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

Percentage Accurate: 90.9% → 95.7%

Time: 10.4s

Alternatives: 7

Speedup: 0.6×

• 48.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

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

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

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

Initial Program: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

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

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

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

Alternative 1: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -5 \cdot 10^{-306} \lor \neg \left(y \cdot 4 \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(y \cdot t\right) + 4 \cdot {\left(z \cdot \sqrt{y}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y 4.0) -5e-306) (not (<= (* y 4.0) 5e-35)))
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (- (* x x) (+ (* -4.0 (* y t)) (* 4.0 (pow (* z (sqrt y)) 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * 4.0) <= -5e-306) || !((y * 4.0) <= 5e-35)) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x * x) - ((-4.0 * (y * t)) + (4.0 * pow((z * sqrt(y)), 2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * 4.0) <= -5e-306) || !(Float64(y * 4.0) <= 5e-35))
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(-4.0 * Float64(y * t)) + Float64(4.0 * (Float64(z * sqrt(y)) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * 4.0), $MachinePrecision], -5e-306], N[Not[LessEqual[N[(y * 4.0), $MachinePrecision], 5e-35]], $MachinePrecision]], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[Power[N[(z * N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y 4) < -4.99999999999999998e-306 or 4.99999999999999964e-35 < (*.f64 y 4)

   1. Initial program 91.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. fma-neg 95.0%

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

      2. distribute-lft-neg-in 95.0%

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

      3. *-commutative 95.0%

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

      4. distribute-rgt-neg-in 95.0%

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

      5. metadata-eval 95.0%

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

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)} \]
   4. Add Preprocessing

   if -4.99999999999999998e-306 < (*.f64 y 4) < 4.99999999999999964e-35

   1. Initial program 82.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 82.2%

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

      1. add-sqr-sqrt 82.2%

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

      2. pow2 82.2%

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

      3. *-commutative 82.2%

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

      4. sqrt-prod 82.2%

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

      5. sqrt-pow1 98.4%

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

      6. metadata-eval 98.4%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot {\left({z}^{\color{blue}{1}} \cdot \sqrt{y}\right)}^{2}\right) \]

      7. pow1 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \color{blue}{{\left(z \cdot \sqrt{y}\right)}^{2}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -5 \cdot 10^{-306} \lor \neg \left(y \cdot 4 \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(y \cdot t\right) + 4 \cdot {\left(z \cdot \sqrt{y}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5e+292)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma x x (* 4.0 (* y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+292) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma(x, x, (4.0 * (y * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5e+292)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(x, x, Float64(4.0 * Float64(y * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+292], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999996e292

   1. Initial program 94.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Step-by-step derivation

      1. fma-neg 94.5%

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

      2. distribute-lft-neg-in 94.5%

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

      3. *-commutative 94.5%

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

      4. distribute-rgt-neg-in 94.5%

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

      5. metadata-eval 94.5%

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

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)} \]
   4. Add Preprocessing

   if 4.9999999999999996e292 < (*.f64 x x)

   1. Initial program 74.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 88.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative 88.1%

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

   5. Simplified 88.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 88.1%

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

      2. fma-define 91.0%

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

      3. metadata-eval 91.0%

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

   7. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5e+292)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (fma x x (* 4.0 (* y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+292) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma(x, x, (4.0 * (y * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5e+292)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(x, x, Float64(4.0 * Float64(y * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+292], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999996e292

   1. Initial program 94.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 4.9999999999999996e292 < (*.f64 x x)

   1. Initial program 74.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 88.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative 88.1%

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

   5. Simplified 88.1%

      \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(y \cdot t\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 88.1%

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

      2. fma-define 91.0%

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

      3. metadata-eval 91.0%

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

   7. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{+299}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.7e+299) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.7e+299) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	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
    real(8) :: tmp
    if ((x * x) <= 1.7d+299) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.7e+299) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.7e+299:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.7e+299)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.7e+299)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.7e+299], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.70000000000000016e299

   1. Initial program 94.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 1.70000000000000016e299 < (*.f64 x x)

   1. Initial program 74.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 74.6%

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

   5. Simplified 91.0%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{+299}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.02 \cdot 10^{+50}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.02e+50) (* 4.0 (* y t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.02e+50) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	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
    real(8) :: tmp
    if ((x * x) <= 1.02d+50) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.02e+50) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.02e+50:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.02e+50)
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.02e+50)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.02e+50], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.01999999999999991e50

   1. Initial program 93.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf 54.0%

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative 54.0%

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

   5. Simplified 54.0%

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

   if 1.01999999999999991e50 < (*.f64 x x)

   1. Initial program 84.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 84.8%

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

   5. Simplified 80.2%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.02 \cdot 10^{+50}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x \cdot x - -4 \cdot \left(y \cdot t\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * x) - (-4.0 * (y * t));
}

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) - ((-4.0d0) * (y * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (-4.0 * (y * t));
}

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.3%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0 74.2%

   \[\leadsto x \cdot x - \color{blue}{-4 \cdot \left(t \cdot y\right)} \]
4. Step-by-step derivation

   1. *-commutative 74.2%

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

5. Simplified 74.2%

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

6. Final simplification 74.2%

   \[\leadsto x \cdot x - -4 \cdot \left(y \cdot t\right) \]
7. Add Preprocessing

Alternative 7: 32.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(y \cdot t\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 4.0 * (y * t);
}

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 = 4.0d0 * (y * t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 4.0 * (y * t);
}

Python

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

Julia

function code(x, y, z, t)
	return Float64(4.0 * Float64(y * t))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.3%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf 36.4%

   \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
4. Step-by-step derivation

   1. *-commutative 36.4%

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

5. Simplified 36.4%

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

6. Final simplification 36.4%

   \[\leadsto 4 \cdot \left(y \cdot t\right) \]
7. Add Preprocessing

Developer target: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

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) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(4.0 * Float64(y * Float64(Float64(z * z) - t))))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024055 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))