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

Percentage Accurate: 97.8% → 100.0%

Time: 6.4s

Alternatives: 5

Speedup: 1.3×

• 20.4% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out-- N/A

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

   4. *-lft-identity N/A

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

   5. associate-+l- N/A

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

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \color{blue}{\left(x \cdot z - x \cdot y\right)}\right) \]

   7. distribute-lft-out-- N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \left(x \cdot \color{blue}{\left(z - y\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{\left(z - y\right)}\right)\right) \]

   9. --lowering--.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -1.0) t_0 (if (<= x 4.6e-5) (+ z (* x y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 4.6e-5) {
		tmp = z + (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 * (y - z)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 4.6d-5) then
        tmp = z + (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 * (y - z);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 4.6e-5) {
		tmp = z + (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 4.6e-5:
		tmp = z + (x * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 4.6e-5)
		tmp = Float64(z + Float64(x * y));
	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 <= -1.0)
		tmp = t_0;
	elseif (x <= 4.6e-5)
		tmp = z + (x * y);
	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, -1.0], t$95$0, If[LessEqual[x, 4.6e-5], N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.6e-5 < x

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(z - y\right)\right)}\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z - \color{blue}{-1 \cdot y}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + y\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 98.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 98.8%

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

   if -1 < x < 4.6e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -1.35e-128) t_0 (if (<= x 2e-8) z t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -1.35e-128) {
		tmp = t_0;
	} else if (x <= 2e-8) {
		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 <= (-1.35d-128)) then
        tmp = t_0
    else if (x <= 2d-8) 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 <= -1.35e-128) {
		tmp = t_0;
	} else if (x <= 2e-8) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -1.35e-128:
		tmp = t_0
	elif x <= 2e-8:
		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 <= -1.35e-128)
		tmp = t_0;
	elseif (x <= 2e-8)
		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 <= -1.35e-128)
		tmp = t_0;
	elseif (x <= 2e-8)
		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, -1.35e-128], t$95$0, If[LessEqual[x, 2e-8], z, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e-128 or 2e-8 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(-1 \cdot \left(z - y\right)\right)}\right) \]

      7. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z - \color{blue}{-1 \cdot y}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \color{blue}{\left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot z + y\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \color{blue}{-1 \cdot z}\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 90.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 90.7%

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

   if -1.35000000000000003e-128 < x < 2e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 71.4%

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

Alternative 4: 55.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-17}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e-12) (* x y) (if (<= y 8.8e-17) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e-12) {
		tmp = x * y;
	} else if (y <= 8.8e-17) {
		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 (y <= (-1.4d-12)) then
        tmp = x * y
    else if (y <= 8.8d-17) 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 (y <= -1.4e-12) {
		tmp = x * y;
	} else if (y <= 8.8e-17) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.4e-12:
		tmp = x * y
	elif y <= 8.8e-17:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e-12)
		tmp = Float64(x * y);
	elseif (y <= 8.8e-17)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.4e-12)
		tmp = x * y;
	elseif (y <= 8.8e-17)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e-12], N[(x * y), $MachinePrecision], If[LessEqual[y, 8.8e-17], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4000000000000001e-12 or 8.8e-17 < y

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 66.1%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 66.1%

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

   if -1.4000000000000001e-12 < y < 8.8e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z} \]
   4. Step-by-step derivation

   5. Simplified 48.8%

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

Alternative 5: 37.5% 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 99.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z} \]
4. Step-by-step derivation

5. Simplified 34.6%

   \[\leadsto \color{blue}{z} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))