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

Average Accuracy: 90.7% → 99.9%

Time: 16.4s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

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

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) {
	return (x * x) + (4.0 * ((y * t) - (z * (z * y))));
}

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
    code = (x * x) + (4.0d0 * ((y * t) - (z * (z * y))))
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) {
	return (x * x) + (4.0 * ((y * t) - (z * (z * y))));
}

Python

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

↓

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

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)
	return Float64(Float64(x * x) + Float64(4.0 * Float64(Float64(y * t) - Float64(z * Float64(z * y)))))
end

MATLAB

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

↓

function tmp = code(x, y, z, t)
	tmp = (x * x) + (4.0 * ((y * t) - (z * (z * y))));
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_] := N[(N[(x * x), $MachinePrecision] + N[(4.0 * N[(N[(y * t), $MachinePrecision] - N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	90.7%
Target	90.7%
Herbie	99.9%

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

Derivation

1. Initial program 90.7%

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

2. Simplified 90.6%

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

   [Start] 90.7	\[ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   associate-*l* [=>] 90.6	\[ x \cdot x - \color{blue}{y \cdot \left(4 \cdot \left(z \cdot z - t\right)\right)} \]

3. Applied egg-rr 90.7%

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

   [Start] 90.6	\[ x \cdot x - y \cdot \left(4 \cdot \left(z \cdot z - t\right)\right) \]
   associate-*r* [=>] 90.7	\[ x \cdot x - \color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]
   sub-neg [=>] 90.7	\[ x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(z \cdot z + \left(-t\right)\right)} \]
   distribute-rgt-in [=>] 90.7	\[ x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-t\right) \cdot \left(y \cdot 4\right)\right)} \]

4. Applied egg-rr 99.9%

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

   [Start] 90.7	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-t\right) \cdot \left(y \cdot 4\right)\right) \]
   associate-*r* [=>] 90.7	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \color{blue}{\left(\left(-t\right) \cdot y\right) \cdot 4}\right) \]
   distribute-lft-neg-out [=>] 90.7	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \color{blue}{\left(-t \cdot y\right)} \cdot 4\right) \]
   add-sqr-sqrt [=>] 45.4	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot y\right) \cdot 4\right) \]
   sqrt-unprod [=>] 59.0	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-\color{blue}{\sqrt{t \cdot t}} \cdot y\right) \cdot 4\right) \]
   sqr-neg [<=] 59.0	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-\sqrt{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} \cdot y\right) \cdot 4\right) \]
   sqrt-unprod [<=] 26.4	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-\color{blue}{\left(\sqrt{-t} \cdot \sqrt{-t}\right)} \cdot y\right) \cdot 4\right) \]
   add-sqr-sqrt [<=] 52.5	\[ x \cdot x - \left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) + \left(-\color{blue}{\left(-t\right)} \cdot y\right) \cdot 4\right) \]
   cancel-sign-sub-inv [<=] 52.5	\[ x \cdot x - \color{blue}{\left(\left(z \cdot z\right) \cdot \left(y \cdot 4\right) - \left(\left(-t\right) \cdot y\right) \cdot 4\right)} \]
   associate-*r* [=>] 52.5	\[ x \cdot x - \left(\color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot 4} - \left(\left(-t\right) \cdot y\right) \cdot 4\right) \]
   distribute-rgt-out-- [=>] 52.5	\[ x \cdot x - \color{blue}{4 \cdot \left(\left(z \cdot z\right) \cdot y - \left(-t\right) \cdot y\right)} \]
   associate-*l* [=>] 61.7	\[ x \cdot x - 4 \cdot \left(\color{blue}{z \cdot \left(z \cdot y\right)} - \left(-t\right) \cdot y\right) \]
   *-commutative [=>] 61.7	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - \color{blue}{y \cdot \left(-t\right)}\right) \]
   add-sqr-sqrt [=>] 30.7	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{\left(\sqrt{-t} \cdot \sqrt{-t}\right)}\right) \]
   sqrt-unprod [=>] 66.1	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}}\right) \]
   sqr-neg [=>] 66.1	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \sqrt{\color{blue}{t \cdot t}}\right) \]
   sqrt-unprod [<=] 50.4	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}\right) \]
   add-sqr-sqrt [<=] 99.9	\[ x \cdot x - 4 \cdot \left(z \cdot \left(z \cdot y\right) - y \cdot \color{blue}{t}\right) \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	98.4%
Cost	2121

\[\begin{array}{l} t_1 := z \cdot z - t\\ t_2 := \left(4 \cdot y\right) \cdot t_1\\ \mathbf{if}\;t_2 \leq -\infty \lor \neg \left(t_2 \leq 2 \cdot 10^{+277}\right):\\ \;\;\;\;x \cdot x + \left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \left(t_1 \cdot -4\right)\\ \end{array} \]

Alternative 2
Accuracy	87.1%
Cost	1496

\[\begin{array}{l} t_1 := y \cdot \left(z \cdot z\right)\\ t_2 := x \cdot x + -4 \cdot t_1\\ t_3 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ t_4 := x \cdot x + t \cdot \left(4 \cdot y\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.48 \cdot 10^{-18}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 210000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+52}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+139}:\\ \;\;\;\;4 \cdot \left(y \cdot t - t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Accuracy	85.1%
Cost	1364

\[\begin{array}{l} t_1 := x \cdot x + t \cdot \left(4 \cdot y\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+26}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+139}:\\ \;\;\;\;4 \cdot \left(y \cdot t - y \cdot \left(z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	88.8%
Cost	1364

\[\begin{array}{l} t_1 := x \cdot x + t \cdot \left(4 \cdot y\right)\\ t_2 := x \cdot x + \left(z \cdot \left(z \cdot y\right)\right) \cdot -4\\ t_3 := y \cdot \left(z \cdot z\right)\\ \mathbf{if}\;z \leq -6.1 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot x + -4 \cdot t_3\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{+133}:\\ \;\;\;\;4 \cdot \left(y \cdot t - t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	57.9%
Cost	1240

\[\begin{array}{l} t_1 := 4 \cdot \left(y \cdot t\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{if}\;z \leq -8 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-47}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1160000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	85.1%
Cost	1236

\[\begin{array}{l} t_1 := x \cdot x + t \cdot \left(4 \cdot y\right)\\ t_2 := z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ t_3 := \left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{+26}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+139}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	57.9%
Cost	850

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+44} \lor \neg \left(x \leq 1000000000000\right) \land \left(x \leq 1.1 \cdot 10^{+50} \lor \neg \left(x \leq 2.05 \cdot 10^{+80}\right)\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 8
Accuracy	74.3%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+44}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-22}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+49}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+80}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 9
Accuracy	36.0%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023133 
(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))))