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

Percentage Accurate: 97.7% → 100.0%

Time: 4.3s

Alternatives: 8

Speedup: 1.3×

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + \left(1 - x\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

C

double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((1.0d0 - x) * z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 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 97.2%

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

   1. *-commutative 97.2%

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

   2. distribute-lft-out-- 97.2%

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

   3. *-rgt-identity 97.2%

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

   4. cancel-sign-sub-inv 97.2%

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

   5. +-commutative 97.2%

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

   6. associate-+r+ 97.2%

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

   7. +-commutative 97.2%

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

   8. *-commutative 97.2%

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

   9. distribute-rgt-out 100.0%

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

   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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 62.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-10}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-46}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+71}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= x -1.6e+107)
     t_0
     (if (<= x -1.05e-10)
       (* x y)
       (if (<= x 1.25e-46) z (if (<= x 9.5e+71) (* x y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (x <= -1.6e+107) {
		tmp = t_0;
	} else if (x <= -1.05e-10) {
		tmp = x * y;
	} else if (x <= 1.25e-46) {
		tmp = z;
	} else if (x <= 9.5e+71) {
		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 <= (-1.6d+107)) then
        tmp = t_0
    else if (x <= (-1.05d-10)) then
        tmp = x * y
    else if (x <= 1.25d-46) then
        tmp = z
    else if (x <= 9.5d+71) 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 <= -1.6e+107) {
		tmp = t_0;
	} else if (x <= -1.05e-10) {
		tmp = x * y;
	} else if (x <= 1.25e-46) {
		tmp = z;
	} else if (x <= 9.5e+71) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if x <= -1.6e+107:
		tmp = t_0
	elif x <= -1.05e-10:
		tmp = x * y
	elif x <= 1.25e-46:
		tmp = z
	elif x <= 9.5e+71:
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (x <= -1.6e+107)
		tmp = t_0;
	elseif (x <= -1.05e-10)
		tmp = Float64(x * y);
	elseif (x <= 1.25e-46)
		tmp = z;
	elseif (x <= 9.5e+71)
		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 <= -1.6e+107)
		tmp = t_0;
	elseif (x <= -1.05e-10)
		tmp = x * y;
	elseif (x <= 1.25e-46)
		tmp = z;
	elseif (x <= 9.5e+71)
		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, -1.6e+107], t$95$0, If[LessEqual[x, -1.05e-10], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.25e-46], z, If[LessEqual[x, 9.5e+71], N[(x * y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.60000000000000015e107 or 9.50000000000000015e71 < x

   1. Initial program 92.6%

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

      1. +-commutative 92.6%

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

      2. *-commutative 92.6%

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

      3. distribute-rgt-out-- 92.6%

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

      4. *-lft-identity 92.6%

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

      5. associate-+l- 92.6%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 65.0%

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

      1. associate-*r* 65.0%

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

      2. mul-1-neg 65.0%

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

   8. Simplified 65.0%

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

   if -1.60000000000000015e107 < x < -1.05e-10 or 1.24999999999999998e-46 < x < 9.50000000000000015e71

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 71.3%

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

   if -1.05e-10 < x < 1.24999999999999998e-46

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 76.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+107}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-10}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-46}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+71}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{-73} \lor \neg \left(x \leq 4.5 \cdot 10^{-45}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.28e-73) (not (<= x 4.5e-45))) (* x (- y z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.28e-73) || !(x <= 4.5e-45)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.28d-73)) .or. (.not. (x <= 4.5d-45))) then
        tmp = x * (y - z)
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.28e-73) || !(x <= 4.5e-45)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.28e-73) or not (x <= 4.5e-45):
		tmp = x * (y - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.28e-73) || !(x <= 4.5e-45))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.28e-73) || ~((x <= 4.5e-45)))
		tmp = x * (y - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.28e-73], N[Not[LessEqual[x, 4.5e-45]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -1.2799999999999999e-73 or 4.4999999999999999e-45 < x

