp = x * y
	elif x <= -1.0:
		tmp = t_0
	elif x <= 5.08e-85:
		tmp = z
	elif (x <= 1.95e+110) or not (x <= 1e+189):
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -5.8e+232)
		tmp = Float64(x * y);
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= 5.08e-85)
		tmp = z;
	elseif ((x <= 1.95e+110) || !(x <= 1e+189))
		tmp = Float64(x * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * -x;
	tmp = 0.0;
	if (x <= -5.8e+232)
		tmp = x * y;
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= 5.08e-85)
		tmp = z;
	elseif ((x <= 1.95e+110) || ~((x <= 1e+189)))
		tmp = x * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * (-x)), $MachinePrecision]}, If[LessEqual[x, -5.8e+232], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 5.08e-85], z, If[Or[LessEqual[x, 1.95e+110], N[Not[LessEqual[x, 1e+189]], $MachinePrecision]], N[(x * y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.80000000000000047e232 or 5.0799999999999997e-85 < x < 1.9500000000000002e110 or 1e189 < x

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 68.3%

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

   if -5.80000000000000047e232 < x < -1 or 1.9500000000000002e110 < x < 1e189

   1. Initial program 97.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. distribute-rgt-neg-out 68.1%

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

   8. Simplified 68.1%

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

   if -1 < x < 5.0799999999999997e-85

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+232}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 5.08 \cdot 10^{-85}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+110} \lor \neg \left(x \leq 10^{+189}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+232}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.08 \cdot 10^{-85}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+191}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -5.6e+232)
     (* x y)
     (if (<= x -1.0)
       t_0
       (if (<= x 5.08e-85)
         z
         (if (<= x 1.35e+108)
           (* x (+ z y))
           (if (<= x 1.35e+191) t_0 (* x y))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -5.6e+232) {
		tmp = x * y;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 5.08e-85) {
		tmp = z;
	} else if (x <= 1.35e+108) {
		tmp = x * (z + y);
	} else if (x <= 1.35e+191) {
		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 <= (-5.6d+232)) then
        tmp = x * y
    else if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 5.08d-85) then
        tmp = z
    else if (x <= 1.35d+108) then
        tmp = x * (z + y)
    else if (x <= 1.35d+191) 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 <= -5.6e+232) {
		tmp = x * y;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 5.08e-85) {
		tmp = z;
	} else if (x <= 1.35e+108) {
		tmp = x * (z + y);
	} else if (x <= 1.35e+191) {
		tmp = t_0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -5.6e+232:
		tmp = x * y
	elif x <= -1.0:
		tmp = t_0
	elif x <= 5.08e-85:
		tmp = z
	elif x <= 1.35e+108:
		tmp = x * (z + y)
	elif x <= 1.35e+191:
		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 <= -5.6e+232)
		tmp = Float64(x * y);
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= 5.08e-85)
		tmp = z;
	elseif (x <= 1.35e+108)
		tmp = Float64(x * Float64(z + y));
	elseif (x <= 1.35e+191)
		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 <= -5.6e+232)
		tmp = x * y;
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= 5.08e-85)
		tmp = z;
	elseif (x <= 1.35e+108)
		tmp = x * (z + y);
	elseif (x <= 1.35e+191)
		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, -5.6e+232], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 5.08e-85], z, If[LessEqual[x, 1.35e+108], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+191], t$95$0, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.5999999999999998e232 or 1.34999999999999998e191 < x

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 72.8%

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

   if -5.5999999999999998e232 < x < -1 or 1.35e108 < x < 1.34999999999999998e191

   1. Initial program 97.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. distribute-rgt-neg-out 68.1%

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

   8. Simplified 68.1%

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

   if -1 < x < 5.0799999999999997e-85

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.0%

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

   if 5.0799999999999997e-85 < x < 1.35e108

   1. Initial program 99.9%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 76.6%

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

      1. mul-1-neg 76.6%

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

      2. distribute-lft-neg-out 76.6%

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

      3. *-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. fma-undefine 76.6%

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

      2. +-commutative 76.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 64.8%

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

      5. sqr-neg 64.8%

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

      6. sqrt-unprod 64.8%

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

      7. add-sqr-sqrt 64.8%

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

   9. Applied egg-rr 64.8%

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

       1. +-commutative 64.8%

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

       2. *-commutative 64.8%

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

       3. distribute-rgt-out 64.8%

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

   11. Simplified 64.8%

       \[\leadsto \color{blue}{x \cdot \left(y + z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+232}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 5.08 \cdot 10^{-85}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+191}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-5} \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3e-5) (not (<= x 5.08e-85))) (* x (- y z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e-5) || !(x <= 5.08e-85)) {
		tmp = x * (y - z);
	} 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 <= (-3d-5)) .or. (.not. (x <= 5.08d-85))) then
        tmp = x * (y - z)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e-5) || !(x <= 5.08e-85)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3e-5) or not (x <= 5.08e-85):
		tmp = x * (y - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3e-5) || !(x <= 5.08e-85))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3e-5) || ~((x <= 5.08e-85)))
		tmp = x * (y - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3e-5], N[Not[LessEqual[x, 5.08e-85]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000008e-5 or 5.0799999999999997e-85 < x

