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

Percentage Accurate: 98.0% → 100.0%

Time: 6.4s

Alternatives: 8

Speedup: 1.3×

• 20.7% 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: 98.0% 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 98.4%

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

   1. sub-neg 98.4%

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

   2. +-commutative 98.4%

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

   3. distribute-lft1-in 98.4%

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

   4. associate-+r+ 98.4%

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

   5. +-commutative 98.4%

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

   6. distribute-lft-neg-out 98.4%

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

   7. distribute-rgt-neg-out 98.4%

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

   8. distribute-lft-out 99.9%

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

   9. fma-define 100.0%

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

   10. +-commutative 100.0%

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

   11. 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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 59.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot z\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+218}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{+45}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-41}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-64}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+43} \lor \neg \left(x \leq 2.8 \cdot 10^{+240}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x z))))
   (if (<= x -1.32e+218)
     t_0
     (if (<= x -1e+178)
       (* x y)
       (if (<= x -7.8e+135)
         t_0
         (if (<= x -5.7e+45)
           (* x y)
           (if (<= x -1.0)
             t_0
             (if (<= x -8.6e-41)
               z
               (if (<= x -2e-64)
                 (* x y)
                 (if (<= x 1.6e-97)
                   z
                   (if (or (<= x 9e+43) (not (<= x 2.8e+240)))
                     (* x y)
                     t_0)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = -(x * z);
	double tmp;
	if (x <= -1.32e+218) {
		tmp = t_0;
	} else if (x <= -1e+178) {
		tmp = x * y;
	} else if (x <= -7.8e+135) {
		tmp = t_0;
	} else if (x <= -5.7e+45) {
		tmp = x * y;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -8.6e-41) {
		tmp = z;
	} else if (x <= -2e-64) {
		tmp = x * y;
	} else if (x <= 1.6e-97) {
		tmp = z;
	} else if ((x <= 9e+43) || !(x <= 2.8e+240)) {
		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 = -(x * z)
    if (x <= (-1.32d+218)) then
        tmp = t_0
    else if (x <= (-1d+178)) then
        tmp = x * y
    else if (x <= (-7.8d+135)) then
        tmp = t_0
    else if (x <= (-5.7d+45)) then
        tmp = x * y
    else if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-8.6d-41)) then
        tmp = z
    else if (x <= (-2d-64)) then
        tmp = x * y
    else if (x <= 1.6d-97) then
        tmp = z
    else if ((x <= 9d+43) .or. (.not. (x <= 2.8d+240))) 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 = -(x * z);
	double tmp;
	if (x <= -1.32e+218) {
		tmp = t_0;
	} else if (x <= -1e+178) {
		tmp = x * y;
	} else if (x <= -7.8e+135) {
		tmp = t_0;
	} else if (x <= -5.7e+45) {
		tmp = x * y;
	} else if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -8.6e-41) {
		tmp = z;
	} else if (x <= -2e-64) {
		tmp = x * y;
	} else if (x <= 1.6e-97) {
		tmp = z;
	} else if ((x <= 9e+43) || !(x <= 2.8e+240)) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(x * z)
	tmp = 0
	if x <= -1.32e+218:
		tmp = t_0
	elif x <= -1e+178:
		tmp = x * y
	elif x <= -7.8e+135:
		tmp = t_0
	elif x <= -5.7e+45:
		tmp = x * y
	elif x <= -1.0:
		tmp = t_0
	elif x <= -8.6e-41:
		tmp = z
	elif x <= -2e-64:
		tmp = x * y
	elif x <= 1.6e-97:
		tmp = z
	elif (x <= 9e+43) or not (x <= 2.8e+240):
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(x * z))
	tmp = 0.0
	if (x <= -1.32e+218)
		tmp = t_0;
	elseif (x <= -1e+178)
		tmp = Float64(x * y);
	elseif (x <= -7.8e+135)
		tmp = t_0;
	elseif (x <= -5.7e+45)
		tmp = Float64(x * y);
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= -8.6e-41)
		tmp = z;
	elseif (x <= -2e-64)
		tmp = Float64(x * y);
	elseif (x <= 1.6e-97)
		tmp = z;
	elseif ((x <= 9e+43) || !(x <= 2.8e+240))
		tmp = Float64(x * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(x * z);
	tmp = 0.0;
	if (x <= -1.32e+218)
		tmp = t_0;
	elseif (x <= -1e+178)
		tmp = x * y;
	elseif (x <= -7.8e+135)
		tmp = t_0;
	elseif (x <= -5.7e+45)
		tmp = x * y;
	elseif (x <= -1.0)
		tmp = t_0;
	elseif (x <= -8.6e-41)
		tmp = z;
	elseif (x <= -2e-64)
		tmp = x * y;
	elseif (x <= 1.6e-97)
		tmp = z;
	elseif ((x <= 9e+43) || ~((x <= 2.8e+240)))
		tmp = x * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(x * z), $MachinePrecision])}, If[LessEqual[x, -1.32e+218], t$95$0, If[LessEqual[x, -1e+178], N[(x * y), $MachinePrecision], If[LessEqual[x, -7.8e+135], t$95$0, If[LessEqual[x, -5.7e+45], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -8.6e-41], z, If[LessEqual[x, -2e-64], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.6e-97], z, If[Or[LessEqual[x, 9e+43], N[Not[LessEqual[x, 2.8e+240]], $MachinePrecision]], N[(x * y), $MachinePrecision], t$95$0]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32e218 or -1.0000000000000001e178 < x < -7.80000000000000064e135 or -5.70000000000000027e45 < x < -1 or 9e43 < x < 2.8000000000000001e240

