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

Average Error: 5.9 → 0.4

Time: 11.4s

Precision: binary64

Cost: 1097

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+154} \lor \neg \left(z \leq 5.8 \cdot 10^{+89}\right):\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \]

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 -1.3e+154) (not (<= z 5.8e+89)))
   (- (* x x) (* z (* z (* 4.0 y))))
   (+ (* x x) (* (* 4.0 y) (- t (* z z))))))

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 <= -1.3e+154) || !(z <= 5.8e+89)) {
		tmp = (x * x) - (z * (z * (4.0 * y)));
	} else {
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)));
	}
	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 <= (-1.3d+154)) .or. (.not. (z <= 5.8d+89))) then
        tmp = (x * x) - (z * (z * (4.0d0 * y)))
    else
        tmp = (x * x) + ((4.0d0 * y) * (t - (z * z)))
    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 <= -1.3e+154) || !(z <= 5.8e+89)) {
		tmp = (x * x) - (z * (z * (4.0 * y)));
	} else {
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)));
	}
	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 <= -1.3e+154) or not (z <= 5.8e+89):
		tmp = (x * x) - (z * (z * (4.0 * y)))
	else:
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)))
	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 <= -1.3e+154) || !(z <= 5.8e+89))
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(4.0 * y))));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(4.0 * y) * Float64(t - Float64(z * z))));
	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 <= -1.3e+154) || ~((z <= 5.8e+89)))
		tmp = (x * x) - (z * (z * (4.0 * y)));
	else
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)));
	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, -1.3e+154], N[Not[LessEqual[z, 5.8e+89]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(N[(4.0 * y), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.9
Target	5.9
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 < -1.29999999999999994e154 or 5.80000000000000051e89 < z

   1. Initial program 43.3

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

   2. Taylor expanded in z around inf 45.5

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

   3. Simplified 2.4

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

      [Start] 45.5	\[ x \cdot x - 4 \cdot \left(y \cdot {z}^{2}\right) \]
      associate-*r* [=>] 45.5	\[ x \cdot x - \color{blue}{\left(4 \cdot y\right) \cdot {z}^{2}} \]
      unpow2 [=>] 45.5	\[ x \cdot x - \left(4 \cdot y\right) \cdot \color{blue}{\left(z \cdot z\right)} \]
      *-commutative [<=] 45.5	\[ x \cdot x - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z\right) \]
      *-commutative [=>] 45.5	\[ x \cdot x - \color{blue}{\left(z \cdot z\right) \cdot \left(y \cdot 4\right)} \]
      associate-*l* [=>] 2.4	\[ x \cdot x - \color{blue}{z \cdot \left(z \cdot \left(y \cdot 4\right)\right)} \]
      *-commutative [=>] 2.4	\[ x \cdot x - z \cdot \left(z \cdot \color{blue}{\left(4 \cdot y\right)}\right) \]

   if -1.29999999999999994e154 < z < 5.80000000000000051e89

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+154} \lor \neg \left(z \leq 5.8 \cdot 10^{+89}\right):\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-107}:\\ \;\;\;\;x \cdot x + \left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+288}:\\ \;\;\;\;y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 2
Error	27.2
Cost	976

\[\begin{array}{l} t_1 := 4 \cdot \left(y \cdot t\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-195}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	6.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-19} \lor \neg \left(z \leq 1.65 \cdot 10^{-34}\right):\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(4 \cdot y\right) \cdot t\\ \end{array} \]

Alternative 4
Error	15.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -16500000000 \lor \neg \left(x \leq 9.8 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \end{array} \]

Alternative 5
Error	15.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -7400000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Error	25.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -19000000000 \lor \neg \left(x \leq 6.8 \cdot 10^{-46}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7
Error	41.6
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023059 
(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))))