   1. Initial program 95.4%

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

      1. +-commutative 95.4%

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

      2. *-commutative 95.4%

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

      3. distribute-rgt-out-- 95.4%

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

      4. *-lft-identity 95.4%

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

      5. associate-+l- 95.4%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 96.2%

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

   if -1.2799999999999999e-73 < x < 4.4999999999999999e-45

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 79.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{-73} \lor \neg \left(x \leq 4.5 \cdot 10^{-45}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-9} \lor \neg \left(x \leq 7.6 \cdot 10^{-47}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5e-9) (not (<= x 7.6e-47))) (* x (- y z)) (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e-9) || !(x <= 7.6e-47)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d-9)) .or. (.not. (x <= 7.6d-47))) then
        tmp = x * (y - z)
    else
        tmp = z * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e-9) || !(x <= 7.6e-47)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.5e-9) or not (x <= 7.6e-47):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.5e-9) || !(x <= 7.6e-47))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.5e-9) || ~((x <= 7.6e-47)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.5e-9], N[Not[LessEqual[x, 7.6e-47]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999999e-9 or 7.60000000000000029e-47 < x

   1. Initial program 95.2%

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

      1. +-commutative 95.2%

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

      2. *-commutative 95.2%

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

      3. distribute-rgt-out-- 95.2%

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

      4. *-lft-identity 95.2%

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

      5. associate-+l- 95.2%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.1%

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

   if -1.49999999999999999e-9 < x < 7.60000000000000029e-47

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 77.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-9} \lor \neg \left(x \leq 7.6 \cdot 10^{-47}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 2e-43))) (* x (- y z)) (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 2e-43)) {
		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 <= 2d-43))) 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 <= 2e-43)) {
		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 <= 2e-43):
		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 <= 2e-43))
		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 <= 2e-43)))
		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, 2e-43]], $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 2.00000000000000015e-43 < x

   1. Initial program 95.1%

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

      1. +-commutative 95.1%

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

      2. *-commutative 95.1%

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

      3. distribute-rgt-out-- 95.1%

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

      4. *-lft-identity 95.1%

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

      5. associate-+l- 95.1%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

   if -1 < x < 2.00000000000000015e-43

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 99.3%

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

      1. associate-*r* 99.3%

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

      2. neg-mul-1 99.3%

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

      3. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. distribute-rgt-neg-out 99.3%

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

      4. remove-double-neg 99.3%

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-10} \lor \neg \left(x \leq 2.55 \cdot 10^{-48}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.1e-10) (not (<= x 2.55e-48))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.1e-10) || !(x <= 2.55e-48)) {
		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 <= (-1.1d-10)) .or. (.not. (x <= 2.55d-48))) 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 <= -1.1e-10) || !(x <= 2.55e-48)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.1e-10) or not (x <= 2.55e-48):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.1e-10) || !(x <= 2.55e-48))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.1e-10) || ~((x <= 2.55e-48)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.1e-10], N[Not[LessEqual[x, 2.55e-48]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -1.09999999999999995e-10 or 2.55000000000000006e-48 < x

   1. Initial program 95.2%

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

      1. +-commutative 95.2%

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

      2. *-commutative 95.2%

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

      3. distribute-rgt-out-- 95.2%

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

      4. *-lft-identity 95.2%

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

      5. associate-+l- 95.2%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 52.8%

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

   if -1.09999999999999995e-10 < x < 2.55000000000000006e-48

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+l- 100.0%

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

      6. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 76.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-10} \lor \neg \left(x \leq 2.55 \cdot 10^{-48}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
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 97.2%

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

   1. +-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. distribute-rgt-out-- 97.2%

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

   4. *-lft-identity 97.2%

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

   5. associate-+l- 97.2%

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

   6. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 36.4% 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 97.2%

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

   1. +-commutative 97.2%

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

   2. *-commutative 97.2%

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

   3. distribute-rgt-out-- 97.2%

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

   4. *-lft-identity 97.2%

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

   5. associate-+l- 97.2%

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

   6. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 34.7%

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

6. Final simplification 34.7%

   \[\leadsto z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))