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. sub-neg 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in y around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. mul-1-neg 68.7%

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

   8. Simplified 68.7%

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

   if -1.32e218 < x < -1.0000000000000001e178 or -7.80000000000000064e135 < x < -5.70000000000000027e45 or -8.5999999999999997e-41 < x < -1.99999999999999993e-64 or 1.5999999999999999e-97 < x < 9e43 or 2.8000000000000001e240 < x

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf 68.2%

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

   if -1 < x < -8.5999999999999997e-41 or -1.99999999999999993e-64 < x < 1.5999999999999999e-97

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+218}:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+135}:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{+45}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;-x \cdot z\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-41}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-64}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+43} \lor \neg \left(x \leq 2.8 \cdot 10^{+240}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -0.0075)
     t_0
     (if (<= x -6.5e-44)
       z
       (if (<= x -1.8e-64) (* x y) (if (<= x 1.6e-97) z t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -0.0075) {
		tmp = t_0;
	} else if (x <= -6.5e-44) {
		tmp = z;
	} else if (x <= -1.8e-64) {
		tmp = x * y;
	} else if (x <= 1.6e-97) {
		tmp = z;
	} 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 = x * (y - z)
    if (x <= (-0.0075d0)) then
        tmp = t_0
    else if (x <= (-6.5d-44)) then
        tmp = z
    else if (x <= (-1.8d-64)) then
        tmp = x * y
    else if (x <= 1.6d-97) then
        tmp = z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -0.0075) {
		tmp = t_0;
	} else if (x <= -6.5e-44) {
		tmp = z;
	} else if (x <= -1.8e-64) {
		tmp = x * y;
	} else if (x <= 1.6e-97) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -0.0075:
		tmp = t_0
	elif x <= -6.5e-44:
		tmp = z
	elif x <= -1.8e-64:
		tmp = x * y
	elif x <= 1.6e-97:
		tmp = z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	tmp = 0.0
	if (x <= -0.0075)
		tmp = t_0;
	elseif (x <= -6.5e-44)
		tmp = z;
	elseif (x <= -1.8e-64)
		tmp = Float64(x * y);
	elseif (x <= 1.6e-97)
		tmp = z;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y - z);
	tmp = 0.0;
	if (x <= -0.0075)
		tmp = t_0;
	elseif (x <= -6.5e-44)
		tmp = z;
	elseif (x <= -1.8e-64)
		tmp = x * y;
	elseif (x <= 1.6e-97)
		tmp = z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0075], t$95$0, If[LessEqual[x, -6.5e-44], z, If[LessEqual[x, -1.8e-64], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.6e-97], z, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0074999999999999997 or 1.5999999999999999e-97 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. sub-neg 93.9%

