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

Percentage Accurate: 90.4% → 96.1%

Time: 9.5s

Alternatives: 9

Speedup: 0.6×

• 49.6% 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: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

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

Alternative 1: 96.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, \mathsf{fma}\left(z, -z, t\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+302)
   (fma (* y 4.0) (fma z (- z) t) (* x x))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+302) {
		tmp = fma((y * 4.0), fma(z, -z, t), (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+302)
		tmp = fma(Float64(y * 4.0), fma(z, Float64(-z), t), 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+302], N[(N[(y * 4.0), $MachinePrecision] * N[(z * (-z) + t), $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) < 2.0000000000000002e302

   1. Initial program 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. distribute-lft-neg-out 99.0%

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

      3. +-commutative 99.0%

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

      4. distribute-lft-neg-out 99.0%

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

      5. distribute-lft-neg-in 99.0%

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

      6. distribute-rgt-neg-in 99.0%

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

      7. fma-define 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(y \cdot 4, \color{blue}{0 - \left(z \cdot z - t\right)}, x \cdot x\right) \]

      9. associate-+l- 100.0%

         \[\leadsto \mathsf{fma}\left(y \cdot 4, \color{blue}{\left(0 - z \cdot z\right) + t}, x \cdot x\right) \]

      10. neg-sub0 100.0%

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

      11. distribute-rgt-neg-out 100.0%

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

      12. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if 2.0000000000000002e302 < (*.f64 z z)

   1. Initial program 71.5%

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

      1. fmm-def 81.9%

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

      2. distribute-lft-neg-in 81.9%

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

      3. *-commutative 81.9%

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

      4. distribute-rgt-neg-in 81.9%

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

      5. metadata-eval 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around inf 81.9%

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

      1. associate-*r* 81.9%

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

      2. *-commutative 81.9%

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

      3. *-commutative 81.9%

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

   7. Simplified 81.9%

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

      1. add-sqr-sqrt 36.9%

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

      2. pow2 36.9%

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

      3. sqrt-prod 36.9%

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

      4. sqrt-pow1 43.5%

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

      5. metadata-eval 43.5%

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

      6. pow1 43.5%

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

   9. Applied egg-rr 43.5%

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

       1. unpow2 43.5%

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

       2. swap-sqr 36.9%

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

       3. add-sqr-sqrt 81.9%

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

       4. associate-*l* 90.1%

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

       5. *-commutative 90.1%

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

   11. Applied egg-rr 90.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, \mathsf{fma}\left(z, -z, t\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+302)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+302) {
		tmp = fma((y * 4.0), (t - (z * z)), (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+302)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), 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+302], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $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) < 2.0000000000000002e302

   1. Initial program 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. distribute-lft-neg-out 99.0%

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

      3. +-commutative 99.0%

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

      4. distribute-lft-neg-out 99.0%

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

      5. distribute-lft-neg-in 99.0%

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

      6. distribute-rgt-neg-in 99.0%

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

      7. fma-define 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 2.0000000000000002e302 < (*.f64 z z)

   1. Initial program 71.5%

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

      1. fmm-def 81.9%

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

      2. distribute-lft-neg-in 81.9%

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

      3. *-commutative 81.9%

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

      4. distribute-rgt-neg-in 81.9%

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

      5. metadata-eval 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around inf 81.9%

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

      1. associate-*r* 81.9%

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

      2. *-commutative 81.9%

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

      3. *-commutative 81.9%

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

   7. Simplified 81.9%

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

      1. add-sqr-sqrt 36.9%

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

      2. pow2 36.9%

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

      3. sqrt-prod 36.9%

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

      4. sqrt-pow1 43.5%

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

      5. metadata-eval 43.5%

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

      6. pow1 43.5%

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

   9. Applied egg-rr 43.5%

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

       1. unpow2 43.5%

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

       2. swap-sqr 36.9%

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

       3. add-sqr-sqrt 81.9%

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

       4. associate-*l* 90.1%

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

       5. *-commutative 90.1%

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

   11. Applied egg-rr 90.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+296)
   (fma x x (* (* y -4.0) (- (* z z) t)))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+296) {
		tmp = fma(x, x, ((y * -4.0) * ((z * z) - t)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+296)
		tmp = fma(x, x, Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t)));
	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+296], N[(x * x + N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $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) < 1.99999999999999996e296

