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

Percentage Accurate: 90.2% → 97.8%

Time: 8.5s

Alternatives: 13

Speedup: 0.7×

• 49.9% 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: 90.2% 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: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;4 \cdot y \leq -5 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -4 \cdot y, x \cdot x\right)\\ \mathbf{elif}\;4 \cdot y \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(t\_1 \cdot y\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= (* 4.0 y) -5e-120)
     (fma t_1 (* -4.0 y) (* x x))
     (if (<= (* 4.0 y) 5e+81)
       (fma (* (* -4.0 y) z) z (fma (* (- t) y) -4.0 (* x x)))
       (fma x x (* (* t_1 y) -4.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if ((4.0 * y) <= -5e-120) {
		tmp = fma(t_1, (-4.0 * y), (x * x));
	} else if ((4.0 * y) <= 5e+81) {
		tmp = fma(((-4.0 * y) * z), z, fma((-t * y), -4.0, (x * x)));
	} else {
		tmp = fma(x, x, ((t_1 * y) * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (Float64(4.0 * y) <= -5e-120)
		tmp = fma(t_1, Float64(-4.0 * y), Float64(x * x));
	elseif (Float64(4.0 * y) <= 5e+81)
		tmp = fma(Float64(Float64(-4.0 * y) * z), z, fma(Float64(Float64(-t) * y), -4.0, Float64(x * x)));
	else
		tmp = fma(x, x, Float64(Float64(t_1 * y) * -4.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(4.0 * y), $MachinePrecision], -5e-120], N[(t$95$1 * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(4.0 * y), $MachinePrecision], 5e+81], N[(N[(N[(-4.0 * y), $MachinePrecision] * z), $MachinePrecision] * z + N[(N[((-t) * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(t$95$1 * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y #s(literal 4 binary64)) < -5.00000000000000007e-120

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.9

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

   4. Applied rewrites 98.9%

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

   if -5.00000000000000007e-120 < (*.f64 y #s(literal 4 binary64)) < 4.9999999999999998e81

   1. Initial program 88.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. +-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%

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

   if 4.9999999999999998e81 < (*.f64 y #s(literal 4 binary64))

   1. Initial program 83.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. *-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 95.8

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

   4. Applied rewrites 95.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;4 \cdot y \leq -5 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, -4 \cdot y, x \cdot x\right)\\ \mathbf{elif}\;4 \cdot y \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(-t, -4 \cdot y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z, -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+36)
   (fma (- t) (* -4.0 y) (* x x))
   (if (<= (* z z) 2e+293)
     (fma (* z z) (* -4.0 y) (* x x))
     (* (* (* -4.0 z) y) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+36) {
		tmp = fma(-t, (-4.0 * y), (x * x));
	} else if ((z * z) <= 2e+293) {
		tmp = fma((z * z), (-4.0 * y), (x * x));
	} else {
		tmp = ((-4.0 * z) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+36)
		tmp = fma(Float64(-t), Float64(-4.0 * y), Float64(x * x));
	elseif (Float64(z * z) <= 2e+293)
		tmp = fma(Float64(z * z), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+36], N[((-t) * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+293], N[(N[(z * z), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.99999999999999977e36

   1. Initial program 98.5%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.3

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

   7. Applied rewrites 93.3%

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

   if 4.99999999999999977e36 < (*.f64 z z) < 1.9999999999999998e293

   1. Initial program 92.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.8

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

   7. Applied rewrites 82.8%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 72.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 90.4%

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

Alternative 3: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(-t, -4 \cdot y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(-4, \left(z \cdot z\right) \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+106)
   (fma (- t) (* -4.0 y) (* x x))
   (if (<= (* z z) 2e+293)
     (fma -4.0 (* (* z z) y) (* x x))
     (* (* (* -4.0 z) y) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+106) {
		tmp = fma(-t, (-4.0 * y), (x * x));
	} else if ((z * z) <= 2e+293) {
		tmp = fma(-4.0, ((z * z) * y), (x * x));
	} else {
		tmp = ((-4.0 * z) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+106)
		tmp = fma(Float64(-t), Float64(-4.0 * y), Float64(x * x));
	elseif (Float64(z * z) <= 2e+293)
		tmp = fma(-4.0, Float64(Float64(z * z) * y), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+106], N[((-t) * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+293], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.00000000000000018e106

