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

Percentage Accurate: 97.8% → 100.0%

Time: 4.4s

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: 97.8% 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 97.6%

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

   1. *-commutative 97.6%

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

   2. distribute-lft-out-- 97.6%

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

   3. *-rgt-identity 97.6%

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

   4. cancel-sign-sub-inv 97.6%

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

   5. +-commutative 97.6%

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

   6. associate-+r+ 97.6%

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

   7. *-commutative 97.6%

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

   8. distribute-rgt-out 100.0%

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

   9. +-commutative 100.0%

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

   10. fma-def 100.0%

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

   11. +-commutative 100.0%

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

   12. 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: 58.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+275} \lor \neg \left(x \leq 1.4 \cdot 10^{+300}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -4.5e+110)
     t_0
     (if (<= x -2.65e-65)
       (* x y)
       (if (<= x 1.06e-159)
         z
         (if (<= x 3.3e-79)
           (* x y)
           (if (<= x 1.0)
             z
             (if (or (<= x 2.65e+275) (not (<= x 1.4e+300)))
               t_0
               (* x y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -4.5e+110) {
		tmp = t_0;
	} else if (x <= -2.65e-65) {
		tmp = x * y;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 3.3e-79) {
		tmp = x * y;
	} else if (x <= 1.0) {
		tmp = z;
	} else if ((x <= 2.65e+275) || !(x <= 1.4e+300)) {
		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 <= (-4.5d+110)) then
        tmp = t_0
    else if (x <= (-2.65d-65)) then
        tmp = x * y
    else if (x <= 1.06d-159) then
        tmp = z
    else if (x <= 3.3d-79) then
        tmp = x * y
    else if (x <= 1.0d0) then
        tmp = z
    else if ((x <= 2.65d+275) .or. (.not. (x <= 1.4d+300))) 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 <= -4.5e+110) {
		tmp = t_0;
	} else if (x <= -2.65e-65) {
		tmp = x * y;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 3.3e-79) {
		tmp = x * y;
	} else if (x <= 1.0) {
		tmp = z;
	} else if ((x <= 2.65e+275) || !(x <= 1.4e+300)) {
		tmp = t_0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -4.5e+110:
		tmp = t_0
	elif x <= -2.65e-65:
		tmp = x * y
	elif x <= 1.06e-159:
		tmp = z
	elif x <= 3.3e-79:
		tmp = x * y
	elif x <= 1.0:
		tmp = z
	elif (x <= 2.65e+275) or not (x <= 1.4e+300):
		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 <= -4.5e+110)
		tmp = t_0;
	elseif (x <= -2.65e-65)
		tmp = Float64(x * y);
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 3.3e-79)
		tmp = Float64(x * y);
	elseif (x <= 1.0)
		tmp = z;
	elseif ((x <= 2.65e+275) || !(x <= 1.4e+300))
		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 <= -4.5e+110)
		tmp = t_0;
	elseif (x <= -2.65e-65)
		tmp = x * y;
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 3.3e-79)
		tmp = x * y;
	elseif (x <= 1.0)
		tmp = z;
	elseif ((x <= 2.65e+275) || ~((x <= 1.4e+300)))
		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, -4.5e+110], t$95$0, If[LessEqual[x, -2.65e-65], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.06e-159], z, If[LessEqual[x, 3.3e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.0], z, If[Or[LessEqual[x, 2.65e+275], N[Not[LessEqual[x, 1.4e+300]], $MachinePrecision]], t$95$0, N[(x * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5000000000000003e110 or 1 < x < 2.64999999999999992e275 or 1.4000000000000001e300 < x

   1. Initial program 94.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. neg-mul-1 99.8%

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

      2. sub-neg 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. mul-1-neg 69.0%

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

   7. Simplified 69.0%

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

   if -4.5000000000000003e110 < x < -2.65000000000000019e-65 or 1.06e-159 < x < 3.2999999999999998e-79 or 2.64999999999999992e275 < x < 1.4000000000000001e300

