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

Percentage Accurate: 98.0% → 100.0%

Time: 3.7s

Alternatives: 8

Speedup: 1.3×

• 20.0% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. +-commutative 98.0%

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

   2. *-commutative 98.0%

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

   3. distribute-rgt-out-- 98.0%

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

   4. *-lft-identity 98.0%

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

   5. associate-+l- 98.0%

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

   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. Final simplification 100.0%

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

Alternative 2: 61.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-90}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+18} \lor \neg \left(x \leq 1.3 \cdot 10^{+93}\right) \land x \leq 8 \cdot 10^{+140}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- x))))
   (if (<= x -9.5e+56)
     t_0
     (if (<= x -4.9e-14)
       (* x y)
       (if (<= x 4e-90)
         z
         (if (or (<= x 2.05e+18) (and (not (<= x 1.3e+93)) (<= x 8e+140)))
           (* x y)
           t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * -x;
	double tmp;
	if (x <= -9.5e+56) {
		tmp = t_0;
	} else if (x <= -4.9e-14) {
		tmp = x * y;
	} else if (x <= 4e-90) {
		tmp = z;
	} else if ((x <= 2.05e+18) || (!(x <= 1.3e+93) && (x <= 8e+140))) {
		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 = z * -x
    if (x <= (-9.5d+56)) then
        tmp = t_0
    else if (x <= (-4.9d-14)) then
        tmp = x * y
    else if (x <= 4d-90) then
        tmp = z
    else if ((x <= 2.05d+18) .or. (.not. (x <= 1.3d+93)) .and. (x <= 8d+140)) 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 = z * -x;
	double tmp;
	if (x <= -9.5e+56) {
		tmp = t_0;
	} else if (x <= -4.9e-14) {
		tmp = x * y;
	} else if (x <= 4e-90) {
		tmp = z;
	} else if ((x <= 2.05e+18) || (!(x <= 1.3e+93) && (x <= 8e+140))) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * -x
	tmp = 0
	if x <= -9.5e+56:
		tmp = t_0
	elif x <= -4.9e-14:
		tmp = x * y
	elif x <= 4e-90:
		tmp = z
	elif (x <= 2.05e+18) or (not (x <= 1.3e+93) and (x <= 8e+140)):
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-x))
	tmp = 0.0
	if (x <= -9.5e+56)
		tmp = t_0;
	elseif (x <= -4.9e-14)
		tmp = Float64(x * y);
	elseif (x <= 4e-90)
		tmp = z;
	elseif ((x <= 2.05e+18) || (!(x <= 1.3e+93) && (x <= 8e+140)))
		tmp = Float64(x * y);
	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 <= -9.5e+56)
		tmp = t_0;
	elseif (x <= -4.9e-14)
		tmp = x * y;
	elseif (x <= 4e-90)
		tmp = z;
	elseif ((x <= 2.05e+18) || (~((x <= 1.3e+93)) && (x <= 8e+140)))
		tmp = x * y;
	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, -9.5e+56], t$95$0, If[LessEqual[x, -4.9e-14], N[(x * y), $MachinePrecision], If[LessEqual[x, 4e-90], z, If[Or[LessEqual[x, 2.05e+18], And[N[Not[LessEqual[x, 1.3e+93]], $MachinePrecision], LessEqual[x, 8e+140]]], N[(x * y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999997e56 or 2.05e18 < x < 1.3e93 or 8.00000000000000047e140 < x

   1. Initial program 94.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 69.4%

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

      1. associate-*r* 69.4%

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

      2. neg-mul-1 69.4%

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

      3. *-commutative 69.4%

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

   7. Simplified 69.4%

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

   if -9.4999999999999997e56 < x < -4.89999999999999995e-14 or 3.99999999999999998e-90 < x < 2.05e18 or 1.3e93 < x < 8.00000000000000047e140

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 81.9%

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

   if -4.89999999999999995e-14 < x < 3.99999999999999998e-90

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+56}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-90}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+18} \lor \neg \left(x \leq 1.3 \cdot 10^{+93}\right) \land x \leq 8 \cdot 10^{+140}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]

Alternative 3: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-14} \lor \neg \left(x \leq 2.4 \cdot 10^{-92}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.8e-14) (not (<= x 2.4e-92))) (* x (- y z)) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.8e-14) || !(x <= 2.4e-92)) {
		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 <= (-4.8d-14)) .or. (.not. (x <= 2.4d-92))) 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 <= -4.8e-14) || !(x <= 2.4e-92)) {
		tmp = x * (y - z);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.8e-14) or not (x <= 2.4e-92):
		tmp = x * (y - z)
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.8e-14) || !(x <= 2.4e-92))
		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 <= -4.8e-14) || ~((x <= 2.4e-92)))
		tmp = x * (y - z);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.8e-14], N[Not[LessEqual[x, 2.4e-92]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e-14 or 2.4000000000000001e-92 < x

