Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Average Error: 6.2 → 0.4

Time: 6.6s

Precision: binary64

Cost: 1097

Math

\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+125} \lor \neg \left(z \leq 2.7 \cdot 10^{+91}\right):\\ \;\;\;\;x \cdot x + \left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.2e+125) (not (<= z 2.7e+91)))
   (+ (* x x) (* (* z (* z y)) -4.0))
   (- (* x x) (* (* 4.0 y) (- (* z z) t)))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.2e+125) || !(z <= 2.7e+91)) {
		tmp = (x * x) + ((z * (z * y)) * -4.0);
	} else {
		tmp = (x * x) - ((4.0 * y) * ((z * z) - t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-6.2d+125)) .or. (.not. (z <= 2.7d+91))) then
        tmp = (x * x) + ((z * (z * y)) * (-4.0d0))
    else
        tmp = (x * x) - ((4.0d0 * y) * ((z * z) - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.2e+125) || !(z <= 2.7e+91)) {
		tmp = (x * x) + ((z * (z * y)) * -4.0);
	} else {
		tmp = (x * x) - ((4.0 * y) * ((z * z) - t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

↓

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.2e+125) or not (z <= 2.7e+91):
		tmp = (x * x) + ((z * (z * y)) * -4.0)
	else:
		tmp = (x * x) - ((4.0 * y) * ((z * z) - t))
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.2e+125) || !(z <= 2.7e+91))
		tmp = Float64(Float64(x * x) + Float64(Float64(z * Float64(z * y)) * -4.0));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(4.0 * y) * Float64(Float64(z * z) - t)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.2e+125) || ~((z <= 2.7e+91)))
		tmp = (x * x) + ((z * (z * y)) * -4.0);
	else
		tmp = (x * x) - ((4.0 * y) * ((z * z) - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.2e+125], N[Not[LessEqual[z, 2.7e+91]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] + N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(4.0 * y), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.2
Target	6.2
Herbie	0.4

\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

1. Split input into 2 regimes

2. if z < -6.2e125 or 2.7e91 < z

   1. Initial program 38.0

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

   2. Taylor expanded in z around inf 40.0

      \[\leadsto x \cdot x - \color{blue}{4 \cdot \left(y \cdot {z}^{2}\right)} \]

   3. Simplified 2.3

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

      [Start] 40.0	\[ x \cdot x - 4 \cdot \left(y \cdot {z}^{2}\right) \]
      *-commutative [=>] 40.0	\[ x \cdot x - 4 \cdot \color{blue}{\left({z}^{2} \cdot y\right)} \]
      unpow2 [=>] 40.0	\[ x \cdot x - 4 \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \]
      associate-*l* [=>] 2.3	\[ x \cdot x - 4 \cdot \color{blue}{\left(z \cdot \left(z \cdot y\right)\right)} \]

   if -6.2e125 < z < 2.7e91

   1. Initial program 0.1

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+125} \lor \neg \left(z \leq 2.7 \cdot 10^{+91}\right):\\ \;\;\;\;x \cdot x + \left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(4 \cdot y\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+113} \lor \neg \left(z \leq 6.2 \cdot 10^{-32}\right):\\ \;\;\;\;x \cdot x + \left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \]

Alternative 2
Error	17.8
Cost	576

\[x \cdot x - y \cdot \left(t \cdot -4\right) \]

Reproduce

herbie shell --seed 2022356 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))