Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 87.8% → 95.5%

Time: 5.0s

Alternatives: 7

Speedup: 0.7×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - 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 * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - 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 * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+50} \lor \neg \left(y \leq 420000000000\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.3e+50) (not (<= y 420000000000.0)))
   (* (/ y z) (- z x))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e+50) || !(y <= 420000000000.0)) {
		tmp = (y / z) * (z - x);
	} else {
		tmp = (x + (y * (z - x))) / 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 ((y <= (-3.3d+50)) .or. (.not. (y <= 420000000000.0d0))) then
        tmp = (y / z) * (z - x)
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e+50) || !(y <= 420000000000.0)) {
		tmp = (y / z) * (z - x);
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.3e+50) or not (y <= 420000000000.0):
		tmp = (y / z) * (z - x)
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.3e+50) || !(y <= 420000000000.0))
		tmp = Float64(Float64(y / z) * Float64(z - x));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.3e+50) || ~((y <= 420000000000.0)))
		tmp = (y / z) * (z - x);
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.3e+50], N[Not[LessEqual[y, 420000000000.0]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e50 or 4.2e11 < y

   1. Initial program 70.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf 70.0%

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

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{y}{\frac{z}{z - x}}} \]

      2. associate-/r/ 93.9%

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

   4. Simplified 93.9%

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

   if -3.3e50 < y < 4.2e11

   1. Initial program 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+50} \lor \neg \left(y \leq 420000000000\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 2: 61.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{-x}{z}\\ \mathbf{if}\;y \leq -1.08 \cdot 10^{+63}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -1550000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-45}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+229}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ (- x) z))))
   (if (<= y -1.08e+63)
     (/ z (/ z y))
     (if (<= y -1550000000.0)
       t_0
       (if (<= y -8.6e-45)
         y
         (if (<= y 1.0) (/ x z) (if (<= y 1.95e+229) t_0 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (-x / z);
	double tmp;
	if (y <= -1.08e+63) {
		tmp = z / (z / y);
	} else if (y <= -1550000000.0) {
		tmp = t_0;
	} else if (y <= -8.6e-45) {
		tmp = y;
	} else if (y <= 1.0) {
		tmp = x / z;
	} else if (y <= 1.95e+229) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = y * (-x / z)
    if (y <= (-1.08d+63)) then
        tmp = z / (z / y)
    else if (y <= (-1550000000.0d0)) then
        tmp = t_0
    else if (y <= (-8.6d-45)) then
        tmp = y
    else if (y <= 1.0d0) then
        tmp = x / z
    else if (y <= 1.95d+229) then
        tmp = t_0
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (-x / z);
	double tmp;
	if (y <= -1.08e+63) {
		tmp = z / (z / y);
	} else if (y <= -1550000000.0) {
		tmp = t_0;
	} else if (y <= -8.6e-45) {
		tmp = y;
	} else if (y <= 1.0) {
		tmp = x / z;
	} else if (y <= 1.95e+229) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (-x / z)
	tmp = 0
	if y <= -1.08e+63:
		tmp = z / (z / y)
	elif y <= -1550000000.0:
		tmp = t_0
	elif y <= -8.6e-45:
		tmp = y
	elif y <= 1.0:
		tmp = x / z
	elif y <= 1.95e+229:
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(-x) / z))
	tmp = 0.0
	if (y <= -1.08e+63)
		tmp = Float64(z / Float64(z / y));
	elseif (y <= -1550000000.0)
		tmp = t_0;
	elseif (y <= -8.6e-45)
		tmp = y;
	elseif (y <= 1.0)
		tmp = Float64(x / z);
	elseif (y <= 1.95e+229)
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (-x / z);
	tmp = 0.0;
	if (y <= -1.08e+63)
		tmp = z / (z / y);
	elseif (y <= -1550000000.0)
		tmp = t_0;
	elseif (y <= -8.6e-45)
		tmp = y;
	elseif (y <= 1.0)
		tmp = x / z;
	elseif (y <= 1.95e+229)
		tmp = t_0;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.08e+63], N[(z / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1550000000.0], t$95$0, If[LessEqual[y, -8.6e-45], y, If[LessEqual[y, 1.0], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.95e+229], t$95$0, y]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.08e63

   1. Initial program 60.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0 20.3%

      \[\leadsto \frac{\color{blue}{y \cdot z}}{z} \]
   3. Step-by-step derivation

      1. associate-/l* 52.1%

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

      2. associate-/r/ 67.1%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot z} \]

   4. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

         \[\leadsto \color{blue}{z \cdot \frac{y}{z}} \]

      2. clear-num 66.8%

         \[\leadsto z \cdot \color{blue}{\frac{1}{\frac{z}{y}}} \]

