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

Percentage Accurate: 90.8% → 93.1%

Time: 12.0s

Alternatives: 8

Speedup: 0.4×

• 48.4% 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: 90.8% 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: 93.1% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1e+289)
   (+ (* 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) <= 1e+289) {
		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) <= 1e+289)
		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], 1e+289], 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) < 1.0000000000000001e289

   1. Initial program 96.0%

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

   if 1.0000000000000001e289 < (*.f64 x x)

   1. Initial program 90.6%

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

      1. cancel-sign-sub-inv 90.6%

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

      2. distribute-lft-neg-out 90.6%

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

      3. +-commutative 90.6%

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

      4. distribute-lft-neg-out 90.6%

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

      5. distribute-lft-neg-in 90.6%

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

      6. distribute-rgt-neg-in 90.6%

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

      7. fma-define 95.3%

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

      8. sub-neg 95.3%

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

      9. +-commutative 95.3%

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

      10. distribute-neg-in 95.3%

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

      11. remove-double-neg 95.3%

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

      12. sub-neg 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in t around inf 96.9%

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

4. Final simplification 96.2%

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

Alternative 2: 91.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. cancel-sign-sub-inv 94.7%

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

   2. distribute-lft-neg-out 94.7%

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

   3. +-commutative 94.7%

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

   4. distribute-lft-neg-out 94.7%

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

   5. distribute-lft-neg-in 94.7%

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

   6. distribute-rgt-neg-in 94.7%

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

   7. fma-define 95.9%

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

   8. neg-sub0 95.9%

      \[\leadsto \mathsf{fma}\left(y \cdot 4, \color{blue}{0 - \left(z \cdot z - t\right)}, x \cdot x\right) \]

   9. associate-+l- 95.9%

      \[\leadsto \mathsf{fma}\left(y \cdot 4, \color{blue}{\left(0 - z \cdot z\right) + t}, x \cdot x\right) \]

   10. neg-sub0 95.9%

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

   11. distribute-rgt-neg-out 95.9%

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

   12. fma-define 95.9%

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

3. Simplified 95.9%

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

Alternative 3: 91.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. cancel-sign-sub-inv 94.7%

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

   2. distribute-lft-neg-out 94.7%

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

   3. +-commutative 94.7%

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

   4. distribute-lft-neg-out 94.7%

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

   5. distribute-lft-neg-in 94.7%

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

   6. distribute-rgt-neg-in 94.7%

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

   7. fma-define 95.9%

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

   8. sub-neg 95.9%

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

   9. +-commutative 95.9%

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

   10. distribute-neg-in 95.9%

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

   11. remove-double-neg 95.9%

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

   12. sub-neg 95.9%

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

3. Simplified 95.9%

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

Alternative 4: 92.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. fmm-def 95.5%

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

   2. distribute-lft-neg-in 95.5%

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

   3. *-commutative 95.5%

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

   4. distribute-rgt-neg-in 95.5%

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

   5. metadata-eval 95.5%

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

3. Simplified 95.5%

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

Alternative 5: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (+ (* x x) (* (* y 4.0) (- (* z z) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x * x) + ((y * 4.0) * ((z * z) - t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (x * x) + ((y * 4.0) * ((z * z) - t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = (x * x) + ((y * 4.0) * ((z * z) - t))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (x * x) + ((y * 4.0) * ((z * z) - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 97.0%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 16.7%

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

      3. swap-sqr 16.7%

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

      4. metadata-eval 16.7%

         \[\leadsto x \cdot x - \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left(z \cdot z - t\right) \]

      5. metadata-eval 16.7%

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

      6. swap-sqr 16.7%

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

      7. sqrt-unprod 16.7%

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

      8. add-sqr-sqrt 66.7%

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

      9. metadata-eval 66.7%

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

      10. distribute-rgt-neg-in 66.7%

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

   4. Applied egg-rr 66.7%

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

4. Final simplification 96.2%

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

Alternative 6: 93.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (- (* x x) (* (* y 4.0) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x * x) - ((y * 4.0) * t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (x * x) - ((y * 4.0) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = (x * x) - ((y * 4.0) * t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(y * 4.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (x * x) - ((y * 4.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 97.0%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 16.7%

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

      3. swap-sqr 16.7%

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

      4. metadata-eval 16.7%

         \[\leadsto x \cdot x - \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}} \cdot \left(z \cdot z - t\right) \]

      5. metadata-eval 16.7%

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

      6. swap-sqr 16.7%

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

      7. sqrt-unprod 16.7%

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

      8. add-sqr-sqrt 66.7%

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

      9. metadata-eval 66.7%

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

      10. distribute-rgt-neg-in 66.7%

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in z around 0 66.7%

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

      1. neg-mul-1 66.7%

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

   7. Simplified 66.7%

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

4. Final simplification 96.2%

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

Alternative 7: 66.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in z around 0 67.0%

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

   1. *-commutative 67.0%

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

   2. *-commutative 67.0%

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

   3. associate-*l* 67.0%

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

5. Simplified 67.0%

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

Alternative 8: 32.3% 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 94.7%

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

   1. fmm-def 95.5%

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

   2. distribute-lft-neg-in 95.5%

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

   3. *-commutative 95.5%

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

   4. distribute-rgt-neg-in 95.5%

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

   5. metadata-eval 95.5%

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

3. Simplified 95.5%

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

5. Taylor expanded in t around inf 32.2%

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

   1. *-commutative 32.2%

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

7. Simplified 32.2%

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

Developer Target 1: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024170 
(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))))