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

Percentage Accurate: 98.2% → 100.0%

Time: 5.4s

Alternatives: 7

Speedup: 1.3×

• 21.1% 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: 98.2% 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 98.4%

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

   1. +-commutative 98.4%

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

   2. remove-double-neg 98.4%

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

   3. distribute-rgt-neg-out 98.4%

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

   4. neg-sub0 98.4%

      \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

   5. neg-sub0 98.4%

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

   6. *-commutative 98.4%

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

   7. distribute-lft-neg-in 98.4%

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

   8. remove-double-neg 98.4%

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

   9. distribute-rgt-out-- 98.4%

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

   10. *-lft-identity 98.4%

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

   11. associate-+l- 98.4%

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

   12. 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 2: 59.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-125}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-152}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 10^{+73} \lor \neg \left(x \leq 1.6 \cdot 10^{+196}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e-125)
   (* x y)
   (if (<= x 3.5e-152)
     z
     (if (<= x 1.95e-115)
       (* x y)
       (if (<= x 7.8e-7)
         z
         (if (or (<= x 1e+73) (not (<= x 1.6e+196))) (* x y) (* z (- x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e-125) {
		tmp = x * y;
	} else if (x <= 3.5e-152) {
		tmp = z;
	} else if (x <= 1.95e-115) {
		tmp = x * y;
	} else if (x <= 7.8e-7) {
		tmp = z;
	} else if ((x <= 1e+73) || !(x <= 1.6e+196)) {
		tmp = x * y;
	} else {
		tmp = 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 <= (-1.8d-125)) then
        tmp = x * y
    else if (x <= 3.5d-152) then
        tmp = z
    else if (x <= 1.95d-115) then
        tmp = x * y
    else if (x <= 7.8d-7) then
        tmp = z
    else if ((x <= 1d+73) .or. (.not. (x <= 1.6d+196))) then
        tmp = x * y
    else
        tmp = z * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e-125) {
		tmp = x * y;
	} else if (x <= 3.5e-152) {
		tmp = z;
	} else if (x <= 1.95e-115) {
		tmp = x * y;
	} else if (x <= 7.8e-7) {
		tmp = z;
	} else if ((x <= 1e+73) || !(x <= 1.6e+196)) {
		tmp = x * y;
	} else {
		tmp = z * -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.8e-125:
		tmp = x * y
	elif x <= 3.5e-152:
		tmp = z
	elif x <= 1.95e-115:
		tmp = x * y
	elif x <= 7.8e-7:
		tmp = z
	elif (x <= 1e+73) or not (x <= 1.6e+196):
		tmp = x * y
	else:
		tmp = z * -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e-125)
		tmp = Float64(x * y);
	elseif (x <= 3.5e-152)
		tmp = z;
	elseif (x <= 1.95e-115)
		tmp = Float64(x * y);
	elseif (x <= 7.8e-7)
		tmp = z;
	elseif ((x <= 1e+73) || !(x <= 1.6e+196))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e-125)
		tmp = x * y;
	elseif (x <= 3.5e-152)
		tmp = z;
	elseif (x <= 1.95e-115)
		tmp = x * y;
	elseif (x <= 7.8e-7)
		tmp = z;
	elseif ((x <= 1e+73) || ~((x <= 1.6e+196)))
		tmp = x * y;
	else
		tmp = z * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e-125], N[(x * y), $MachinePrecision], If[LessEqual[x, 3.5e-152], z, If[LessEqual[x, 1.95e-115], N[(x * y), $MachinePrecision], If[LessEqual[x, 7.8e-7], z, If[Or[LessEqual[x, 1e+73], N[Not[LessEqual[x, 1.6e+196]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(z * (-x)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8000000000000001e-125 or 3.5000000000000001e-152 < x < 1.9499999999999999e-115 or 7.80000000000000049e-7 < x < 9.99999999999999983e72 or 1.59999999999999996e196 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf 66.6%

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

   if -1.8000000000000001e-125 < x < 3.5000000000000001e-152 or 1.9499999999999999e-115 < x < 7.80000000000000049e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 80.0%

