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

Percentage Accurate: 97.9% → 100.0%

Time: 3.6s

Alternatives: 7

Speedup: 1.3×

• 21.0% 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: 97.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. sub-neg 96.1%

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

   2. +-commutative 96.1%

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

   3. distribute-lft1-in 96.1%

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

   4. associate-+r+ 96.1%

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

   5. +-commutative 96.1%

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

   6. *-commutative 96.1%

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

   7. neg-mul-1 96.1%

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

   8. associate-*r* 96.1%

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

   9. *-commutative 96.1%

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

   10. distribute-rgt-out 100.0%

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

   11. fma-def 100.0%

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

   12. +-commutative 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+155}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -250:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-114}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-128}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-44}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+111} \lor \neg \left(x \leq 1.22 \cdot 10^{+219}\right) \land x \leq 4 \cdot 10^{+254}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -4.2e+155)
     (* y x)
     (if (<= x -250.0)
       t_0
       (if (<= x -1.55e-114)
         z
         (if (<= x -6e-128)
           (* y x)
           (if (<= x 2.7e-44)
             z
             (if (or (<= x 4.1e+111)
                     (and (not (<= x 1.22e+219)) (<= x 4e+254)))
               (* y x)
               t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -4.2e+155) {
		tmp = y * x;
	} else if (x <= -250.0) {
		tmp = t_0;
	} else if (x <= -1.55e-114) {
		tmp = z;
	} else if (x <= -6e-128) {
		tmp = y * x;
	} else if (x <= 2.7e-44) {
		tmp = z;
	} else if ((x <= 4.1e+111) || (!(x <= 1.22e+219) && (x <= 4e+254))) {
		tmp = y * 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 = z * -x
    if (x <= (-4.2d+155)) then
        tmp = y * x
    else if (x <= (-250.0d0)) then
        tmp = t_0
    else if (x <= (-1.55d-114)) then
        tmp = z
    else if (x <= (-6d-128)) then
        tmp = y * x
    else if (x <= 2.7d-44) then
        tmp = z
    else if ((x <= 4.1d+111) .or. (.not. (x <= 1.22d+219)) .and. (x <= 4d+254)) then
        tmp = y * 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 = z * -x;
	double tmp;
	if (x <= -4.2e+155) {
		tmp = y * x;
	} else if (x <= -250.0) {
		tmp = t_0;
	} else if (x <= -1.55e-114) {
		tmp = z;
	} else if (x <= -6e-128) {
		tmp = y * x;
	} else if (x <= 2.7e-44) {
		tmp = z;
	} else if ((x <= 4.1e+111) || (!(x <= 1.22e+219) && (x <= 4e+254))) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -4.2e+155:
		tmp = y * x
	elif x <= -250.0:
		tmp = t_0
	elif x <= -1.55e-114:
		tmp = z
	elif x <= -6e-128:
		tmp = y * x
	elif x <= 2.7e-44:
		tmp = z
	elif (x <= 4.1e+111) or (not (x <= 1.22e+219) and (x <= 4e+254)):
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -4.2e+155)
		tmp = Float64(y * x);
	elseif (x <= -250.0)
		tmp = t_0;
	elseif (x <= -1.55e-114)
		tmp = z;
	elseif (x <= -6e-128)
		tmp = Float64(y * x);
	elseif (x <= 2.7e-44)
		tmp = z;
	elseif ((x <= 4.1e+111) || (!(x <= 1.22e+219) && (x <= 4e+254)))
		tmp = Float64(y * x);
	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 <= -4.2e+155)
		tmp = y * x;
	elseif (x <= -250.0)
		tmp = t_0;
	elseif (x <= -1.55e-114)
		tmp = z;
	elseif (x <= -6e-128)
		tmp = y * x;
	elseif (x <= 2.7e-44)
		tmp = z;
	elseif ((x <= 4.1e+111) || (~((x <= 1.22e+219)) && (x <= 4e+254)))
		tmp = y * x;
	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, -4.2e+155], N[(y * x), $MachinePrecision], If[LessEqual[x, -250.0], t$95$0, If[LessEqual[x, -1.55e-114], z, If[LessEqual[x, -6e-128], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.7e-44], z, If[Or[LessEqual[x, 4.1e+111], And[N[Not[LessEqual[x, 1.22e+219]], $MachinePrecision], LessEqual[x, 4e+254]]], N[(y * x), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2e155 or -1.55e-114 < x < -5.99999999999999956e-128 or 2.6999999999999999e-44 < x < 4.09999999999999986e111 or 1.22000000000000004e219 < x < 3.9999999999999997e254

   1. Initial program 89.6%

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

   2. Taylor expanded in y around inf 73.7%

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

   if -4.2e155 < x < -250 or 4.09999999999999986e111 < x < 1.22000000000000004e219 or 3.9999999999999997e254 < x

