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

Average Accuracy: 90.9% → 99.6%

Time: 16.3s

Precision: binary64

Cost: 1348

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+238}:\\ \;\;\;\;x \cdot x + \left(\left(y \cdot 4\right) \cdot t - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

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+238)
   (+ (* x x) (- (* (* y 4.0) t) (* (* z z) (* 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+238) {
		tmp = (x * x) + (((y * 4.0) * t) - ((z * z) * (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+238) then
        tmp = (x * x) + (((y * 4.0d0) * t) - ((z * z) * (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+238) {
		tmp = (x * x) + (((y * 4.0) * t) - ((z * z) * (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+238:
		tmp = (x * x) + (((y * 4.0) * t) - ((z * z) * (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+238)
		tmp = Float64(Float64(x * x) + Float64(Float64(Float64(y * 4.0) * t) - Float64(Float64(z * z) * 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+238)
		tmp = (x * x) + (((y * 4.0) * t) - ((z * z) * (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+238], N[(N[(x * x), $MachinePrecision] + N[(N[(N[(y * 4.0), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * z), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	90.9%
Target	90.9%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e238

   1. Initial program 99.9%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
      sub-neg [=>] 99.9	\[ x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z + \left(-t\right)\right)} \]
      distribute-rgt-in [=>] 99.9	\[ x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-t\right) \cdot \left(y \cdot 4\right)\right)} \]

   if 2.0000000000000001e238 < (*.f64 z z)

   1. Initial program 32.1%

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

   2. Taylor expanded in z around inf 30.5%

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

   3. Simplified 98.1%

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

      [Start] 30.5	\[ x \cdot x - 4 \cdot \left(y \cdot {z}^{2}\right) \]
      unpow2 [=>] 30.5	\[ x \cdot x - 4 \cdot \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right) \]
      associate-*r* [=>] 30.5	\[ x \cdot x - \color{blue}{\left(4 \cdot y\right) \cdot \left(z \cdot z\right)} \]
      *-commutative [<=] 30.5	\[ x \cdot x - \color{blue}{\left(y \cdot 4\right)} \cdot \left(z \cdot z\right) \]
      associate-*r* [=>] 98.1	\[ x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot z\right) \cdot z} \]
      *-commutative [=>] 98.1	\[ x \cdot x - \left(\color{blue}{\left(4 \cdot y\right)} \cdot z\right) \cdot z \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+238}:\\ \;\;\;\;x \cdot x + \left(\left(y \cdot 4\right) \cdot t - \left(z \cdot z\right) \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+238}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	59.1%
Cost	976

\[\begin{array}{l} t_1 := \left(y \cdot 4\right) \cdot t\\ t_2 := \left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-230}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 12600000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	85.8%
Cost	972

\[\begin{array}{l} t_1 := \left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-10}:\\ \;\;\;\;4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+67}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	90.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-32} \lor \neg \left(z \leq 10200000000\right):\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \end{array} \]

Alternative 5
Accuracy	76.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.65 \cdot 10^{+62}:\\ \;\;\;\;4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Accuracy	59.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.32 \cdot 10^{+58}:\\ \;\;\;\;\left(y \cdot 4\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 7
Accuracy	34.6%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023146 
(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))))