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

   if 9.99999999999999983e72 < x < 1.59999999999999996e196

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. neg-mul-1 99.9%

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

      2. unsub-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 66.7%

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

      1. mul-1-neg 66.7%

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

      2. *-commutative 66.7%

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

      3. distribute-rgt-neg-in 66.7%

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

   8. Simplified 66.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-125}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-152}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 10^{+73} \lor \neg \left(x \leq 1.6 \cdot 10^{+196}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-150}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-114}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))))
   (if (<= x -2.05e-125)
     t_0
     (if (<= x 1.18e-150)
       z
       (if (<= x 2.25e-114) (* x y) (if (<= x 9.5e-8) z t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double tmp;
	if (x <= -2.05e-125) {
		tmp = t_0;
	} else if (x <= 1.18e-150) {
		tmp = z;
	} else if (x <= 2.25e-114) {
		tmp = x * y;
	} else if (x <= 9.5e-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 <= (-2.05d-125)) then
        tmp = t_0
    else if (x <= 1.18d-150) then
        tmp = z
    else if (x <= 2.25d-114) then
        tmp = x * y
    else if (x <= 9.5d-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 <= -2.05e-125) {
		tmp = t_0;
	} else if (x <= 1.18e-150) {
		tmp = z;
	} else if (x <= 2.25e-114) {
		tmp = x * y;
	} else if (x <= 9.5e-8) {
		tmp = z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	tmp = 0
	if x <= -2.05e-125:
		tmp = t_0
	elif x <= 1.18e-150:
		tmp = z
	elif x <= 2.25e-114:
		tmp = x * y
	elif x <= 9.5e-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 <= -2.05e-125)
		tmp = t_0;
	elseif (x <= 1.18e-150)
		tmp = z;
	elseif (x <= 2.25e-114)
		tmp = Float64(x * y);
	elseif (x <= 9.5e-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 <= -2.05e-125)
		tmp = t_0;
	elseif (x <= 1.18e-150)
		tmp = z;
	elseif (x <= 2.25e-114)
		tmp = x * y;
	elseif (x <= 9.5e-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, -2.05e-125], t$95$0, If[LessEqual[x, 1.18e-150], z, If[LessEqual[x, 2.25e-114], N[(x * y), $MachinePrecision], If[LessEqual[x, 9.5e-8], z, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0499999999999999e-125 or 9.50000000000000036e-8 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf 94.1%

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

      1. neg-mul-1 94.1%

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

      2. unsub-neg 94.1%

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

   5. Simplified 94.1%

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

   if -2.0499999999999999e-125 < x < 1.18e-150 or 2.24999999999999984e-114 < x < 9.50000000000000036e-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 80.0%

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

   if 1.18e-150 < x < 2.24999999999999984e-114

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-150}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-114}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ t_1 := z - z \cdot x\\ \mathbf{if}\;x \leq -1.06 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-150}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.00022:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- y z))) (t_1 (- z (* z x))))
   (if (<= x -1.06e-129)
     t_0
     (if (<= x 1.18e-150)
       t_1
       (if (<= x 2.2e-115) (* x y) (if (<= x 0.00022) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y - z);
	double t_1 = z - (z * x);
	double tmp;
	if (x <= -1.06e-129) {
		tmp = t_0;
	} else if (x <= 1.18e-150) {
		tmp = t_1;
	} else if (x <= 2.2e-115) {
		tmp = x * y;
	} else if (x <= 0.00022) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * (y - z)
    t_1 = z - (z * x)
    if (x <= (-1.06d-129)) then
        tmp = t_0
    else if (x <= 1.18d-150) then
        tmp = t_1
    else if (x <= 2.2d-115) then
        tmp = x * y
    else if (x <= 0.00022d0) then
        tmp = t_1
    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 t_1 = z - (z * x);
	double tmp;
	if (x <= -1.06e-129) {
		tmp = t_0;
	} else if (x <= 1.18e-150) {
		tmp = t_1;
	} else if (x <= 2.2e-115) {
		tmp = x * y;
	} else if (x <= 0.00022) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y - z)
	t_1 = z - (z * x)
	tmp = 0
	if x <= -1.06e-129:
		tmp = t_0
	elif x <= 1.18e-150:
		tmp = t_1
	elif x <= 2.2e-115:
		tmp = x * y
	elif x <= 0.00022:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y - z))
	t_1 = Float64(z - Float64(z * x))
	tmp = 0.0
	if (x <= -1.06e-129)
		tmp = t_0;
	elseif (x <= 1.18e-150)
		tmp = t_1;
	elseif (x <= 2.2e-115)
		tmp = Float64(x * y);
	elseif (x <= 0.00022)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y - z);
	t_1 = z - (z * x);
	tmp = 0.0;
	if (x <= -1.06e-129)
		tmp = t_0;
	elseif (x <= 1.18e-150)
		tmp = t_1;
	elseif (x <= 2.2e-115)
		tmp = x * y;
	elseif (x <= 0.00022)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(z - N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.06e-129], t$95$0, If[LessEqual[x, 1.18e-150], t$95$1, If[LessEqual[x, 2.2e-115], N[(x * y), $MachinePrecision], If[LessEqual[x, 0.00022], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0600000000000001e-129 or 2.20000000000000008e-4 < x

