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

Percentage Accurate: 90.5% → 98.0%

Time: 8.3s

Alternatives: 8

Speedup: 0.8×

• 48.9% 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: 90.5% 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: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, -t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+248)
   (fma x x (* y (* (fma z z (- t)) -4.0)))
   (fma x x (* -4.0 (* z (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+248) {
		tmp = fma(x, x, (y * (fma(z, z, -t) * -4.0)));
	} else {
		tmp = fma(x, x, (-4.0 * (z * (z * y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+248)
		tmp = fma(x, x, Float64(y * Float64(fma(z, z, Float64(-t)) * -4.0)));
	else
		tmp = fma(x, x, Float64(-4.0 * Float64(z * Float64(z * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+248], N[(x * x + N[(y * N[(N[(z * z + (-t)), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000018e248

   1. Initial program 99.3%

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

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   if 4.00000000000000018e248 < (*.f64 z z)

   1. Initial program 72.8%

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

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. metadata-eval 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.9

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

   10. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, -t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(-4, \left(z \cdot z\right) \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-96)
   (fma y (* t 4.0) (* x x))
   (if (<= (* z z) 4e+248)
     (fma -4.0 (* (* z z) y) (* x x))
     (* z (* z (* y -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-96) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else if ((z * z) <= 4e+248) {
		tmp = fma(-4.0, ((z * z) * y), (x * x));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-96)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	elseif (Float64(z * z) <= 4e+248)
		tmp = fma(-4.0, Float64(Float64(z * z) * y), Float64(x * x));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-96], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 4e+248], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.99999999999999995e-96

   1. Initial program 98.9%

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 4.99999999999999995e-96 < (*.f64 z z) < 4.00000000000000018e248

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 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. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if 4.00000000000000018e248 < (*.f64 z z)

   1. Initial program 72.8%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 91.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(-4, \left(z \cdot z\right) \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+215}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -1e+18)
     (* y (* t 4.0))
     (if (<= t_1 1e+215) (* x x) (* z (* z (* y -4.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = y * (t * 4.0);
	} else if (t_1 <= 1e+215) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -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) :: t_1
    real(8) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= (-1d+18)) then
        tmp = y * (t * 4.0d0)
    else if (t_1 <= 1d+215) then
        tmp = x * x
    else
        tmp = z * (z * (y * (-4.0d0)))
    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 <= -1e+18) {
		tmp = y * (t * 4.0);
	} else if (t_1 <= 1e+215) {
		tmp = x * x;
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -1e+18:
		tmp = y * (t * 4.0)
	elif t_1 <= 1e+215:
		tmp = x * x
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = Float64(y * Float64(t * 4.0));
	elseif (t_1 <= 1e+215)
		tmp = Float64(x * x);
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = y * (t * 4.0);
	elseif (t_1 <= 1e+215)
		tmp = x * x;
	else
		tmp = z * (z * (y * -4.0));
	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, -1e+18], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+215], N[(x * x), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -1e18

   1. Initial program 96.3%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -1e18 < (-.f64 (*.f64 z z) t) < 9.99999999999999907e214

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 50.5

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

   5. Applied rewrites 50.5%

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

   if 9.99999999999999907e214 < (-.f64 (*.f64 z z) t)

   1. Initial program 77.1%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 84.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -1 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;z \cdot z - t \leq 10^{+215}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* -4.0 (* (* z z) y))))
   (if (<= x 4.5e-284)
     t_1
     (if (<= x 5e-166) (* y (* t 4.0)) (if (<= x 1.2e+61) t_1 (* x x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * ((z * z) * y);
	double tmp;
	if (x <= 4.5e-284) {
		tmp = t_1;
	} else if (x <= 5e-166) {
		tmp = y * (t * 4.0);
	} else if (x <= 1.2e+61) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * ((z * z) * y)
    if (x <= 4.5d-284) then
        tmp = t_1
    else if (x <= 5d-166) then
        tmp = y * (t * 4.0d0)
    else if (x <= 1.2d+61) then
        tmp = t_1
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * ((z * z) * y);
	double tmp;
	if (x <= 4.5e-284) {
		tmp = t_1;
	} else if (x <= 5e-166) {
		tmp = y * (t * 4.0);
	} else if (x <= 1.2e+61) {
		tmp = t_1;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -4.0 * ((z * z) * y)
	tmp = 0
	if x <= 4.5e-284:
		tmp = t_1
	elif x <= 5e-166:
		tmp = y * (t * 4.0)
	elif x <= 1.2e+61:
		tmp = t_1
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-4.0 * Float64(Float64(z * z) * y))
	tmp = 0.0
	if (x <= 4.5e-284)
		tmp = t_1;
	elseif (x <= 5e-166)
		tmp = Float64(y * Float64(t * 4.0));
	elseif (x <= 1.2e+61)
		tmp = t_1;
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -4.0 * ((z * z) * y);
	tmp = 0.0;
	if (x <= 4.5e-284)
		tmp = t_1;
	elseif (x <= 5e-166)
		tmp = y * (t * 4.0);
	elseif (x <= 1.2e+61)
		tmp = t_1;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.5e-284], t$95$1, If[LessEqual[x, 5e-166], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e+61], t$95$1, N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.4999999999999999e-284 or 5e-166 < x < 1.1999999999999999e61

   1. Initial program 91.0%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   if 4.4999999999999999e-284 < x < 5e-166

   1. Initial program 91.7%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if 1.1999999999999999e61 < x

   1. Initial program 86.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-284}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+61}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-96)
   (fma y (* t 4.0) (* x x))
   (fma x x (* -4.0 (* z (* z y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-96) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else {
		tmp = fma(x, x, (-4.0 * (z * (z * y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-96)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	else
		tmp = fma(x, x, Float64(-4.0 * Float64(z * Float64(z * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-96], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(-4.0 * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999995e-96

   1. Initial program 98.9%

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 4.99999999999999995e-96 < (*.f64 z z)

   1. Initial program 85.5%

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

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. metadata-eval 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 89.0

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

   10. Applied rewrites 89.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(z \cdot \left(z \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+212) (fma y (* t 4.0) (* x x)) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+212) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+212)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+212], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999998e212

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 1.9999999999999998e212 < (*.f64 z z)

   1. Initial program 75.4%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

   7. Applied rewrites 89.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.75e+16) (* y (* t 4.0)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.75e+16) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 2.75d+16) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.75e+16) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.75e+16:
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.75e+16)
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.75e+16)
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.75e+16], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.75e16

   1. Initial program 91.1%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 33.7

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

   5. Applied rewrites 33.7%

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

   if 2.75e16 < x

   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 x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.0

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

   5. Applied rewrites 69.0%

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

Alternative 8: 40.7% 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 90.4%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 35.6

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

5. Applied rewrites 35.6%

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

Developer Target 1: 90.5% 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 2024232 
(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))))