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

   5. Simplified 93.9%

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

   if -0.0074999999999999997 < x < -6.5e-44 or -1.7999999999999999e-64 < x < 1.5999999999999999e-97

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.0%

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

   if -6.5e-44 < x < -1.7999999999999999e-64

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-64}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ t_1 := z \cdot \left(1 - x\right)\\ \mathbf{if}\;x \leq -33000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-64}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))) (t_1 (* z (- 1.0 x))))
   (if (<= x -33000000000.0)
     t_0
     (if (<= x -8.5e-44)
       t_1
       (if (<= x -1.3e-64) (* x y) (if (<= x 1.6e-97) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double t_1 = z * (1.0 - x);
	double tmp;
	if (x <= -33000000000.0) {
		tmp = t_0;
	} else if (x <= -8.5e-44) {
		tmp = t_1;
	} else if (x <= -1.3e-64) {
		tmp = x * y;
	} else if (x <= 1.6e-97) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * (y - z)
    t_1 = z * (1.0d0 - x)
    if (x <= (-33000000000.0d0)) then
        tmp = t_0
    else if (x <= (-8.5d-44)) then
        tmp = t_1
    else if (x <= (-1.3d-64)) then
        tmp = x * y
    else if (x <= 1.6d-97) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double t_1 = z * (1.0 - x);
	double tmp;
	if (x <= -33000000000.0) {
		tmp = t_0;
	} else if (x <= -8.5e-44) {
		tmp = t_1;
	} else if (x <= -1.3e-64) {
		tmp = x * y;
	} else if (x <= 1.6e-97) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	t_1 = z * (1.0 - x)
	tmp = 0
	if x <= -33000000000.0:
		tmp = t_0
	elif x <= -8.5e-44:
		tmp = t_1
	elif x <= -1.3e-64:
		tmp = x * y
	elif x <= 1.6e-97:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	t_1 = Float64(z * Float64(1.0 - x))
	tmp = 0.0
	if (x <= -33000000000.0)
		tmp = t_0;
	elseif (x <= -8.5e-44)
		tmp = t_1;
	elseif (x <= -1.3e-64)
		tmp = Float64(x * y);
	elseif (x <= 1.6e-97)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y - z);
	t_1 = z * (1.0 - x);
	tmp = 0.0;
	if (x <= -33000000000.0)
		tmp = t_0;
	elseif (x <= -8.5e-44)
		tmp = t_1;
	elseif (x <= -1.3e-64)
		tmp = x * y;
	elseif (x <= 1.6e-97)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -33000000000.0], t$95$0, If[LessEqual[x, -8.5e-44], t$95$1, If[LessEqual[x, -1.3e-64], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.6e-97], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3e10 or 1.5999999999999999e-97 < x

   1. Initial program 97.3%

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

   3. Taylor expanded in x around inf 94.7%

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

      1. mul-1-neg 94.7%

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

      2. sub-neg 94.7%

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

   5. Simplified 94.7%

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

   if -3.3e10 < x < -8.5000000000000002e-44 or -1.3e-64 < x < 1.5999999999999999e-97