   1. Initial program 98.1%

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

   2. Taylor expanded in y around inf 65.8%

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

   if -2.65000000000000019e-65 < x < 1.06e-159 or 3.2999999999999998e-79 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+110}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+275} \lor \neg \left(x \leq 1.4 \cdot 10^{+300}\right):\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-9}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -2.6e-65)
     t_0
     (if (<= x 1.06e-159)
       z
       (if (<= x 4.2e-79) (* x y) (if (<= x 1.35e-9) z t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -2.6e-65) {
		tmp = t_0;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 4.2e-79) {
		tmp = x * y;
	} else if (x <= 1.35e-9) {
		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 <= (-2.6d-65)) then
        tmp = t_0
    else if (x <= 1.06d-159) then
        tmp = z
    else if (x <= 4.2d-79) then
        tmp = x * y
    else if (x <= 1.35d-9) 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 <= -2.6e-65) {
		tmp = t_0;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 4.2e-79) {
		tmp = x * y;
	} else if (x <= 1.35e-9) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -2.6e-65:
		tmp = t_0
	elif x <= 1.06e-159:
		tmp = z
	elif x <= 4.2e-79:
		tmp = x * y
	elif x <= 1.35e-9:
		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 <= -2.6e-65)
		tmp = t_0;
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 4.2e-79)
		tmp = Float64(x * y);
	elseif (x <= 1.35e-9)
		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 <= -2.6e-65)
		tmp = t_0;
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 4.2e-79)
		tmp = x * y;
	elseif (x <= 1.35e-9)
		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, -2.6e-65], t$95$0, If[LessEqual[x, 1.06e-159], z, If[LessEqual[x, 4.2e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.35e-9], z, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000001e-65 or 1.3500000000000001e-9 < x

   1. Initial program 95.6%

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

   2. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. sub-neg 98.0%

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

   4. Simplified 98.0%

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

   if -2.6000000000000001e-65 < x < 1.06e-159 or 4.1999999999999999e-79 < x < 1.3500000000000001e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.4%

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

   if 1.06e-159 < x < 4.1999999999999999e-79

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 66.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-9}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]

Alternative 4: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1020000000000:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -2.65e-65)
     t_0
     (if (<= x 1.06e-159)
       z
       (if (<= x 3.3e-79)
         (* x y)
         (if (<= x 1020000000000.0) (* z (- 1.0 x)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -2.65e-65) {
		tmp = t_0;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 3.3e-79) {
		tmp = x * y;
	} else if (x <= 1020000000000.0) {
		tmp = z * (1.0 - x);
	} 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 <= (-2.65d-65)) then
        tmp = t_0
    else if (x <= 1.06d-159) then
        tmp = z
    else if (x <= 3.3d-79) then
        tmp = x * y
    else if (x <= 1020000000000.0d0) then
        tmp = z * (1.0d0 - x)
    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 <= -2.65e-65) {
		tmp = t_0;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 3.3e-79) {
		tmp = x * y;
	} else if (x <= 1020000000000.0) {
		tmp = z * (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -2.65e-65:
		tmp = t_0
	elif x <= 1.06e-159:
		tmp = z
	elif x <= 3.3e-79:
		tmp = x * y
	elif x <= 1020000000000.0:
		tmp = z * (1.0 - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	tmp = 0.0
	if (x <= -2.65e-65)
		tmp = t_0;
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 3.3e-79)
		tmp = Float64(x * y);
	elseif (x <= 1020000000000.0)
		tmp = Float64(z * Float64(1.0 - x));
	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 <= -2.65e-65)
		tmp = t_0;
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 3.3e-79)
		tmp = x * y;
	elseif (x <= 1020000000000.0)
		tmp = z * (1.0 - x);
	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, -2.65e-65], t$95$0, If[LessEqual[x, 1.06e-159], z, If[LessEqual[x, 3.3e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 1020000000000.0], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.65000000000000019e-65 or 1.02e12 < x

