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

Percentage Accurate: 91.1% → 93.4%

Time: 10.9s

Alternatives: 6

Speedup: 0.5×

• 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: 91.1% 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: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot \frac{z}{t} + -1\right) \cdot \left(t \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* (- (* z z) t) (* y 4.0)) 5e+306)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* -4.0 (* (+ (* z (/ z t)) -1.0) (* t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((z * z) - t) * (y * 4.0)) <= 5e+306) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (((z * (z / t)) + -1.0) * (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 ((((z * z) - t) * (y * 4.0d0)) <= 5d+306) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (-4.0d0) * (((z * (z / t)) + (-1.0d0)) * (t * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((((z * z) - t) * (y * 4.0)) <= 5e+306) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = -4.0 * (((z * (z / t)) + -1.0) * (t * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((z * z) - t) * (y * 4.0)) <= 5e+306:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = -4.0 * (((z * (z / t)) + -1.0) * (t * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(z * z) - t) * Float64(y * 4.0)) <= 5e+306)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(z * Float64(z / t)) + -1.0) * Float64(t * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((z * z) - t) * (y * 4.0)) <= 5e+306)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = -4.0 * (((z * (z / t)) + -1.0) * (t * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 5e+306], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 4.99999999999999993e306

   1. Initial program 96.0%

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

   if 4.99999999999999993e306 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 70.2%

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

   3. Taylor expanded in t around inf 70.2%

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

      1. unpow2 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. times-frac 70.2%

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

   5. Applied egg-rr 70.2%

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

   6. Taylor expanded in x around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*r* 83.1%

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

      3. sub-neg 83.1%

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

      4. metadata-eval 83.1%

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

   8. Simplified 83.1%

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

      1. unpow2 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. times-frac 70.2%

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

   10. Applied egg-rr 85.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z - t\right) \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot \frac{z}{t} + -1\right) \cdot \left(t \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.4% 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 91.3%

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

   1. fma-neg 93.7%

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

   2. distribute-lft-neg-in 93.7%

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

   3. *-commutative 93.7%

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

   4. distribute-rgt-neg-in 93.7%

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

   5. metadata-eval 93.7%

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

3. Simplified 93.7%

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

Alternative 3: 93.4% accurate, 0.4× speedup

Math

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

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)))))

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;
	}
	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;
	}
	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
	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(x * Float64(-x));
	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;
	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[(x * (-x)), $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 94.3%

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

      1. cancel-sign-sub-inv 0.0%

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

      2. distribute-lft-neg-out 0.0%

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

      3. +-commutative 0.0%

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

      4. distribute-lft-neg-out 0.0%

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

      5. distribute-lft-neg-in 0.0%

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

      6. distribute-rgt-neg-in 0.0%

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

      7. fma-define 12.5%

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

      8. sub-neg 12.5%

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

      9. +-commutative 12.5%

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

      10. distribute-neg-in 12.5%

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

      11. remove-double-neg 12.5%

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

      12. sub-neg 12.5%

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

   3. Simplified 12.5%

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

      1. fma-undefine 0.0%

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

      2. flip-+ 0.0%

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

      3. pow2 0.0%

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

      4. pow2 0.0%

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

      5. pow2 0.0%

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

      6. pow2 0.0%

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

      7. pow-prod-up 0.0%

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

      8. metadata-eval 0.0%

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

      9. pow2 0.0%

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

      10. pow2 0.0%

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

   6. Applied egg-rr 0.0%

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

   8. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-{x}^{4}}{-{x}^{2}}} \]
   9. Step-by-step derivation

      1. neg-sub0 0.0%

         \[\leadsto \frac{\color{blue}{0 - {x}^{4}}}{-{x}^{2}} \]

      2. metadata-eval 0.0%

         \[\leadsto \frac{\color{blue}{0 \cdot 0} - {x}^{4}}{-{x}^{2}} \]

      3. metadata-eval 0.0%

         \[\leadsto \frac{0 \cdot 0 - {x}^{\color{blue}{\left(2 + 2\right)}}}{-{x}^{2}} \]

      4. pow-prod-up 0.0%

         \[\leadsto \frac{0 \cdot 0 - \color{blue}{{x}^{2} \cdot {x}^{2}}}{-{x}^{2}} \]

      5. pow-prod-down 0.0%

         \[\leadsto \frac{0 \cdot 0 - \color{blue}{{\left(x \cdot x\right)}^{2}}}{-{x}^{2}} \]

      6. pow2 0.0%

         \[\leadsto \frac{0 \cdot 0 - \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{-{x}^{2}} \]

      7. pow2 0.0%

         \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{-\color{blue}{x \cdot x}} \]