   1. Initial program 97.4%

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

   3. Taylor expanded in x around inf 94.7%

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

      1. neg-mul-1 94.7%

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

      2. unsub-neg 94.7%

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

   5. Simplified 94.7%

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

   if -1.0600000000000001e-129 < x < 1.18e-150 or 2.1999999999999999e-115 < x < 2.20000000000000008e-4

   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. remove-double-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

      5. neg-sub0 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. *-lft-identity 100.0%

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

      11. associate-+l- 100.0%

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

      12. 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 79.9%

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

      1. *-commutative 79.9%

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

   7. Simplified 79.9%

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

   if 1.18e-150 < x < 2.1999999999999999e-115

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-150}:\\ \;\;\;\;z - z \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.00022:\\ \;\;\;\;z - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-125} \lor \neg \left(x \leq 2.1 \cdot 10^{-151} \lor \neg \left(x \leq 1.9 \cdot 10^{-115}\right) \land x \leq 9.2 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.05e-125)
         (not (or (<= x 2.1e-151) (and (not (<= x 1.9e-115)) (<= x 9.2e-8)))))
   (* x y)
   z))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.05e-125) || !((x <= 2.1e-151) || (!(x <= 1.9e-115) && (x <= 9.2e-8)))) {
		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 <= (-2.05d-125)) .or. (.not. (x <= 2.1d-151) .or. (.not. (x <= 1.9d-115)) .and. (x <= 9.2d-8))) 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 <= -2.05e-125) || !((x <= 2.1e-151) || (!(x <= 1.9e-115) && (x <= 9.2e-8)))) {
		tmp = x * y;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.05e-125) or not ((x <= 2.1e-151) or (not (x <= 1.9e-115) and (x <= 9.2e-8))):
		tmp = x * y
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.05e-125) || !((x <= 2.1e-151) || (!(x <= 1.9e-115) && (x <= 9.2e-8))))
		tmp = Float64(x * y);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.05e-125) || ~(((x <= 2.1e-151) || (~((x <= 1.9e-115)) && (x <= 9.2e-8)))))
		tmp = x * y;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.05e-125], N[Not[Or[LessEqual[x, 2.1e-151], And[N[Not[LessEqual[x, 1.9e-115]], $MachinePrecision], LessEqual[x, 9.2e-8]]]], $MachinePrecision]], N[(x * y), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < -2.0499999999999999e-125 or 2.0999999999999999e-151 < x < 1.89999999999999996e-115 or 9.2000000000000003e-8 < x

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf 62.9%

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

   if -2.0499999999999999e-125 < x < 2.0999999999999999e-151 or 1.89999999999999996e-115 < x < 9.2000000000000003e-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 80.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-125} \lor \neg \left(x \leq 2.1 \cdot 10^{-151} \lor \neg \left(x \leq 1.9 \cdot 10^{-115}\right) \land x \leq 9.2 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = x * (y - z);
	} else {
		tmp = z + (x * y);
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(x * Float64(y - z));
	else
		tmp = Float64(z + Float64(x * y));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf 99.3%

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

      1. neg-mul-1 99.3%

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

      2. unsub-neg 99.3%

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

   5. Simplified 99.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(1 - x\right) \cdot \left(-z\right)\right)} + x \cdot y \]

      5. neg-sub0 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-rgt-out-- 100.0%

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

      10. *-lft-identity 100.0%

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

      11. associate-+l- 100.0%

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

      12. 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 99.2%

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

      1. associate-*r* 99.2%

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

      2. neg-mul-1 99.2%

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

      3. *-commutative 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.2%

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

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 98.4%

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

3. Taylor expanded in x around 0 34.0%

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

4. Final simplification 34.0%

   \[\leadsto z \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))