      3. un-div-inv 67.4%

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

   6. Applied egg-rr 67.4%

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

   if -1.08e63 < y < -1.55e9 or 1 < y < 1.9499999999999999e229

   1. Initial program 83.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around inf 56.3%

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

      1. associate-/l* 62.3%

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot y}{\frac{z}{x}}} \]

      2. mul-1-neg 62.3%

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

      3. unsub-neg 62.3%

         \[\leadsto \frac{\color{blue}{1 - y}}{\frac{z}{x}} \]

   4. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{1 - y}{\frac{z}{x}}} \]

   5. Taylor expanded in y around inf 55.4%

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

      1. mul-1-neg 55.4%

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

      2. associate-*r/ 61.4%

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

      3. *-commutative 61.4%

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

      4. distribute-rgt-neg-out 61.4%

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

   7. Simplified 61.4%

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

   if -1.55e9 < y < -8.5999999999999998e-45 or 1.9499999999999999e229 < y

   1. Initial program 69.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0 68.0%

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

   if -8.5999999999999998e-45 < y < 1

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around 0 72.6%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.08 \cdot 10^{+63}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -1550000000:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-45}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+229}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -0.0122:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{z}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e+66)
   (/ z (/ z y))
   (if (<= y -0.0122)
     (/ (* y (- x)) z)
     (if (<= y -7.5e-45)
       y
       (if (<= y 1.0) (/ x z) (if (<= y 6.5e+230) (* y (/ (- x) z)) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e+66) {
		tmp = z / (z / y);
	} else if (y <= -0.0122) {
		tmp = (y * -x) / z;
	} else if (y <= -7.5e-45) {
		tmp = y;
	} else if (y <= 1.0) {
		tmp = x / z;
	} else if (y <= 6.5e+230) {
		tmp = y * (-x / z);
	} else {
		tmp = 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 <= (-5.5d+66)) then
        tmp = z / (z / y)
    else if (y <= (-0.0122d0)) then
        tmp = (y * -x) / z
    else if (y <= (-7.5d-45)) then
        tmp = y
    else if (y <= 1.0d0) then
        tmp = x / z
    else if (y <= 6.5d+230) then
        tmp = y * (-x / z)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e+66) {
		tmp = z / (z / y);
	} else if (y <= -0.0122) {
		tmp = (y * -x) / z;
	} else if (y <= -7.5e-45) {
		tmp = y;
	} else if (y <= 1.0) {
		tmp = x / z;
	} else if (y <= 6.5e+230) {
		tmp = y * (-x / z);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.5e+66:
		tmp = z / (z / y)
	elif y <= -0.0122:
		tmp = (y * -x) / z
	elif y <= -7.5e-45:
		tmp = y
	elif y <= 1.0:
		tmp = x / z
	elif y <= 6.5e+230:
		tmp = y * (-x / z)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.5e+66)
		tmp = Float64(z / Float64(z / y));
	elseif (y <= -0.0122)
		tmp = Float64(Float64(y * Float64(-x)) / z);
	elseif (y <= -7.5e-45)
		tmp = y;
	elseif (y <= 1.0)
		tmp = Float64(x / z);
	elseif (y <= 6.5e+230)
		tmp = Float64(y * Float64(Float64(-x) / z));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.5e+66)
		tmp = z / (z / y);
	elseif (y <= -0.0122)
		tmp = (y * -x) / z;
	elseif (y <= -7.5e-45)
		tmp = y;
	elseif (y <= 1.0)
		tmp = x / z;
	elseif (y <= 6.5e+230)
		tmp = y * (-x / z);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.5e+66], N[(z / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -0.0122], N[(N[(y * (-x)), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, -7.5e-45], y, If[LessEqual[y, 1.0], N[(x / z), $MachinePrecision], If[LessEqual[y, 6.5e+230], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], y]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.5e66

   1. Initial program 60.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0 20.3%

      \[\leadsto \frac{\color{blue}{y \cdot z}}{z} \]
   3. Step-by-step derivation

      1. associate-/l* 52.1%

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

      2. associate-/r/ 67.1%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot z} \]

   4. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

         \[\leadsto \color{blue}{z \cdot \frac{y}{z}} \]

      2. clear-num 66.8%

         \[\leadsto z \cdot \color{blue}{\frac{1}{\frac{z}{y}}} \]

      3. un-div-inv 67.4%

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

   6. Applied egg-rr 67.4%

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

   if -5.5e66 < y < -0.0122000000000000008

   1. Initial program 91.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around inf 73.0%

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

      1. +-commutative 73.0%

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

      2. distribute-rgt1-in 73.0%

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

      3. mul-1-neg 73.0%

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

      4. cancel-sign-sub-inv 73.0%

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

   4. Simplified 73.0%

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

   5. Taylor expanded in y around inf 68.9%

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

      1. mul-1-neg 68.9%

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

      2. distribute-rgt-neg-in 68.9%

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

   7. Simplified 68.9%

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

   if -0.0122000000000000008 < y < -7.5000000000000006e-45 or 6.4999999999999997e230 < y

