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

Percentage Accurate: 90.6% → 92.5%

Time: 11.4s

Alternatives: 8

Speedup: 0.6×

• 49.1% 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: 90.6% 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: 92.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+263)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (fma (* y 4.0) t (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+263) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = fma((y * 4.0), t, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+263)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = fma(Float64(y * 4.0), t, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+263], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 4.0), $MachinePrecision] * t + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000003e263

   1. Initial program 94.2%

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

      1. fma-neg 94.2%

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

      2. distribute-lft-neg-in 94.2%

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

      3. *-commutative 94.2%

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

      4. distribute-rgt-neg-in 94.2%

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

      5. metadata-eval 94.2%

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

   3. Simplified 94.2%

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

   if 2.00000000000000003e263 < (*.f64 x x)

   1. Initial program 79.5%

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

      1. cancel-sign-sub-inv 79.5%

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

      2. distribute-lft-neg-out 79.5%

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

      3. +-commutative 79.5%

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

      4. distribute-lft-neg-out 79.5%

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

      5. distribute-lft-neg-in 79.5%

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

      6. distribute-rgt-neg-in 79.5%

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

      7. fma-define 80.8%

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

      8. sub-neg 80.8%

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

      9. +-commutative 80.8%

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

      10. distribute-neg-in 80.8%

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

      11. remove-double-neg 80.8%

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

      12. sub-neg 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in t around inf 94.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.9 \cdot 10^{+270}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.9e+270)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (fma (* y 4.0) t (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.9e+270) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma((y * 4.0), t, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.9e+270)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(Float64(y * 4.0), t, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.9e+270], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 4.0), $MachinePrecision] * t + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.90000000000000009e270

   1. Initial program 94.2%

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

   if 1.90000000000000009e270 < (*.f64 x x)

   1. Initial program 79.5%

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

      1. cancel-sign-sub-inv 79.5%

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

      2. distribute-lft-neg-out 79.5%

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

      3. +-commutative 79.5%

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

      4. distribute-lft-neg-out 79.5%

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

      5. distribute-lft-neg-in 79.5%

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

      6. distribute-rgt-neg-in 79.5%

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

      7. fma-define 80.8%

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

      8. sub-neg 80.8%

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

      9. +-commutative 80.8%

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

      10. distribute-neg-in 80.8%

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

      11. remove-double-neg 80.8%

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

      12. sub-neg 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in t around inf 94.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.9 \cdot 10^{+270}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.9 \cdot 10^{-10}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \frac{z \cdot -4}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2.9e-10)
   (* t (* y (+ 4.0 (/ (* z -4.0) (/ t z)))))
   (- (* x x) (* -4.0 (* t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.9e-10) {
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	} else {
		tmp = (x * x) - (-4.0 * (t * y));
	}
	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) <= 2.9d-10) then
        tmp = t * (y * (4.0d0 + ((z * (-4.0d0)) / (t / z))))
    else
        tmp = (x * x) - ((-4.0d0) * (t * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2.9e-10) {
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	} else {
		tmp = (x * x) - (-4.0 * (t * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2.9e-10:
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))))
	else:
		tmp = (x * x) - (-4.0 * (t * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2.9e-10)
		tmp = Float64(t * Float64(y * Float64(4.0 + Float64(Float64(z * -4.0) / Float64(t / z)))));
	else
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 2.9e-10)
		tmp = t * (y * (4.0 + ((z * -4.0) / (t / z))));
	else
		tmp = (x * x) - (-4.0 * (t * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.9e-10], N[(t * N[(y * N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.89999999999999981e-10

   1. Initial program 92.8%

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

   3. Taylor expanded in t around inf 89.4%

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

      1. +-commutative 89.4%

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

      2. fma-define 89.4%

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

      3. associate-/l* 88.6%

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

      4. *-commutative 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around -inf 82.8%

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

      1. add-sqr-sqrt 40.9%

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

      2. sqrt-prod 51.8%

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

      3. sqr-neg 51.8%

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

      4. sqrt-prod 24.0%

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

      5. add-sqr-sqrt 45.2%

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

      6. unpow2 45.2%

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

      7. *-un-lft-identity 45.2%

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

      8. times-frac 45.2%

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

      9. add-sqr-sqrt 24.0%

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

      10. sqrt-prod 55.4%

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

      11. sqr-neg 55.4%

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

      12. sqrt-prod 42.5%

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

      13. add-sqr-sqrt 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. /-rgt-identity 85.9%

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

      2. associate-*r* 85.9%

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

      3. clear-num 85.9%

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

      4. un-div-inv 85.9%

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

   10. Applied egg-rr 85.9%

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

   if 2.89999999999999981e-10 < (*.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 81.8%

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

      1. *-commutative 81.8%

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

   5. Simplified 81.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.9 \cdot 10^{-10}:\\ \;\;\;\;t \cdot \left(y \cdot \left(4 + \frac{z \cdot -4}{\frac{t}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.95 \cdot 10^{+270}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.95e+270) (+ (* 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) <= 1.95e+270) {
		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) <= 1.95d+270) 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) <= 1.95e+270) {
		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) <= 1.95e+270:
		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) <= 1.95e+270)
		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) <= 1.95e+270)
		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], 1.95e+270], 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) < 1.95e270

   1. Initial program 94.2%

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

   if 1.95e270 < (*.f64 x x)

   1. Initial program 79.5%

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

   3. Taylor expanded in y around 0 79.5%

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

   5. Simplified 94.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.95 \cdot 10^{+270}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 3.6e-11) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 3.6e-11) {
		tmp = y * (t * 4.0);
	} 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) <= 3.6d-11) then
        tmp = y * (t * 4.0d0)
    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) <= 3.6e-11) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 3.6e-11:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 3.6e-11)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 3.6e-11)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.6e-11], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.59999999999999985e-11

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf 49.3%

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

      1. associate-*r* 49.3%

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

   5. Simplified 49.3%

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

   if 3.59999999999999985e-11 < (*.f64 x x)

   1. Initial program 87.4%

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

   3. Taylor expanded in y around 0 87.4%

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

   5. Simplified 74.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((-4.0d0) * (t * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in z around 0 69.7%

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

   1. *-commutative 69.7%

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

5. Simplified 69.7%

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

6. Final simplification 69.7%

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

Alternative 7: 6.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-4.0d0) * (t * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

   1. sub-neg 89.7%

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

   2. flip-+ 58.0%

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

   3. pow2 58.0%

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

   4. pow2 58.0%

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

   5. pow-prod-up 58.0%

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

   6. metadata-eval 58.0%

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

   7. pow2 58.0%

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

4. Applied egg-rr 58.0%

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

   1. clear-num 57.9%

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

   2. un-div-inv 57.9%

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

   3. clear-num 57.9%

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

   4. metadata-eval 57.9%

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

   5. pow-prod-up 57.9%

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

   6. flip-+ 89.6%

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

   7. unpow2 89.6%

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

   8. fma-define 89.6%

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

   9. add-sqr-sqrt 42.2%

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

   10. sqrt-prod 72.7%

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

   11. sqr-neg 72.7%

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

   12. sqrt-prod 36.4%

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

   13. add-sqr-sqrt 66.4%

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

6. Applied egg-rr 66.4%

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

7. Taylor expanded in t around inf 5.5%

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

   1. *-commutative 5.5%

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

9. Simplified 5.5%

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

10. Final simplification 5.5%

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

Alternative 8: 32.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 * (t * 4.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in t around inf 29.2%

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

   1. associate-*r* 29.2%

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

5. Simplified 29.2%

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

6. Final simplification 29.2%

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

Developer target: 90.6% 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 2024081 
(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))))