   1. Initial program 95.5%

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

   2. Taylor expanded in x around inf 98.1%

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

      1. neg-mul-1 98.1%

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

      2. sub-neg 98.1%

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

   4. Simplified 98.1%

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

   if -2.65000000000000019e-65 < x < 1.06e-159

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.6%

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

   if 1.06e-159 < x < 3.2999999999999998e-79

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 66.9%

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

   if 3.2999999999999998e-79 < x < 1.02e12

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.5%

      \[\leadsto \color{blue}{z \cdot \left(1 - x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1020000000000:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]

Alternative 5: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-66}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-10}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e-66)
   (* x y)
   (if (<= x 1.06e-159)
     z
     (if (<= x 3.3e-79) (* x y) (if (<= x 2.7e-10) z (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e-66) {
		tmp = x * y;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 3.3e-79) {
		tmp = x * y;
	} else if (x <= 2.7e-10) {
		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 <= (-9.5d-66)) then
        tmp = x * y
    else if (x <= 1.06d-159) then
        tmp = z
    else if (x <= 3.3d-79) then
        tmp = x * y
    else if (x <= 2.7d-10) 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 <= -9.5e-66) {
		tmp = x * y;
	} else if (x <= 1.06e-159) {
		tmp = z;
	} else if (x <= 3.3e-79) {
		tmp = x * y;
	} else if (x <= 2.7e-10) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e-66:
		tmp = x * y
	elif x <= 1.06e-159:
		tmp = z
	elif x <= 3.3e-79:
		tmp = x * y
	elif x <= 2.7e-10:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e-66)
		tmp = Float64(x * y);
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 3.3e-79)
		tmp = Float64(x * y);
	elseif (x <= 2.7e-10)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e-66)
		tmp = x * y;
	elseif (x <= 1.06e-159)
		tmp = z;
	elseif (x <= 3.3e-79)
		tmp = x * y;
	elseif (x <= 2.7e-10)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e-66], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.06e-159], z, If[LessEqual[x, 3.3e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 2.7e-10], z, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000004e-66 or 1.06e-159 < x < 3.2999999999999998e-79 or 2.7e-10 < x

   1. Initial program 95.9%

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

   2. Taylor expanded in y around inf 46.5%

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

   if -9.5000000000000004e-66 < x < 1.06e-159 or 3.2999999999999998e-79 < x < 2.7e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-66}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-159}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-10}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1820 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1820.0) (not (<= x 1.0))) (* x (- y z)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1820.0) || !(x <= 1.0)) {
		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 <= (-1820.0d0)) .or. (.not. (x <= 1.0d0))) 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 <= -1820.0) || !(x <= 1.0)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1820.0) or not (x <= 1.0):
		tmp = x * (y - z)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1820.0) || !(x <= 1.0))
		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 <= -1820.0) || ~((x <= 1.0)))
		tmp = x * (y - z);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1820.0], N[Not[LessEqual[x, 1.0]], $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 < -1820 or 1 < x

   1. Initial program 95.3%

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

   2. Taylor expanded in x around inf 98.6%

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

      1. neg-mul-1 98.6%

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

      2. sub-neg 98.6%

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

   4. Simplified 98.6%

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

   if -1820 < x < 1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-out-- 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+r+ 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-out 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-def 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Step-by-step derivation

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around inf 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1820 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

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 97.6%

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

   1. *-commutative 97.6%

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

   2. distribute-lft-out-- 97.6%

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

   3. *-rgt-identity 97.6%

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

   4. cancel-sign-sub-inv 97.6%

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

   5. +-commutative 97.6%

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

   6. associate-+r+ 97.6%

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

   7. *-commutative 97.6%

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

   8. distribute-rgt-out 100.0%

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

   9. +-commutative 100.0%

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

   10. fma-def 100.0%

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

   11. +-commutative 100.0%

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

   12. 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. Step-by-step derivation

   1. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

6. 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 97.6%

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

2. Taylor expanded in x around 0 36.1%

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

3. Final simplification 36.1%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))