   1. Initial program 95.3%

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

      1. sub-neg 95.3%

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

      2. +-commutative 95.3%

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

      3. distribute-lft1-in 95.3%

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

      4. associate-+r+ 95.3%

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

      5. +-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. neg-mul-1 95.3%

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

      8. associate-*r* 95.3%

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

      9. *-commutative 95.3%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.1%

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

   5. Taylor expanded in y around 0 67.3%

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

      1. mul-1-neg 67.3%

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

      2. distribute-rgt-neg-in 67.3%

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

   7. Simplified 67.3%

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

   if -250 < x < -1.55e-114 or -5.99999999999999956e-128 < x < 2.6999999999999999e-44

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.5%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+155}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -250:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-114}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-128}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-44}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+111} \lor \neg \left(x \leq 1.22 \cdot 10^{+219}\right) \land x \leq 4 \cdot 10^{+254}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - z\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-115}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-128}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-44}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y z) x)))
   (if (<= x -2.3e-45)
     t_0
     (if (<= x -1.85e-115)
       z
       (if (<= x -6e-128) (* y x) (if (<= x 3e-44) (* z (- 1.0 x)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (y - z) * x;
	double tmp;
	if (x <= -2.3e-45) {
		tmp = t_0;
	} else if (x <= -1.85e-115) {
		tmp = z;
	} else if (x <= -6e-128) {
		tmp = y * x;
	} else if (x <= 3e-44) {
		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 = (y - z) * x
    if (x <= (-2.3d-45)) then
        tmp = t_0
    else if (x <= (-1.85d-115)) then
        tmp = z
    else if (x <= (-6d-128)) then
        tmp = y * x
    else if (x <= 3d-44) 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 = (y - z) * x;
	double tmp;
	if (x <= -2.3e-45) {
		tmp = t_0;
	} else if (x <= -1.85e-115) {
		tmp = z;
	} else if (x <= -6e-128) {
		tmp = y * x;
	} else if (x <= 3e-44) {
		tmp = z * (1.0 - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - z) * x
	tmp = 0
	if x <= -2.3e-45:
		tmp = t_0
	elif x <= -1.85e-115:
		tmp = z
	elif x <= -6e-128:
		tmp = y * x
	elif x <= 3e-44:
		tmp = z * (1.0 - x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - z) * x)
	tmp = 0.0
	if (x <= -2.3e-45)
		tmp = t_0;
	elseif (x <= -1.85e-115)
		tmp = z;
	elseif (x <= -6e-128)
		tmp = Float64(y * x);
	elseif (x <= 3e-44)
		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 = (y - z) * x;
	tmp = 0.0;
	if (x <= -2.3e-45)
		tmp = t_0;
	elseif (x <= -1.85e-115)
		tmp = z;
	elseif (x <= -6e-128)
		tmp = y * x;
	elseif (x <= 3e-44)
		tmp = z * (1.0 - x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e-45], t$95$0, If[LessEqual[x, -1.85e-115], z, If[LessEqual[x, -6e-128], N[(y * x), $MachinePrecision], If[LessEqual[x, 3e-44], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.29999999999999992e-45 or 3.0000000000000002e-44 < x

   1. Initial program 92.5%

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

      1. sub-neg 92.5%

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

      2. +-commutative 92.5%

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

      3. distribute-lft1-in 92.5%

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

      4. associate-+r+ 92.5%

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

      5. +-commutative 92.5%

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

      6. *-commutative 92.5%

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

      7. neg-mul-1 92.5%

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

      8. associate-*r* 92.5%

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

      9. *-commutative 92.5%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 96.8%

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

   if -2.29999999999999992e-45 < x < -1.85e-115

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.6%

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

   if -1.85e-115 < x < -5.99999999999999956e-128

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

   if -5.99999999999999956e-128 < x < 3.0000000000000002e-44

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-45}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-115}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-128}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-44}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \end{array} \]