   1. Initial program 99.5%

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

      1. fmm-def 99.5%

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

      2. distribute-lft-neg-in 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   if 1.99999999999999996e296 < (*.f64 z z)

   1. Initial program 70.3%

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

      1. fmm-def 80.5%

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

      2. distribute-lft-neg-in 80.5%

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

      3. *-commutative 80.5%

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

      4. distribute-rgt-neg-in 80.5%

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

      5. metadata-eval 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in z around inf 80.5%

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

      1. associate-*r* 80.5%

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

      2. *-commutative 80.5%

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

      3. *-commutative 80.5%

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

   7. Simplified 80.5%

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

      1. add-sqr-sqrt 36.3%

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

      2. pow2 36.3%

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

      3. sqrt-prod 36.3%

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

      4. sqrt-pow1 42.7%

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

      5. metadata-eval 42.7%

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

      6. pow1 42.7%

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

   9. Applied egg-rr 42.7%

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

       1. unpow2 42.7%

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

       2. swap-sqr 36.3%

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

       3. add-sqr-sqrt 80.5%

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

       4. associate-*l* 88.5%

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

       5. *-commutative 88.5%

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

   11. Applied egg-rr 88.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+296}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+296)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+296) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} 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) :: tmp
    if ((z * z) <= 1d+296) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    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 tmp;
	if ((z * z) <= 1e+296) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+296:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+296)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+296)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+296], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $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) < 9.99999999999999981e295

   1. Initial program 100.0%

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

   if 9.99999999999999981e295 < (*.f64 z z)

   1. Initial program 69.1%

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

      1. fmm-def 79.1%

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

      2. distribute-lft-neg-in 79.1%

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

      3. *-commutative 79.1%

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

      4. distribute-rgt-neg-in 79.1%

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

      5. metadata-eval 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in z around inf 79.1%

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

      1. associate-*r* 79.1%

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

      2. *-commutative 79.1%

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

      3. *-commutative 79.1%

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

   7. Simplified 79.1%

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

      1. add-sqr-sqrt 35.7%

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

      2. pow2 35.7%

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

      3. sqrt-prod 35.7%

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

      4. sqrt-pow1 42.0%

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

      5. metadata-eval 42.0%

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

      6. pow1 42.0%

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

   9. Applied egg-rr 42.0%

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

       1. unpow2 42.0%

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

       2. swap-sqr 35.7%

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

       3. add-sqr-sqrt 79.1%

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

       4. associate-*l* 87.0%

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

       5. *-commutative 87.0%

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

   11. Applied egg-rr 87.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+296}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+130}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+130) (- (* x x) (* y (* t -4.0))) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+130) {
		tmp = (x * x) - (y * (t * -4.0));
	} 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) :: tmp
    if ((z * z) <= 4d+130) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    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 tmp;
	if ((z * z) <= 4e+130) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 4e+130:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+130)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4e+130)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+130], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $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) < 4.0000000000000002e130

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 94.0%

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

      1. *-commutative 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*l* 94.0%

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

   5. Simplified 94.0%

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

   if 4.0000000000000002e130 < (*.f64 z z)