   1. Initial program 97.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if 2.00000000000000018e106 < (*.f64 z z) < 1.9999999999999998e293

   1. Initial program 93.3%

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

   3. Taylor expanded in t around 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. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 72.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 90.4%

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

Alternative 4: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(z \cdot y, z, \left(-t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.8e-95)
   (fma x x (* (fma (* z y) z (* (- t) y)) -4.0))
   (fma x x (* (* (- (* z z) t) y) -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.8e-95) {
		tmp = fma(x, x, (fma((z * y), z, (-t * y)) * -4.0));
	} else {
		tmp = fma(x, x, ((((z * z) - t) * y) * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.8e-95)
		tmp = fma(x, x, Float64(fma(Float64(z * y), z, Float64(Float64(-t) * y)) * -4.0));
	else
		tmp = fma(x, x, Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.8e-95], N[(x * x + N[(N[(N[(z * y), $MachinePrecision] * z + N[((-t) * y), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.7999999999999999e-95

   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. 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 92.9

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

   4. Applied rewrites 92.9%

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

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 97.5

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

   6. Applied rewrites 97.5%

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

   if 2.7999999999999999e-95 < t

   1. Initial program 90.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. *-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 94.7

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

   4. Applied rewrites 94.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(z \cdot y, z, \left(-t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+293)
   (fma (- (* z z) t) (* -4.0 y) (* x x))
   (* (* (* -4.0 z) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+293) {
		tmp = fma(((z * z) - t), (-4.0 * y), (x * x));
	} else {
		tmp = ((-4.0 * z) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+293)
		tmp = fma(Float64(Float64(z * z) - t), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+293], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999998e293

   1. Initial program 96.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.4

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

   4. Applied rewrites 98.4%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 72.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 90.4%

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

Alternative 6: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+293)
   (fma x x (* (* (- (* z z) t) y) -4.0))
   (* (* (* -4.0 z) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+293) {
		tmp = fma(x, x, ((((z * z) - t) * y) * -4.0));
	} else {
		tmp = ((-4.0 * z) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+293)
		tmp = fma(x, x, Float64(Float64(Float64(Float64(z * z) - t) * y) * -4.0));
	else
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+293], N[(x * x + N[(N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999998e293

   1. Initial program 96.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. 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 98.4

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

   4. Applied rewrites 98.4%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 72.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 90.4%

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

Alternative 7: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(-t, -4 \cdot y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+200)
   (fma (- t) (* -4.0 y) (* x x))
   (* (* (* -4.0 z) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+200) {
		tmp = fma(-t, (-4.0 * y), (x * x));
	} else {
		tmp = ((-4.0 * z) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+200)
		tmp = fma(Float64(-t), Float64(-4.0 * y), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+200], N[((-t) * N[(-4.0 * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e200

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if 1.9999999999999999e200 < (*.f64 z z)

   1. Initial program 79.2%

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

   3. Taylor expanded in z around 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 84.4%

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

Alternative 8: 45.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+94}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.4e-200)
   (* t (* 4.0 y))
   (if (<= x 1.52e+94) (* (* (* -4.0 z) y) z) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.4e-200) {
		tmp = t * (4.0 * y);
	} else if (x <= 1.52e+94) {
		tmp = ((-4.0 * z) * y) * z;
	} 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 <= 3.4d-200) then
        tmp = t * (4.0d0 * y)
    else if (x <= 1.52d+94) then
        tmp = (((-4.0d0) * z) * y) * z
    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 <= 3.4e-200) {
		tmp = t * (4.0 * y);
	} else if (x <= 1.52e+94) {
		tmp = ((-4.0 * z) * y) * z;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 3.4e-200:
		tmp = t * (4.0 * y)
	elif x <= 1.52e+94:
		tmp = ((-4.0 * z) * y) * z
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.4e-200)
		tmp = Float64(t * Float64(4.0 * y));
	elseif (x <= 1.52e+94)
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 3.4e-200)
		tmp = t * (4.0 * y);
	elseif (x <= 1.52e+94)
		tmp = ((-4.0 * z) * y) * z;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 3.4e-200], N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.52e+94], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4000000000000003e-200

