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

Percentage Accurate: 90.1% → 96.6%

Time: 8.6s

Alternatives: 7

Speedup: 0.8×

• 48.9% 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: 96.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+308)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+308) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+308)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+308], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e308

   1. Initial program 97.8%

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

      1. cancel-sign-sub-inv 97.8%

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

      2. distribute-lft-neg-out 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-lft-neg-out 97.8%

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

      5. distribute-lft-neg-in 97.8%

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

      6. distribute-rgt-neg-in 97.8%

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

      7. fma-def 98.9%

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

      8. sub-neg 98.9%

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

      9. +-commutative 98.9%

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

      10. distribute-neg-in 98.9%

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

      11. remove-double-neg 98.9%

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

      12. sub-neg 98.9%

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

   3. Simplified 98.9%

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

   if 1e308 < (*.f64 z z)

   1. Initial program 74.3%

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

   2. Taylor expanded in z around inf 79.9%

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

      1. associate-*r* 79.9%

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

   4. Simplified 79.9%

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

      1. add-cbrt-cube 79.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}} \]

      2. pow3 79.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}^{3}}} \]

      3. *-commutative 79.9%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot -4\right)} \cdot {z}^{2}\right)}^{3}} \]

      4. associate-*l* 79.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}}^{3}} \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}^{3}}} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 79.9%

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

      2. associate-*r* 79.9%

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

      3. unpow2 79.9%

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

      4. associate-*r* 91.1%

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

   8. Applied egg-rr 91.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 2: 96.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+308)
   (fma x x (* (* y -4.0) (- (* z z) t)))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+308) {
		tmp = fma(x, x, ((y * -4.0) * ((z * z) - t)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+308)
		tmp = fma(x, x, Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+308], N[(x * x + N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e308

   1. Initial program 97.8%

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

      1. fma-neg 98.9%

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

      2. distribute-lft-neg-in 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   if 1e308 < (*.f64 z z)

   1. Initial program 74.3%

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

   2. Taylor expanded in z around inf 79.9%

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

      1. associate-*r* 79.9%

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

   4. Simplified 79.9%

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

      1. add-cbrt-cube 79.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}} \]

      2. pow3 79.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}^{3}}} \]

      3. *-commutative 79.9%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot -4\right)} \cdot {z}^{2}\right)}^{3}} \]

      4. associate-*l* 79.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}}^{3}} \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}^{3}}} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 79.9%

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

      2. associate-*r* 79.9%

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

      3. unpow2 79.9%

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

      4. associate-*r* 91.1%

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

   8. Applied egg-rr 91.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 3: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+308)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+308) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = 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
    real(8) :: tmp
    if ((z * z) <= 1d+308) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+308) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+308:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+308)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+308)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+308], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e308

   1. Initial program 97.8%

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

   if 1e308 < (*.f64 z z)

   1. Initial program 74.3%

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

   2. Taylor expanded in z around inf 79.9%

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

      1. associate-*r* 79.9%

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

   4. Simplified 79.9%

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

      1. add-cbrt-cube 79.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}} \]

      2. pow3 79.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}^{3}}} \]

      3. *-commutative 79.9%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot -4\right)} \cdot {z}^{2}\right)}^{3}} \]

      4. associate-*l* 79.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}}^{3}} \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}^{3}}} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 79.9%

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

      2. associate-*r* 79.9%

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

      3. unpow2 79.9%

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

      4. associate-*r* 91.1%

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

   8. Applied egg-rr 91.1%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+76}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.3e+76) (- (* x x) (* -4.0 (* y t))) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.3e+76) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = 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
    real(8) :: tmp
    if (z <= 1.3d+76) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.3e+76) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.3e+76:
		tmp = (x * x) - (-4.0 * (y * t))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.3e+76)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.3e+76)
		tmp = (x * x) - (-4.0 * (y * t));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.3e+76], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.3e76

   1. Initial program 94.2%

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

   2. Taylor expanded in z around 0 74.0%

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

      1. *-commutative 74.0%

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

   4. Simplified 74.0%

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

   if 1.3e76 < z

   1. Initial program 79.3%

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

   2. Taylor expanded in z around inf 79.4%

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

      1. associate-*r* 79.4%

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

   4. Simplified 79.4%

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

      1. add-cbrt-cube 77.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}} \]

      2. pow3 77.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}^{3}}} \]

      3. *-commutative 77.6%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot -4\right)} \cdot {z}^{2}\right)}^{3}} \]

      4. associate-*l* 77.6%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}}^{3}} \]

   6. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}^{3}}} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 79.4%

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

      2. associate-*r* 79.4%

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

      3. unpow2 79.4%

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

      4. associate-*r* 87.9%

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

   8. Applied egg-rr 87.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{+76}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 5: 45.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.48 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.48e+60) (* y (* 4.0 t)) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.48e+60) {
		tmp = y * (4.0 * t);
	} else {
		tmp = 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
    real(8) :: tmp
    if (z <= 1.48d+60) then
        tmp = y * (4.0d0 * t)
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.48e+60) {
		tmp = y * (4.0 * t);
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.48e+60:
		tmp = y * (4.0 * t)
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.48e+60)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.48e+60)
		tmp = y * (4.0 * t);
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.48e+60], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.47999999999999999e60

   1. Initial program 94.1%

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

   2. Taylor expanded in t around inf 34.6%

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

      1. associate-*r* 34.7%

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

      2. *-commutative 34.7%

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

   4. Simplified 34.7%

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

   if 1.47999999999999999e60 < z

   1. Initial program 81.2%

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

   2. Taylor expanded in z around inf 76.0%

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

      1. associate-*r* 76.0%

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

   4. Simplified 76.0%

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

      1. add-cbrt-cube 72.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)\right) \cdot \left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}} \]

      2. pow3 72.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\left(-4 \cdot y\right) \cdot {z}^{2}\right)}^{3}}} \]

      3. *-commutative 72.9%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot -4\right)} \cdot {z}^{2}\right)}^{3}} \]

      4. associate-*l* 72.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}}^{3}} \]

   6. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(y \cdot \left(-4 \cdot {z}^{2}\right)\right)}^{3}}} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 76.0%

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

      2. associate-*r* 76.0%

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

      3. unpow2 76.0%

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

      4. associate-*r* 83.7%

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

   8. Applied egg-rr 83.7%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.48 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 6: 30.4% 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 91.3%

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

2. Taylor expanded in t around inf 29.5%

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

   1. *-commutative 29.5%

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

4. Simplified 29.5%

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

5. Final simplification 29.5%

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

Alternative 7: 30.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return y * (4.0 * 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 = y * (4.0d0 * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

2. Taylor expanded in t around inf 29.5%

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

   1. associate-*r* 29.5%

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

   2. *-commutative 29.5%

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

4. Simplified 29.5%

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

5. Final simplification 29.5%

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

Developer target: 90.1% 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 2023310 
(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))))