Alternative 4: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-19}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-115}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-128}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.25e-19)
   (* y x)
   (if (<= x -1.85e-115)
     z
     (if (<= x -6e-128) (* y x) (if (<= x 8.5e-43) z (* y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e-19) {
		tmp = y * x;
	} else if (x <= -1.85e-115) {
		tmp = z;
	} else if (x <= -6e-128) {
		tmp = y * x;
	} else if (x <= 8.5e-43) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.25d-19)) then
        tmp = y * x
    else if (x <= (-1.85d-115)) then
        tmp = z
    else if (x <= (-6d-128)) then
        tmp = y * x
    else if (x <= 8.5d-43) then
        tmp = z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e-19) {
		tmp = y * x;
	} else if (x <= -1.85e-115) {
		tmp = z;
	} else if (x <= -6e-128) {
		tmp = y * x;
	} else if (x <= 8.5e-43) {
		tmp = z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.25e-19:
		tmp = y * x
	elif x <= -1.85e-115:
		tmp = z
	elif x <= -6e-128:
		tmp = y * x
	elif x <= 8.5e-43:
		tmp = z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.25e-19)
		tmp = Float64(y * x);
	elseif (x <= -1.85e-115)
		tmp = z;
	elseif (x <= -6e-128)
		tmp = Float64(y * x);
	elseif (x <= 8.5e-43)
		tmp = z;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.25e-19)
		tmp = y * x;
	elseif (x <= -1.85e-115)
		tmp = z;
	elseif (x <= -6e-128)
		tmp = y * x;
	elseif (x <= 8.5e-43)
		tmp = z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.25e-19], N[(y * x), $MachinePrecision], If[LessEqual[x, -1.85e-115], z, If[LessEqual[x, -6e-128], N[(y * x), $MachinePrecision], If[LessEqual[x, 8.5e-43], z, N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2500000000000001e-19 or -1.85e-115 < x < -5.99999999999999956e-128 or 8.50000000000000056e-43 < x

   1. Initial program 92.6%

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

   2. Taylor expanded in y around inf 57.4%

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

   if -1.2500000000000001e-19 < x < -1.85e-115 or -5.99999999999999956e-128 < x < 8.50000000000000056e-43

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.2%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-19}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-115}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-128}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-43}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-140} \lor \neg \left(z \leq 1.05 \cdot 10^{-101}\right):\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3e-140) (not (<= z 1.05e-101))) (* z (- 1.0 x)) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e-140) || !(z <= 1.05e-101)) {
		tmp = z * (1.0 - x);
	} else {
		tmp = y * 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 ((z <= (-1.3d-140)) .or. (.not. (z <= 1.05d-101))) then
        tmp = z * (1.0d0 - x)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e-140) || !(z <= 1.05e-101)) {
		tmp = z * (1.0 - x);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.3e-140) or not (z <= 1.05e-101):
		tmp = z * (1.0 - x)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.3e-140) || !(z <= 1.05e-101))
		tmp = Float64(z * Float64(1.0 - x));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.3e-140) || ~((z <= 1.05e-101)))
		tmp = z * (1.0 - x);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.3e-140], N[Not[LessEqual[z, 1.05e-101]], $MachinePrecision]], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e-140 or 1.05000000000000008e-101 < z

   1. Initial program 94.7%

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

   2. Taylor expanded in y around 0 79.9%

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

   if -1.2999999999999999e-140 < z < 1.05000000000000008e-101

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 74.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-140} \lor \neg \left(z \leq 1.05 \cdot 10^{-101}\right):\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -250 \lor \neg \left(x \leq 4.2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -250.0) (not (<= x 4.2e-11))) (* (- y z) x) (+ z (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -250.0) || !(x <= 4.2e-11)) {
		tmp = (y - z) * x;
	} else {
		tmp = z + (y * 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 <= (-250.0d0)) .or. (.not. (x <= 4.2d-11))) then
        tmp = (y - z) * x
    else
        tmp = z + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -250.0) || !(x <= 4.2e-11)) {
		tmp = (y - z) * x;
	} else {
		tmp = z + (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -250.0) or not (x <= 4.2e-11):
		tmp = (y - z) * x
	else:
		tmp = z + (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -250.0) || !(x <= 4.2e-11))
		tmp = Float64(Float64(y - z) * x);
	else
		tmp = Float64(z + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -250.0) || ~((x <= 4.2e-11)))
		tmp = (y - z) * x;
	else
		tmp = z + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -250.0], N[Not[LessEqual[x, 4.2e-11]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision], N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -250 or 4.1999999999999997e-11 < x

   1. Initial program 91.6%

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

      1. sub-neg 91.6%

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

      2. +-commutative 91.6%

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

      3. distribute-lft1-in 91.6%

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

      4. associate-+r+ 91.6%

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

      5. +-commutative 91.6%

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

      6. *-commutative 91.6%

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

      7. neg-mul-1 91.6%

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

      8. associate-*r* 91.6%

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

      9. *-commutative 91.6%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.5%

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

   if -250 < x < 4.1999999999999997e-11

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft1-in 100.0%

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

      4. associate-+r+ 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. neg-mul-1 100.0%

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

      8. associate-*r* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. fma-def 100.0%

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

      12. +-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in y around inf 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250 \lor \neg \left(x \leq 4.2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 7: 36.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 96.1%

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

2. Taylor expanded in x around 0 41.5%

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

3. Final simplification 41.5%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023215 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))