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

Percentage Accurate: 90.8% → 96.3%

Time: 8.2s

Alternatives: 6

Speedup: 0.8×

• 49.5% 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: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot -4\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+285)
   (fma (- (* z z) t) (* y -4.0) (* x x))
   (* (* (* y -4.0) z) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+285) {
		tmp = fma(((z * z) - t), (y * -4.0), (x * x));
	} else {
		tmp = ((y * -4.0) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+285)
		tmp = fma(Float64(Float64(z * z) - t), Float64(y * -4.0), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * -4.0) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+285], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000016e285

   1. Initial program 96.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.4

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

   4. Applied rewrites 98.4%

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

   if 5.00000000000000016e285 < (*.f64 z z)

   1. Initial program 62.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 89.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot -4\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+237}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+235}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot -4\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -5e+237)
     (* (* 4.0 y) t)
     (if (<= t_1 2e+235) (* x x) (* (* (* y -4.0) z) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5e+237) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+235) {
		tmp = x * x;
	} else {
		tmp = ((y * -4.0) * z) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= (-5d+237)) then
        tmp = (4.0d0 * y) * t
    else if (t_1 <= 2d+235) then
        tmp = x * x
    else
        tmp = ((y * (-4.0d0)) * z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5e+237) {
		tmp = (4.0 * y) * t;
	} else if (t_1 <= 2e+235) {
		tmp = x * x;
	} else {
		tmp = ((y * -4.0) * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -5e+237:
		tmp = (4.0 * y) * t
	elif t_1 <= 2e+235:
		tmp = x * x
	else:
		tmp = ((y * -4.0) * z) * z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= -5e+237)
		tmp = Float64(Float64(4.0 * y) * t);
	elseif (t_1 <= 2e+235)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(y * -4.0) * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= -5e+237)
		tmp = (4.0 * y) * t;
	elseif (t_1 <= 2e+235)
		tmp = x * x;
	else
		tmp = ((y * -4.0) * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+237], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+235], N[(x * x), $MachinePrecision], N[(N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -5.0000000000000002e237

   1. Initial program 93.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 81.9%

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

   if -5.0000000000000002e237 < (-.f64 (*.f64 z z) t) < 2.0000000000000001e235

   1. Initial program 97.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

   9. Applied rewrites 87.6%

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

   10. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 61.8

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

   12. Applied rewrites 61.8%

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

   if 2.0000000000000001e235 < (-.f64 (*.f64 z z) t)

   1. Initial program 70.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 76.1%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -5 \cdot 10^{+237}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;z \cdot z - t \leq 2 \cdot 10^{+235}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot -4\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+29)
   (fma (* 4.0 t) y (* x x))
   (fma (* (* y -4.0) z) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+29) {
		tmp = fma((4.0 * t), y, (x * x));
	} else {
		tmp = fma(((y * -4.0) * z), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+29)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	else
		tmp = fma(Float64(Float64(y * -4.0) * z), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+29], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999914e28

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 93.7%

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

   if 9.99999999999999914e28 < (*.f64 z z)

   1. Initial program 75.3%

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

   3. Taylor expanded in t around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 86.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot -4\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+230) (fma (* 4.0 t) y (* x x)) (* (* (* y -4.0) z) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+230) {
		tmp = fma((4.0 * t), y, (x * x));
	} else {
		tmp = ((y * -4.0) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+230)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * -4.0) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+230], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * -4.0), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e230

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 88.6%

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

   if 1.0000000000000001e230 < (*.f64 z z)

   1. Initial program 66.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 86.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot -4\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3600000:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 3600000.0) (* (* 4.0 y) t) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 3600000.0) {
		tmp = (4.0 * y) * t;
	} 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) <= 3600000.0d0) then
        tmp = (4.0d0 * y) * t
    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) <= 3600000.0) {
		tmp = (4.0 * y) * t;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 3600000.0:
		tmp = (4.0 * y) * t
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 3600000.0)
		tmp = Float64(Float64(4.0 * y) * t);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 3600000.0)
		tmp = (4.0 * y) * t;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 3600000.0], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.6e6

   1. Initial program 90.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 49.0

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

   5. Applied rewrites 49.0%

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

   7. Applied rewrites 49.8%

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

   if 3.6e6 < (*.f64 x x)

   1. Initial program 85.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 80.3

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 79.5%

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

   10. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 76.2

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

   12. Applied rewrites 76.2%

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

Alternative 6: 41.4% accurate, 4.5× 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 88.2%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 88.0

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in t around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/l* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 84.3

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

7. Applied rewrites 84.3%

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

9. Applied rewrites 79.6%

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

10. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 45.5

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

12. Applied rewrites 45.5%

    \[\leadsto \color{blue}{x \cdot x} \]
13. 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 2024285 
(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))))