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

Average Error: 6.3 → 0.5

Time: 14.6s

Precision: binary64

Cost: 1092

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+184}:\\ \;\;\;\;x \cdot x + \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+184)
   (+ (* x x) (* (- (* z z) t) (* y -4.0)))
   (+ (* x x) (* z (* z (* y -4.0))))))

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 * z) <= 2e+184) {
		tmp = (x * x) + (((z * z) - t) * (y * -4.0));
	} else {
		tmp = (x * x) + (z * (z * (y * -4.0)));
	}
	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 * z) <= 2d+184) then
        tmp = (x * x) + (((z * z) - t) * (y * (-4.0d0)))
    else
        tmp = (x * x) + (z * (z * (y * (-4.0d0))))
    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 * z) <= 2e+184) {
		tmp = (x * x) + (((z * z) - t) * (y * -4.0));
	} else {
		tmp = (x * x) + (z * (z * (y * -4.0)));
	}
	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 * z) <= 2e+184:
		tmp = (x * x) + (((z * z) - t) * (y * -4.0))
	else:
		tmp = (x * x) + (z * (z * (y * -4.0)))
	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 (Float64(z * z) <= 2e+184)
		tmp = Float64(Float64(x * x) + Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) + Float64(z * Float64(z * Float64(y * -4.0))));
	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 * z) <= 2e+184)
		tmp = (x * x) + (((z * z) - t) * (y * -4.0));
	else
		tmp = (x * x) + (z * (z * (y * -4.0)));
	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[LessEqual[N[(z * z), $MachinePrecision], 2e+184], N[(N[(x * x), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.3
Target	6.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e184

   1. Initial program 0.1

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

   if 2.00000000000000003e184 < (*.f64 z z)

   1. Initial program 35.0

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

   2. Taylor expanded in z around inf 37.3

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

   3. Simplified 2.5

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

      [Start] 37.3	\[ x \cdot x - 4 \cdot \left(y \cdot {z}^{2}\right) \]
      associate-*r* [=>] 37.3	\[ x \cdot x - \color{blue}{\left(4 \cdot y\right) \cdot {z}^{2}} \]
      unpow2 [=>] 37.3	\[ x \cdot x - \left(4 \cdot y\right) \cdot \color{blue}{\left(z \cdot z\right)} \]
      *-commutative [<=] 37.3	\[ x \cdot x - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z\right) \]
      *-commutative [=>] 37.3	\[ x \cdot x - \color{blue}{\left(z \cdot z\right) \cdot \left(y \cdot 4\right)} \]
      associate-*l* [=>] 2.5	\[ x \cdot x - \color{blue}{z \cdot \left(z \cdot \left(y \cdot 4\right)\right)} \]
      *-commutative [=>] 2.5	\[ x \cdot x - z \cdot \left(z \cdot \color{blue}{\left(4 \cdot y\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+184}:\\ \;\;\;\;x \cdot x + \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	27.1
Cost	1376

\[\begin{array}{l} t_1 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ t_2 := t \cdot \left(4 \cdot y\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-36}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 88000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	6.7
Cost	1232

\[\begin{array}{l} t_1 := y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ t_2 := x \cdot x + z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-48}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	8.5
Cost	1104

\[\begin{array}{l} t_1 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+95}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.1
Cost	960

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

Alternative 5
Error	15.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -84000000000000 \lor \neg \left(x \leq 56000000000\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(4 \cdot \left(t - z \cdot z\right)\right)\\ \end{array} \]

Alternative 6
Error	15.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+18} \lor \neg \left(x \leq 26500000000\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]

Alternative 7
Error	25.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 78000000000:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 8
Error	41.4
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023055 
(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))))