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

Percentage Accurate: 90.8% → 98.5%

Time: 7.4s

Alternatives: 7

Speedup: 0.9×

• 49.0% 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: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -2 \cdot 10^{+43} \lor \neg \left(y \cdot 4 \leq 2000000\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y 4.0) -2e+43) (not (<= (* y 4.0) 2000000.0)))
   (fma x x (* (* (- (* z z) t) y) -4.0))
   (fma (* (* -4.0 y) z) z (fma (* (- t) y) -4.0 (* x x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * 4.0) <= -2e+43) || !((y * 4.0) <= 2000000.0)) {
		tmp = fma(x, x, ((((z * z) - t) * y) * -4.0));
	} else {
		tmp = fma(((-4.0 * y) * z), z, fma((-t * y), -4.0, (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * 4.0) <= -2e+43) || !(Float64(y * 4.0) <= 2000000.0))
		tmp = fma(x, x, Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0));
	else
		tmp = fma(Float64(Float64(-4.0 * y) * z), z, fma(Float64(Float64(-t) * y), -4.0, Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * 4.0), $MachinePrecision], -2e+43], N[Not[LessEqual[N[(y * 4.0), $MachinePrecision], 2000000.0]], $MachinePrecision]], N[(x * x + N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(N[((-t) * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 4 binary64)) < -2.00000000000000003e43 or 2e6 < (*.f64 y #s(literal 4 binary64))

   1. Initial program 90.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. 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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.1

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

   4. Applied rewrites 98.1%

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

   if -2.00000000000000003e43 < (*.f64 y #s(literal 4 binary64)) < 2e6

   1. Initial program 91.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. distribute-lft-neg-in N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -2 \cdot 10^{+43} \lor \neg \left(y \cdot 4 \leq 2000000\right):\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.75 \cdot 10^{-205}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+34}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot z\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.75e-205)
   (* x x)
   (if (<= z 6.5e-161)
     (* (* t 4.0) y)
     (if (<= z 8.5e+34) (* x x) (* (* (* z y) z) -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.75e-205) {
		tmp = x * x;
	} else if (z <= 6.5e-161) {
		tmp = (t * 4.0) * y;
	} else if (z <= 8.5e+34) {
		tmp = x * x;
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 3.75d-205) then
        tmp = x * x
    else if (z <= 6.5d-161) then
        tmp = (t * 4.0d0) * y
    else if (z <= 8.5d+34) then
        tmp = x * x
    else
        tmp = ((z * y) * z) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.75e-205) {
		tmp = x * x;
	} else if (z <= 6.5e-161) {
		tmp = (t * 4.0) * y;
	} else if (z <= 8.5e+34) {
		tmp = x * x;
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.75e-205:
		tmp = x * x
	elif z <= 6.5e-161:
		tmp = (t * 4.0) * y
	elif z <= 8.5e+34:
		tmp = x * x
	else:
		tmp = ((z * y) * z) * -4.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.75e-205)
		tmp = Float64(x * x);
	elseif (z <= 6.5e-161)
		tmp = Float64(Float64(t * 4.0) * y);
	elseif (z <= 8.5e+34)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(z * y) * z) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.75e-205)
		tmp = x * x;
	elseif (z <= 6.5e-161)
		tmp = (t * 4.0) * y;
	elseif (z <= 8.5e+34)
		tmp = x * x;
	else
		tmp = ((z * y) * z) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.75e-205], N[(x * x), $MachinePrecision], If[LessEqual[z, 6.5e-161], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 8.5e+34], N[(x * x), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.7499999999999998e-205 or 6.50000000000000008e-161 < z < 8.5000000000000003e34