   1. Initial program 98.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. sub-neg 92.7%

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

   5. Simplified 92.7%

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

   if -3.00000000000000008e-5 < x < 5.0799999999999997e-85

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-5} \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-5} \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8e-5) (not (<= x 5.08e-85))) (* x (- y z)) (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8e-5) || !(x <= 5.08e-85)) {
		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 <= (-8d-5)) .or. (.not. (x <= 5.08d-85))) 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 <= -8e-5) || !(x <= 5.08e-85)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8e-5) or not (x <= 5.08e-85):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8e-5) || !(x <= 5.08e-85))
		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 <= -8e-5) || ~((x <= 5.08e-85)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8e-5], N[Not[LessEqual[x, 5.08e-85]], $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 < -8.00000000000000065e-5 or 5.0799999999999997e-85 < x

   1. Initial program 98.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. sub-neg 92.7%

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

   5. Simplified 92.7%

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

   if -8.00000000000000065e-5 < x < 5.0799999999999997e-85

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-5} \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.45 \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.45) (not (<= x 5.08e-85))) (* x (- y z)) (- z (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.45) || !(x <= 5.08e-85)) {
		tmp = x * (y - z);
	} else {
		tmp = z - (z * 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 <= (-0.45d0)) .or. (.not. (x <= 5.08d-85))) then
        tmp = x * (y - z)
    else
        tmp = z - (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.45) || !(x <= 5.08e-85)) {
		tmp = x * (y - z);
	} else {
		tmp = z - (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.45) or not (x <= 5.08e-85):
		tmp = x * (y - z)
	else:
		tmp = z - (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.45) || !(x <= 5.08e-85))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z - Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.45) || ~((x <= 5.08e-85)))
		tmp = x * (y - z);
	else
		tmp = z - (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.45], N[Not[LessEqual[x, 5.08e-85]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z - N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.450000000000000011 or 5.0799999999999997e-85 < x

   1. Initial program 98.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. sub-neg 92.7%

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

   5. Simplified 92.7%

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

   if -0.450000000000000011 < x < 5.0799999999999997e-85

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 77.6%

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

      1. *-commutative 77.6%

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

   7. Simplified 77.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.45 \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15500000000000 \lor \neg \left(x \leq 0.000118\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -15500000000000.0) (not (<= x 0.000118)))
   (* x (- y z))
   (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -15500000000000.0) || !(x <= 0.000118)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (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 <= (-15500000000000.0d0)) .or. (.not. (x <= 0.000118d0))) then
        tmp = x * (y - z)
    else
        tmp = z + (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -15500000000000.0) || !(x <= 0.000118)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -15500000000000.0) or not (x <= 0.000118):
		tmp = x * (y - z)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -15500000000000.0) || !(x <= 0.000118))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -15500000000000.0) || ~((x <= 0.000118)))
		tmp = x * (y - z);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -15500000000000.0], N[Not[LessEqual[x, 0.000118]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e13 or 1.18e-4 < x

   1. Initial program 98.4%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. sub-neg 99.4%

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

   5. Simplified 99.4%

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

   if -1.55e13 < x < 1.18e-4

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15500000000000 \lor \neg \left(x \leq 0.000118\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-5} \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3e-5) (not (<= x 5.08e-85))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e-5) || !(x <= 5.08e-85)) {
		tmp = x * y;
	} 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 <= (-3d-5)) .or. (.not. (x <= 5.08d-85))) then
        tmp = x * y
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e-5) || !(x <= 5.08e-85)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3e-5) or not (x <= 5.08e-85):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3e-5) || !(x <= 5.08e-85))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3e-5) || ~((x <= 5.08e-85)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3e-5], N[Not[LessEqual[x, 5.08e-85]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000008e-5 or 5.0799999999999997e-85 < x

   1. Initial program 98.7%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 53.3%

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

   if -3.00000000000000008e-5 < x < 5.0799999999999997e-85

   1. Initial program 100.0%

      \[x \cdot y + \left(1 - x\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-5} \lor \neg \left(x \leq 5.08 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.3% 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 99.2%

   \[x \cdot y + \left(1 - x\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0 36.0%

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

4. Final simplification 36.0%

   \[\leadsto z \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))