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.7%

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

   if -8.5000000000000002e-44 < x < -1.3e-64

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -33000000000:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-44}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-64}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-97}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -33000000000 \lor \neg \left(x \leq -1.7 \cdot 10^{-42}\right) \land \left(x \leq -1.8 \cdot 10^{-64} \lor \neg \left(x \leq 1.5 \cdot 10^{-97}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -33000000000.0)
         (and (not (<= x -1.7e-42)) (or (<= x -1.8e-64) (not (<= x 1.5e-97)))))
   (* x y)
   z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -33000000000.0) || (!(x <= -1.7e-42) && ((x <= -1.8e-64) || !(x <= 1.5e-97)))) {
		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 <= (-33000000000.0d0)) .or. (.not. (x <= (-1.7d-42))) .and. (x <= (-1.8d-64)) .or. (.not. (x <= 1.5d-97))) 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 <= -33000000000.0) || (!(x <= -1.7e-42) && ((x <= -1.8e-64) || !(x <= 1.5e-97)))) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -33000000000.0) or (not (x <= -1.7e-42) and ((x <= -1.8e-64) or not (x <= 1.5e-97))):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -33000000000.0) || (!(x <= -1.7e-42) && ((x <= -1.8e-64) || !(x <= 1.5e-97))))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -33000000000.0) || (~((x <= -1.7e-42)) && ((x <= -1.8e-64) || ~((x <= 1.5e-97)))))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -33000000000.0], And[N[Not[LessEqual[x, -1.7e-42]], $MachinePrecision], Or[LessEqual[x, -1.8e-64], N[Not[LessEqual[x, 1.5e-97]], $MachinePrecision]]]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -3.3e10 or -1.70000000000000011e-42 < x < -1.7999999999999999e-64 or 1.50000000000000012e-97 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in y around inf 55.2%

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

   if -3.3e10 < x < -1.70000000000000011e-42 or -1.7999999999999999e-64 < x < 1.50000000000000012e-97

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -33000000000 \lor \neg \left(x \leq -1.7 \cdot 10^{-42}\right) \land \left(x \leq -1.8 \cdot 10^{-64} \lor \neg \left(x \leq 1.5 \cdot 10^{-97}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.46\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.46))) (* x (- y z)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.46)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 0.46d0))) 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 <= -1.0) || !(x <= 0.46)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.46):
		tmp = x * (y - z)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.46))
		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 <= -1.0) || ~((x <= 0.46)))
		tmp = x * (y - z);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.46]], $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 or 0.46000000000000002 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf 97.5%

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

      1. mul-1-neg 97.5%

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

      2. sub-neg 97.5%

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

   5. Simplified 97.5%

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

   if -1 < x < 0.46000000000000002

   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. remove-double-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

      5. neg-sub0 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. *-lft-identity 100.0%

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

      11. associate-+l- 100.0%

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

      12. 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 y around inf 96.9%

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

      1. neg-mul-1 96.9%

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

      2. +-commutative 96.9%

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

      3. distribute-rgt-in 96.9%

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

      4. *-commutative 96.9%

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

      5. neg-mul-1 96.9%

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

      6. associate-*r* 96.9%

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

      7. associate-/l* 92.1%

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

      8. associate-*r* 91.8%

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

      9. *-commutative 91.8%

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

      10. *-commutative 91.8%

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

      11. distribute-lft-out 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in y around inf 97.6%

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

      1. associate-*r* 97.6%

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

      2. mul-1-neg 97.6%

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

   10. Simplified 97.6%

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

       1. cancel-sign-sub 97.6%

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

       2. +-commutative 97.6%

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

   12. Applied egg-rr 97.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.46\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

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 98.4%

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

   1. +-commutative 98.4%

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

   2. remove-double-neg 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. neg-sub0 98.4%

      \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

   5. neg-sub0 98.4%

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

   6. *-commutative 98.4%

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

   7. distribute-lft-neg-in 98.4%

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

   8. remove-double-neg 98.4%

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

   9. distribute-rgt-out-- 98.4%

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

   10. *-lft-identity 98.4%

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

   11. associate-+l- 98.4%

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

   12. distribute-lft-out-- 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 36.5% 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 98.4%

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

3. Taylor expanded in x around 0 35.3%

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

4. Final simplification 35.3%

   \[\leadsto z \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))