   1. Initial program 69.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0 68.0%

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

   if -7.5000000000000006e-45 < y < 1

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around 0 72.6%

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

   if 1 < y < 6.4999999999999997e230

   1. Initial program 82.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around inf 53.5%

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

      1. associate-/l* 60.5%

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot y}{\frac{z}{x}}} \]

      2. mul-1-neg 60.5%

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

      3. unsub-neg 60.5%

         \[\leadsto \frac{\color{blue}{1 - y}}{\frac{z}{x}} \]

   4. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{1 - y}{\frac{z}{x}}} \]

   5. Taylor expanded in y around inf 53.1%

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

      1. mul-1-neg 53.1%

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

      2. associate-*r/ 60.1%

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

      3. *-commutative 60.1%

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

      4. distribute-rgt-neg-out 60.1%

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

   7. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(-y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -0.0122:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{z}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-44} \lor \neg \left(y \leq 3.4 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e-44) (not (<= y 3.4e-64))) (* (/ y z) (- z x)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-44) || !(y <= 3.4e-64)) {
		tmp = (y / z) * (z - x);
	} else {
		tmp = x / 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 ((y <= (-1d-44)) .or. (.not. (y <= 3.4d-64))) then
        tmp = (y / z) * (z - x)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-44) || !(y <= 3.4e-64)) {
		tmp = (y / z) * (z - x);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e-44) or not (y <= 3.4e-64):
		tmp = (y / z) * (z - x)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e-44) || !(y <= 3.4e-64))
		tmp = Float64(Float64(y / z) * Float64(z - x));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e-44) || ~((y <= 3.4e-64)))
		tmp = (y / z) * (z - x);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e-44], N[Not[LessEqual[y, 3.4e-64]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999953e-45 or 3.40000000000000012e-64 < y

   1. Initial program 76.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around inf 71.1%

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

      1. associate-/l* 94.9%

         \[\leadsto \color{blue}{\frac{y}{\frac{z}{z - x}}} \]

      2. associate-/r/ 88.3%

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

   4. Simplified 88.3%

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

   if -9.99999999999999953e-45 < y < 3.40000000000000012e-64

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around 0 76.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-44} \lor \neg \left(y \leq 3.4 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5: 59.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-44}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-44) y (if (<= y 3.1e-72) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-44) {
		tmp = y;
	} else if (y <= 3.1e-72) {
		tmp = x / z;
	} else {
		tmp = 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 <= (-1d-44)) then
        tmp = y
    else if (y <= 3.1d-72) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-44) {
		tmp = y;
	} else if (y <= 3.1e-72) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-44:
		tmp = y
	elif y <= 3.1e-72:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-44)
		tmp = y;
	elseif (y <= 3.1e-72)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-44)
		tmp = y;
	elseif (y <= 3.1e-72)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-44], y, If[LessEqual[y, 3.1e-72], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999953e-45 or 3.0999999999999998e-72 < y

   1. Initial program 76.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0 49.8%

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

   if -9.99999999999999953e-45 < y < 3.0999999999999998e-72

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around 0 78.0%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-44}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-45}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e-45) (* z (/ y z)) (if (<= y 1.9e-72) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e-45) {
		tmp = z * (y / z);
	} else if (y <= 1.9e-72) {
		tmp = x / z;
	} else {
		tmp = 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 <= (-2.7d-45)) then
        tmp = z * (y / z)
    else if (y <= 1.9d-72) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e-45) {
		tmp = z * (y / z);
	} else if (y <= 1.9e-72) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.7e-45:
		tmp = z * (y / z)
	elif y <= 1.9e-72:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e-45)
		tmp = Float64(z * Float64(y / z));
	elseif (y <= 1.9e-72)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.7e-45)
		tmp = z * (y / z);
	elseif (y <= 1.9e-72)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.7e-45], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-72], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 3 regimes

2. if y < -2.69999999999999985e-45

   1. Initial program 71.6%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0 28.3%

      \[\leadsto \frac{\color{blue}{y \cdot z}}{z} \]
   3. Step-by-step derivation

      1. associate-/l* 51.6%

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

      2. associate-/r/ 60.1%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot z} \]

   4. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot z} \]

   if -2.69999999999999985e-45 < y < 1.90000000000000001e-72

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in y around 0 78.0%

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

   if 1.90000000000000001e-72 < y

   1. Initial program 79.7%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]

   2. Taylor expanded in x around 0 48.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-45}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 40.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 85.3%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]

2. Taylor expanded in x around 0 40.2%

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

3. Final simplification 40.2%

   \[\leadsto y \]

Developer target: 93.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))