   1. Initial program 91.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.1

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

   4. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 48.0

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

   9. Applied rewrites 48.0%

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

   if 3.7499999999999998e-205 < z < 6.50000000000000008e-161

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 69.2

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

   5. Applied rewrites 69.2%

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

   7. Applied rewrites 69.2%

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

   if 8.5000000000000003e34 < z

   1. Initial program 86.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 81.1%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -4, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.75e+176)
   (fma x x (* (* (- (* z z) t) y) -4.0))
   (fma (* (* y z) -4.0) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.75e+176) {
		tmp = fma(x, x, ((((z * z) - t) * y) * -4.0));
	} else {
		tmp = fma(((y * z) * -4.0), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.75e+176)
		tmp = fma(x, x, Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0));
	else
		tmp = fma(Float64(Float64(y * z) * -4.0), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.75e+176], N[(x * x + N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.75000000000000001e176

   1. Initial program 92.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. 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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.7

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

   4. Applied rewrites 95.7%

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

   if 1.75000000000000001e176 < 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 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 2.1

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

   5. Applied rewrites 2.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{{x}^{2} - 4 \cdot \left(y \cdot {z}^{2}\right)} \]
   7. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.3

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

   8. Applied rewrites 96.3%

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

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot t\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot -4, z, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.15e-9)
   (fma x x (* (* y t) 4.0))
   (fma (* (* y z) -4.0) z (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.15e-9) {
		tmp = fma(x, x, ((y * t) * 4.0));
	} else {
		tmp = fma(((y * z) * -4.0), z, (x * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.15e-9)
		tmp = fma(x, x, Float64(Float64(y * t) * 4.0));
	else
		tmp = fma(Float64(Float64(y * z) * -4.0), z, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.15e-9], N[(x * x + N[(N[(y * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z), $MachinePrecision] * -4.0), $MachinePrecision] * z + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.15e-9

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 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 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.9%

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

   if 1.15e-9 < z

   1. Initial program 86.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   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 7.5

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

   5. Applied rewrites 7.5%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{{x}^{2} - 4 \cdot \left(y \cdot {z}^{2}\right)} \]
   7. 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. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 94.4

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

   8. Applied rewrites 94.4%

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

Alternative 5: 76.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.96 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot t\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot z\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.96e+51) (fma x x (* (* y t) 4.0)) (* (* (* z y) z) -4.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.96e+51) {
		tmp = fma(x, x, ((y * t) * 4.0));
	} else {
		tmp = ((z * y) * z) * -4.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.96e+51)
		tmp = fma(x, x, Float64(Float64(y * t) * 4.0));
	else
		tmp = Float64(Float64(Float64(z * y) * z) * -4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.96e+51], N[(x * x + N[(N[(y * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.9600000000000001e51

   1. Initial program 91.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 75.8%

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

   if 1.9600000000000001e51 < z

   1. Initial program 86.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 84.1%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 45.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-44}:\\ \;\;\;\;\left(t \cdot 4\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.95e-44) (* (* t 4.0) y) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.95e-44) {
		tmp = (t * 4.0) * y;
	} 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 <= 1.95d-44) then
        tmp = (t * 4.0d0) * y
    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 <= 1.95e-44) {
		tmp = (t * 4.0) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.95e-44:
		tmp = (t * 4.0) * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.95e-44)
		tmp = Float64(Float64(t * 4.0) * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.95e-44)
		tmp = (t * 4.0) * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.95e-44], N[(N[(t * 4.0), $MachinePrecision] * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9500000000000001e-44

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   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 34.0

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

   5. Applied rewrites 34.0%

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

   7. Applied rewrites 34.0%

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

   if 1.9500000000000001e-44 < 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. 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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 93.0

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

   4. Applied rewrites 93.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 93.0

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

   6. Applied rewrites 93.0%

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

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.9

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

   9. Applied rewrites 70.9%

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

Alternative 7: 40.8% 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 90.8%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
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. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 93.9

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

4. Applied rewrites 93.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. flip-- N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. clear-num N/A

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

   13. flip-- N/A

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

   14. lift--.f64 N/A

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

   15. lower-/.f64 93.9

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

6. Applied rewrites 93.9%

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

7. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 43.9

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

9. Applied rewrites 43.9%

   \[\leadsto \color{blue}{x \cdot x} \]
10. 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 2024318 
(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))))