   1. Initial program 93.0%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   if 3.4000000000000003e-200 < x < 1.5199999999999999e94

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 44.3%

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

   if 1.5199999999999999e94 < x

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+94}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-200}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 5e-200)
   (* t (* 4.0 y))
   (if (<= x 4.4e+93) (* (* (* z z) y) -4.0) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 5e-200) {
		tmp = t * (4.0 * y);
	} else if (x <= 4.4e+93) {
		tmp = ((z * z) * y) * -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 <= 5d-200) then
        tmp = t * (4.0d0 * y)
    else if (x <= 4.4d+93) then
        tmp = ((z * z) * y) * (-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 <= 5e-200) {
		tmp = t * (4.0 * y);
	} else if (x <= 4.4e+93) {
		tmp = ((z * z) * y) * -4.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 5e-200:
		tmp = t * (4.0 * y)
	elif x <= 4.4e+93:
		tmp = ((z * z) * y) * -4.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 5e-200)
		tmp = Float64(t * Float64(4.0 * y));
	elseif (x <= 4.4e+93)
		tmp = Float64(Float64(Float64(z * z) * y) * -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 <= 5e-200)
		tmp = t * (4.0 * y);
	elseif (x <= 4.4e+93)
		tmp = ((z * z) * y) * -4.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 5e-200], N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+93], N[(N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.99999999999999991e-200

   1. Initial program 93.0%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 37.7

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

   5. Applied rewrites 37.7%

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

   if 4.99999999999999991e-200 < x < 4.40000000000000042e93

   1. Initial program 93.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 37.4

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

   5. Applied rewrites 37.4%

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

   if 4.40000000000000042e93 < x

   1. Initial program 82.2%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-200}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(z \cdot z\right) \cdot y\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(t \cdot y\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+200) (fma x x (* (* t y) 4.0)) (* (* (* -4.0 z) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+200) {
		tmp = fma(x, x, ((t * y) * 4.0));
	} else {
		tmp = ((-4.0 * z) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+200)
		tmp = fma(x, x, Float64(Float64(t * y) * 4.0));
	else
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+200], N[(x * x + N[(N[(t * y), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e200

   1. Initial program 96.9%

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

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if 1.9999999999999999e200 < (*.f64 z z)

   1. Initial program 79.2%

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

   3. Taylor expanded in z around 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 84.4%

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

Alternative 11: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot z\right) \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+200) (fma (* t y) 4.0 (* x x)) (* (* (* -4.0 z) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+200) {
		tmp = fma((t * y), 4.0, (x * x));
	} else {
		tmp = ((-4.0 * z) * y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+200)
		tmp = fma(Float64(t * y), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * z) * y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+200], N[(N[(t * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * z), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e200

   1. Initial program 96.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 87.0

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

   5. Applied rewrites 87.0%

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

   if 1.9999999999999999e200 < (*.f64 z z)

   1. Initial program 79.2%

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

   3. Taylor expanded in z around 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 84.4%

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

Alternative 12: 43.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-79}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 6.6e-79) (* t (* 4.0 y)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 6.6e-79) {
		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 <= 6.6d-79) 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 <= 6.6e-79) {
		tmp = t * (4.0 * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 6.6e-79:
		tmp = t * (4.0 * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 6.6e-79)
		tmp = Float64(t * Float64(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 <= 6.6e-79)
		tmp = t * (4.0 * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 6.6e-79], N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5999999999999996e-79

   1. Initial program 92.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 38.6

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

   5. Applied rewrites 38.6%

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

   if 6.5999999999999996e-79 < x

   1. Initial program 87.1%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{-79}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.3% 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. Taylor expanded in y around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 44.0

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

5. Applied rewrites 44.0%

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

Developer Target 1: 90.2% 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 2024271 
(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))))