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

Percentage Accurate: 90.1% → 94.8%

Time: 12.1s

Alternatives: 6

Speedup: 0.6×

• 49.6% 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: 90.1% 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: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+289}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(y \cdot -4\right) \cdot {\left(\frac{z}{x}\right)}^{2} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+289)
   (- (* x x) (* (* y 4.0) (fma z z (- t))))
   (* (* x x) (+ (* (* y -4.0) (pow (/ z x) 2.0)) 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+289) {
		tmp = (x * x) - ((y * 4.0) * fma(z, z, -t));
	} else {
		tmp = (x * x) * (((y * -4.0) * pow((z / x), 2.0)) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+289)
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * fma(z, z, Float64(-t))));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(Float64(y * -4.0) * (Float64(z / x) ^ 2.0)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+289], N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(z * z + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(y * -4.0), $MachinePrecision] * N[Power[N[(z / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e289

   1. Initial program 93.0%

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

      1. fma-neg 93.1%

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

   4. Applied egg-rr 93.1%

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

   if 2.0000000000000001e289 < (*.f64 x x)

   1. Initial program 87.5%

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

      1. fma-neg 92.2%

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

      2. distribute-lft-neg-in 92.2%

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

      3. *-commutative 92.2%

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

      4. distribute-rgt-neg-in 92.2%

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

      5. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in x around inf 62.5%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + -4 \cdot \frac{y \cdot \left({z}^{2} - t\right)}{{x}^{2}}\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 73.4%

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

   7. Simplified 73.4%

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

   9. Applied egg-rr 62.5%

      \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{\frac{y \cdot \left(4 \cdot \mathsf{fma}\left(z, z, t\right)\right)}{-{x}^{2}}}\right) \]
   10. Step-by-step derivation

       1. associate-/l* 73.4%

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

       2. neg-mul-1 73.4%

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

       3. times-frac 73.4%

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

       4. metadata-eval 73.4%

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

       5. rem-square-sqrt 48.4%

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

       6. fma-undefine 48.4%

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

       7. rem-square-sqrt 35.9%

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

       8. hypot-undefine 35.9%

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

       9. fma-undefine 35.9%

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

       10. rem-square-sqrt 35.9%

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

       11. hypot-undefine 35.9%

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

       12. unpow2 35.9%

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

       13. times-frac 45.3%

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

       14. unpow2 45.3%

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

       15. associate-*r* 45.3%

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

   11. Simplified 45.3%

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

   12. Taylor expanded in z around inf 98.4%

       \[\leadsto {x}^{2} \cdot \left(1 + \left(y \cdot -4\right) \cdot {\color{blue}{\left(\frac{z}{x}\right)}}^{2}\right) \]
   13. Step-by-step derivation

       1. pow2 98.4%

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

   14. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(y \cdot -4\right) \cdot {\left(\frac{z}{x}\right)}^{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+289}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(y \cdot -4\right) \cdot {\left(\frac{z}{x}\right)}^{2} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. fma-neg 92.8%

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

   2. distribute-lft-neg-in 92.8%

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

   3. *-commutative 92.8%

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

   4. distribute-rgt-neg-in 92.8%

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

   5. metadata-eval 92.8%

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

3. Simplified 92.8%

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

5. Final simplification 92.8%

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

Alternative 3: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8.5 \cdot 10^{+291}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 8.5e+291)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* y (* t -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 8.5e+291) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (y * (t * -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
    real(8) :: tmp
    if ((x * x) <= 8.5d+291) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - (y * (t * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 8.5e+291) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 8.5e+291:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (y * (t * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 8.5e+291)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 8.5e+291)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (y * (t * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 8.5e+291], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 8.5000000000000003e291

   1. Initial program 93.1%

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

   if 8.5000000000000003e291 < (*.f64 x x)

   1. Initial program 87.3%

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

   3. Taylor expanded in z around 0 93.9%

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

      1. *-commutative 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*l* 93.9%

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

   5. Simplified 93.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8.5 \cdot 10^{+291}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.7e+65) (- (* x x) (* y (* t -4.0))) (* (* y -4.0) (* z z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.7e+65) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	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 (z <= 3.7d+65) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (y * (-4.0d0)) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.7e+65) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.7e+65:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (y * -4.0) * (z * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.7e+65)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(y * -4.0) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.7e+65)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (y * -4.0) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.7e+65], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.69999999999999995e65

   1. Initial program 94.0%

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

   3. Taylor expanded in z around 0 72.8%

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

      1. *-commutative 72.8%

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

      2. *-commutative 72.8%

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

      3. associate-*l* 72.8%

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

   5. Simplified 72.8%

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

   if 3.69999999999999995e65 < z

   1. Initial program 81.0%

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

      1. fma-neg 83.2%

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

      2. distribute-lft-neg-in 83.2%

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

      3. *-commutative 83.2%

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

      4. distribute-rgt-neg-in 83.2%

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

      5. metadata-eval 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in z around inf 72.3%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

   7. Simplified 72.3%

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

      1. pow2 72.3%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2100000:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2100000.0) (* 4.0 (* y t)) (* (* y -4.0) (* z z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2100000.0) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	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 (z <= 2100000.0d0) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = (y * (-4.0d0)) * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2100000.0) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = (y * -4.0) * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 2100000.0:
		tmp = 4.0 * (y * t)
	else:
		tmp = (y * -4.0) * (z * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2100000.0)
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(Float64(y * -4.0) * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2100000.0)
		tmp = 4.0 * (y * t);
	else
		tmp = (y * -4.0) * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2100000.0], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.1e6

   1. Initial program 93.7%

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

      1. fma-neg 94.7%

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

      2. distribute-lft-neg-in 94.7%

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

      3. *-commutative 94.7%

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

      4. distribute-rgt-neg-in 94.7%

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

      5. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in t around inf 35.2%

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

      1. *-commutative 35.2%

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

   7. Simplified 35.2%

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

   if 2.1e6 < z

   1. Initial program 84.6%

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

      1. fma-neg 86.4%

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

      2. distribute-lft-neg-in 86.4%

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

      3. *-commutative 86.4%

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

      4. distribute-rgt-neg-in 86.4%

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

      5. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in z around inf 63.8%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. *-commutative 63.8%

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

   7. Simplified 63.8%

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

      1. pow2 63.8%

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

   9. Applied egg-rr 63.8%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2100000:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.5% 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 91.7%

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

   1. fma-neg 92.8%

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

   2. distribute-lft-neg-in 92.8%

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

   3. *-commutative 92.8%

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

   4. distribute-rgt-neg-in 92.8%

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

   5. metadata-eval 92.8%

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

3. Simplified 92.8%

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

5. Taylor expanded in t around inf 31.7%

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

   1. *-commutative 31.7%

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

7. Simplified 31.7%

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

Developer Target 1: 90.2% 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 2024146 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

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