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

Average Error: 6.1 → 5.4

Time: 12.6s

Precision: binary64

Cost: 7684

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(t \cdot -4\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x - \left(t_1 + 4 \cdot \left({z}^{2} \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* t -4.0))))
   (if (<= (* z z) 5e+305)
     (- (* x x) (+ t_1 (* 4.0 (* (pow z 2.0) y))))
     (- (* x x) t_1))))

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 t_1 = y * (t * -4.0);
	double tmp;
	if ((z * z) <= 5e+305) {
		tmp = (x * x) - (t_1 + (4.0 * (pow(z, 2.0) * y)));
	} else {
		tmp = (x * x) - t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = y * (t * (-4.0d0))
    if ((z * z) <= 5d+305) then
        tmp = (x * x) - (t_1 + (4.0d0 * ((z ** 2.0d0) * y)))
    else
        tmp = (x * x) - t_1
    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 t_1 = y * (t * -4.0);
	double tmp;
	if ((z * z) <= 5e+305) {
		tmp = (x * x) - (t_1 + (4.0 * (Math.pow(z, 2.0) * y)));
	} else {
		tmp = (x * x) - t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = y * (t * -4.0)
	tmp = 0
	if (z * z) <= 5e+305:
		tmp = (x * x) - (t_1 + (4.0 * (math.pow(z, 2.0) * y)))
	else:
		tmp = (x * x) - t_1
	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)
	t_1 = Float64(y * Float64(t * -4.0))
	tmp = 0.0
	if (Float64(z * z) <= 5e+305)
		tmp = Float64(Float64(x * x) - Float64(t_1 + Float64(4.0 * Float64((z ^ 2.0) * y))));
	else
		tmp = Float64(Float64(x * x) - t_1);
	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)
	t_1 = y * (t * -4.0);
	tmp = 0.0;
	if ((z * z) <= 5e+305)
		tmp = (x * x) - (t_1 + (4.0 * ((z ^ 2.0) * y)));
	else
		tmp = (x * x) - t_1;
	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_] := Block[{t$95$1 = N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 5e+305], N[(N[(x * x), $MachinePrecision] - N[(t$95$1 + N[(4.0 * N[(N[Power[z, 2.0], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Target

Original	6.1
Target	6.0
Herbie	5.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000009e305

   1. Initial program 0.1

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

   2. Taylor expanded in z around 0 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ x \cdot x - \left(4 \cdot \left(y \cdot {z}^{2}\right) + -4 \cdot \left(y \cdot t\right)\right) \]
      rational.json-simplify-1 [<=] 0.1	\[ x \cdot x - \color{blue}{\left(-4 \cdot \left(y \cdot t\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right)} \]
      rational.json-simplify-43 [=>] 0.1	\[ x \cdot x - \left(\color{blue}{y \cdot \left(t \cdot -4\right)} + 4 \cdot \left(y \cdot {z}^{2}\right)\right) \]
      rational.json-simplify-2 [=>] 0.1	\[ x \cdot x - \left(y \cdot \left(t \cdot -4\right) + 4 \cdot \color{blue}{\left({z}^{2} \cdot y\right)}\right) \]

   if 5.00000000000000009e305 < (*.f64 z z)

   1. Initial program 63.2

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

   2. Taylor expanded in z around 0 55.8

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

   3. Simplified 55.9

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

      [Start] 55.8	\[ x \cdot x - -4 \cdot \left(y \cdot t\right) \]
      rational.json-simplify-43 [=>] 55.9	\[ x \cdot x - \color{blue}{y \cdot \left(t \cdot -4\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x - \left(y \cdot \left(t \cdot -4\right) + 4 \cdot \left({z}^{2} \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.4
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \]

Alternative 2
Error	17.6
Cost	576

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

Alternative 3
Error	36.7
Cost	320

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

Alternative 4
Error	60.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023075 
(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))))