   1. Initial program 96.4%

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

   2. Taylor expanded in x around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. sub-neg 95.2%

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

   4. Simplified 95.2%

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

   if -4.8e-14 < x < 2.4000000000000001e-92

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-14} \lor \neg \left(x \leq 2.4 \cdot 10^{-92}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-14} \lor \neg \left(x \leq 4 \cdot 10^{-90}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4e-14) (not (<= x 4e-90))) (* x (- y z)) (* z (- 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4e-14) || !(x <= 4e-90)) {
		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 <= (-4d-14)) .or. (.not. (x <= 4d-90))) 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 <= -4e-14) || !(x <= 4e-90)) {
		tmp = x * (y - z);
	} else {
		tmp = z * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4e-14) or not (x <= 4e-90):
		tmp = x * (y - z)
	else:
		tmp = z * (1.0 - x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4e-14) || !(x <= 4e-90))
		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 <= -4e-14) || ~((x <= 4e-90)))
		tmp = x * (y - z);
	else
		tmp = z * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4e-14], N[Not[LessEqual[x, 4e-90]], $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 < -4e-14 or 3.99999999999999998e-90 < x

   1. Initial program 96.4%

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

   2. Taylor expanded in x around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. sub-neg 95.2%

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

   4. Simplified 95.2%

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

   if -4e-14 < x < 3.99999999999999998e-90

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-14} \lor \neg \left(x \leq 4 \cdot 10^{-90}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-14} \lor \neg \left(x \leq 1.85 \cdot 10^{-94}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.6e-14) (not (<= x 1.85e-94))) (* x (- y z)) (- z (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6e-14) || !(x <= 1.85e-94)) {
		tmp = x * (y - z);
	} else {
		tmp = z - (z * 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 <= (-1.6d-14)) .or. (.not. (x <= 1.85d-94))) then
        tmp = x * (y - z)
    else
        tmp = z - (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6e-14) || !(x <= 1.85e-94)) {
		tmp = x * (y - z);
	} else {
		tmp = z - (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.6e-14) or not (x <= 1.85e-94):
		tmp = x * (y - z)
	else:
		tmp = z - (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.6e-14) || !(x <= 1.85e-94))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z - Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.6e-14) || ~((x <= 1.85e-94)))
		tmp = x * (y - z);
	else
		tmp = z - (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.6e-14], N[Not[LessEqual[x, 1.85e-94]], $MachinePrecision]], N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z - N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001e-14 or 1.8499999999999999e-94 < x

   1. Initial program 96.4%

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

   2. Taylor expanded in x around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. sub-neg 95.2%

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

   4. Simplified 95.2%

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

   if -1.6000000000000001e-14 < x < 1.8499999999999999e-94

   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. Taylor expanded in z around inf 78.6%

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

      1. *-commutative 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-14} \lor \neg \left(x \leq 1.85 \cdot 10^{-94}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z - z \cdot x\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1950000000 \lor \neg \left(x \leq 1.5 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1950000000.0) (not (<= x 1.5e-13)))
   (* x (- y z))
   (+ z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1950000000.0) || !(x <= 1.5e-13)) {
		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 <= (-1950000000.0d0)) .or. (.not. (x <= 1.5d-13))) 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 <= -1950000000.0) || !(x <= 1.5e-13)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1950000000.0) or not (x <= 1.5e-13):
		tmp = x * (y - z)
	else:
		tmp = z + (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1950000000.0) || !(x <= 1.5e-13))
		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 <= -1950000000.0) || ~((x <= 1.5e-13)))
		tmp = x * (y - z);
	else
		tmp = z + (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1950000000.0], N[Not[LessEqual[x, 1.5e-13]], $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 < -1.95e9 or 1.49999999999999992e-13 < x

   1. Initial program 95.8%

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

   2. Taylor expanded in x around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. sub-neg 99.4%

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

   4. Simplified 99.4%

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

   if -1.95e9 < x < 1.49999999999999992e-13

   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. Taylor expanded in z around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-out 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1950000000 \lor \neg \left(x \leq 1.5 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 7: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-14} \lor \neg \left(x \leq 4 \cdot 10^{-90}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e-14) (not (<= x 4e-90))) (* x y) z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e-14) || !(x <= 4e-90)) {
		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 <= (-1.95d-14)) .or. (.not. (x <= 4d-90))) 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 <= -1.95e-14) || !(x <= 4e-90)) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.95e-14) or not (x <= 4e-90):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e-14) || !(x <= 4e-90))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.95e-14) || ~((x <= 4e-90)))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e-14], N[Not[LessEqual[x, 4e-90]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -1.9499999999999999e-14 or 3.99999999999999998e-90 < x

   1. Initial program 96.4%

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

   2. Taylor expanded in y around inf 49.8%

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

   if -1.9499999999999999e-14 < x < 3.99999999999999998e-90

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 78.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-14} \lor \neg \left(x \leq 4 \cdot 10^{-90}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 8: 36.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0 39.5%

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

3. Final simplification 39.5%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))