      8. neg-sub0 0.0%

         \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{0 - x \cdot x}} \]

      9. sub-neg 0.0%

         \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{0 + \left(-x \cdot x\right)}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{-x \cdot x} \cdot \sqrt{-x \cdot x}}} \]

      11. sqrt-unprod 0.0%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{\left(-x \cdot x\right) \cdot \left(-x \cdot x\right)}}} \]

      12. sqr-neg 0.0%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}} \]

      13. sqrt-prod 0.0%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{x \cdot x} \cdot \sqrt{x \cdot x}}} \]

      14. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{x \cdot x}} \]

      15. flip-- 75.0%

          \[\leadsto \color{blue}{0 - x \cdot x} \]

      16. neg-sub0 75.0%

          \[\leadsto \color{blue}{-x \cdot x} \]

      17. distribute-lft-neg-in 75.0%

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

   10. Applied egg-rr 75.0%

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

4. Final simplification 93.7%

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

Alternative 4: 45.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 7.2e+16) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 7.2e+16) {
		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 <= 7.2d+16) 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 <= 7.2e+16) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 7.2e+16:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 7.2e+16)
		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 <= 7.2e+16)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 7.2e+16], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e16

   1. Initial program 92.9%

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

   3. Taylor expanded in t around inf 38.5%

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

      1. associate-*r* 38.5%

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

   5. Simplified 38.5%

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

   if 7.2e16 < x

   1. Initial program 88.0%

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

      1. cancel-sign-sub-inv 88.0%

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

      2. distribute-lft-neg-out 88.0%

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

      3. +-commutative 88.0%

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

      4. distribute-lft-neg-out 88.0%

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

      5. distribute-lft-neg-in 88.0%

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

      6. distribute-rgt-neg-in 88.0%

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

      7. fma-define 89.3%

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

      8. sub-neg 89.3%

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

      9. +-commutative 89.3%

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

      10. distribute-neg-in 89.3%

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

      11. remove-double-neg 89.3%

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

      12. sub-neg 89.3%

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

   3. Simplified 89.3%

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

      1. fma-undefine 88.0%

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

      2. flip-+ 8.6%

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

      3. pow2 8.6%

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

      4. pow2 8.6%

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

      5. pow2 8.6%

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

      6. pow2 8.6%

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

      7. pow-prod-up 8.5%

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

      8. metadata-eval 8.5%

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

      9. pow2 8.5%

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

      10. pow2 8.5%

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

   6. Applied egg-rr 8.5%

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

   8. Simplified 11.5%

      \[\leadsto \color{blue}{\frac{-{x}^{4}}{-{x}^{2}}} \]
   9. Step-by-step derivation

      1. neg-sub0 11.5%

         \[\leadsto \frac{\color{blue}{0 - {x}^{4}}}{-{x}^{2}} \]

      2. metadata-eval 11.5%

         \[\leadsto \frac{\color{blue}{0 \cdot 0} - {x}^{4}}{-{x}^{2}} \]

      3. metadata-eval 11.5%

         \[\leadsto \frac{0 \cdot 0 - {x}^{\color{blue}{\left(2 + 2\right)}}}{-{x}^{2}} \]

      4. pow-prod-up 11.5%

         \[\leadsto \frac{0 \cdot 0 - \color{blue}{{x}^{2} \cdot {x}^{2}}}{-{x}^{2}} \]

      5. pow-prod-down 11.5%

         \[\leadsto \frac{0 \cdot 0 - \color{blue}{{\left(x \cdot x\right)}^{2}}}{-{x}^{2}} \]

      6. pow2 11.5%

         \[\leadsto \frac{0 \cdot 0 - \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{-{x}^{2}} \]

      7. pow2 11.5%

         \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{-\color{blue}{x \cdot x}} \]

      8. neg-sub0 11.5%

         \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{0 - x \cdot x}} \]

      9. sub-neg 11.5%

         \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{0 + \left(-x \cdot x\right)}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{-x \cdot x} \cdot \sqrt{-x \cdot x}}} \]

      11. sqrt-unprod 0.6%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{\left(-x \cdot x\right) \cdot \left(-x \cdot x\right)}}} \]

      12. sqr-neg 0.6%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}} \]

      13. sqrt-prod 4.4%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{x \cdot x} \cdot \sqrt{x \cdot x}}} \]

      14. add-sqr-sqrt 4.4%

          \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{x \cdot x}} \]

      15. flip-- 4.7%

          \[\leadsto \color{blue}{0 - x \cdot x} \]

      16. neg-sub0 4.7%

          \[\leadsto \color{blue}{-x \cdot x} \]

      17. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{-x \cdot x} \cdot \sqrt{-x \cdot x}} \]

      18. sqrt-unprod 56.6%

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

      19. sqr-neg 56.6%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}} \]

      20. sqrt-prod 68.0%

          \[\leadsto \color{blue}{\sqrt{x \cdot x} \cdot \sqrt{x \cdot x}} \]

      21. add-sqr-sqrt 68.0%

          \[\leadsto \color{blue}{x \cdot x} \]

   10. Applied egg-rr 68.0%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% 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 91.3%

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

3. Taylor expanded in z around 0 66.8%

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

   1. *-commutative 66.8%

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

5. Simplified 66.8%

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

6. Final simplification 66.8%

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

Alternative 6: 41.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. cancel-sign-sub-inv 91.3%

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

   2. distribute-lft-neg-out 91.3%

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

   3. +-commutative 91.3%

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

   4. distribute-lft-neg-out 91.3%

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

   5. distribute-lft-neg-in 91.3%

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

   6. distribute-rgt-neg-in 91.3%

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

   7. fma-define 91.7%

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

   8. sub-neg 91.7%

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

   9. +-commutative 91.7%

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

   10. distribute-neg-in 91.7%

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

   11. remove-double-neg 91.7%

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

   12. sub-neg 91.7%

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

3. Simplified 91.7%

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

   1. fma-undefine 91.3%

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

   2. flip-+ 20.8%

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

   3. pow2 20.8%

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

   4. pow2 20.8%

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

   5. pow2 20.8%

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

   6. pow2 20.8%

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

   7. pow-prod-up 20.8%

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

   8. metadata-eval 20.8%

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

   9. pow2 20.8%

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

   10. pow2 20.8%

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

6. Applied egg-rr 20.8%

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

8. Simplified 8.5%

   \[\leadsto \color{blue}{\frac{-{x}^{4}}{-{x}^{2}}} \]
9. Step-by-step derivation

   1. neg-sub0 8.5%

      \[\leadsto \frac{\color{blue}{0 - {x}^{4}}}{-{x}^{2}} \]

   2. metadata-eval 8.5%

      \[\leadsto \frac{\color{blue}{0 \cdot 0} - {x}^{4}}{-{x}^{2}} \]

   3. metadata-eval 8.5%

      \[\leadsto \frac{0 \cdot 0 - {x}^{\color{blue}{\left(2 + 2\right)}}}{-{x}^{2}} \]

   4. pow-prod-up 8.6%

      \[\leadsto \frac{0 \cdot 0 - \color{blue}{{x}^{2} \cdot {x}^{2}}}{-{x}^{2}} \]

   5. pow-prod-down 8.6%

      \[\leadsto \frac{0 \cdot 0 - \color{blue}{{\left(x \cdot x\right)}^{2}}}{-{x}^{2}} \]

   6. pow2 8.6%

      \[\leadsto \frac{0 \cdot 0 - \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{-{x}^{2}} \]

   7. pow2 8.6%

      \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{-\color{blue}{x \cdot x}} \]

   8. neg-sub0 8.6%

      \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{0 - x \cdot x}} \]

   9. sub-neg 8.6%

      \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{0 + \left(-x \cdot x\right)}} \]

   10. add-sqr-sqrt 0.0%

       \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{-x \cdot x} \cdot \sqrt{-x \cdot x}}} \]

   11. sqrt-unprod 0.6%

       \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{\left(-x \cdot x\right) \cdot \left(-x \cdot x\right)}}} \]

   12. sqr-neg 0.6%

       \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}} \]

   13. sqrt-prod 3.4%

       \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{\sqrt{x \cdot x} \cdot \sqrt{x \cdot x}}} \]

   14. add-sqr-sqrt 3.4%

       \[\leadsto \frac{0 \cdot 0 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{0 + \color{blue}{x \cdot x}} \]

   15. flip-- 5.9%

       \[\leadsto \color{blue}{0 - x \cdot x} \]

   16. neg-sub0 5.9%

       \[\leadsto \color{blue}{-x \cdot x} \]

   17. add-sqr-sqrt 2.3%

       \[\leadsto \color{blue}{\sqrt{-x \cdot x} \cdot \sqrt{-x \cdot x}} \]

   18. sqrt-unprod 34.7%

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

   19. sqr-neg 34.7%

       \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}} \]

   20. sqrt-prod 41.6%

       \[\leadsto \color{blue}{\sqrt{x \cdot x} \cdot \sqrt{x \cdot x}} \]

   21. add-sqr-sqrt 41.6%

       \[\leadsto \color{blue}{x \cdot x} \]

10. Applied egg-rr 41.6%

    \[\leadsto \color{blue}{x \cdot x} \]
11. Add Preprocessing

Developer target: 91.1% 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 2024110 
(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))))