   1. Initial program 80.9%

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

      1. fmm-def 87.1%

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

      2. distribute-lft-neg-in 87.1%

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

      3. *-commutative 87.1%

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

      4. distribute-rgt-neg-in 87.1%

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

      5. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in z around inf 75.0%

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

      1. associate-*r* 75.0%

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

      2. *-commutative 75.0%

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

      3. *-commutative 75.0%

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

   7. Simplified 75.0%

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

      1. add-sqr-sqrt 35.7%

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

      2. pow2 35.7%

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

      3. sqrt-prod 35.7%

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

      4. sqrt-pow1 39.6%

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

      5. metadata-eval 39.6%

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

      6. pow1 39.6%

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

   9. Applied egg-rr 39.6%

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

       1. unpow2 39.6%

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

       2. swap-sqr 35.7%

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

       3. add-sqr-sqrt 75.0%

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

       4. associate-*l* 79.9%

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

       5. *-commutative 79.9%

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

   11. Applied egg-rr 79.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+130}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+82}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+82) (* y (* 4.0 t)) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+82) {
		tmp = y * (4.0 * t);
	} 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) :: tmp
    if ((z * z) <= 1d+82) then
        tmp = y * (4.0d0 * t)
    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 tmp;
	if ((z * z) <= 1e+82) {
		tmp = y * (4.0 * t);
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+82:
		tmp = y * (4.0 * t)
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+82)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+82)
		tmp = y * (4.0 * t);
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+82], N[(y * N[(4.0 * t), $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) < 9.9999999999999996e81

   1. Initial program 100.0%

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

      1. fmm-def 100.0%

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

      2. distribute-lft-neg-in 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 48.2%

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

      1. *-commutative 48.2%

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

   7. Simplified 48.2%

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

      1. *-commutative 48.2%

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

      2. metadata-eval 48.2%

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

      3. distribute-rgt-neg-in 48.2%

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

      4. associate-*r* 48.9%

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

      5. add-sqr-sqrt 25.4%

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

      6. sqrt-unprod 19.3%

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

      7. associate-*r* 19.3%

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

      8. associate-*r* 19.3%

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

      9. swap-sqr 19.3%

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

      10. metadata-eval 19.3%

          \[\leadsto -\sqrt{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) \cdot \color{blue}{16}} \]

      11. metadata-eval 19.3%

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

      12. swap-sqr 19.3%

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

      13. *-commutative 19.3%

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

      14. *-commutative 19.3%

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

      15. sqrt-unprod 6.5%

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

      16. add-sqr-sqrt 7.8%

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

      17. *-commutative 7.8%

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

      18. metadata-eval 7.8%

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

      19. distribute-rgt-neg-in 7.8%

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

      20. associate-*r* 7.7%

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

      21. distribute-rgt-neg-in 7.7%

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

      22. *-commutative 7.7%

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

      23. distribute-lft-neg-in 7.7%

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

   9. Applied egg-rr 7.7%

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

       1. add-sqr-sqrt 7.2%

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

       2. sqrt-unprod 26.3%

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

       3. sqr-neg 26.3%

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

       4. sqrt-unprod 29.2%

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

       5. add-sqr-sqrt 48.9%

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

       6. *-commutative 48.9%

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

   11. Applied egg-rr 48.9%

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

   if 9.9999999999999996e81 < (*.f64 z z)

   1. Initial program 82.3%

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

      1. fmm-def 88.0%

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

      2. distribute-lft-neg-in 88.0%

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

      3. *-commutative 88.0%

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

      4. distribute-rgt-neg-in 88.0%

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

      5. metadata-eval 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in z around inf 73.1%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

   7. Simplified 73.1%

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

      1. add-sqr-sqrt 35.8%

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

      2. pow2 35.8%

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

      3. sqrt-prod 35.8%

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

      4. sqrt-pow1 39.4%

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

      5. metadata-eval 39.4%

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

      6. pow1 39.4%

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

   9. Applied egg-rr 39.4%

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

       1. unpow2 39.4%

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

       2. swap-sqr 35.8%

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

       3. add-sqr-sqrt 73.1%

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

       4. associate-*l* 77.7%

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

       5. *-commutative 77.7%

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

   11. Applied egg-rr 77.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+82}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 6.5e+81) (* y (* 4.0 t)) (* (* z z) (* y -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 6.5e+81) {
		tmp = y * (4.0 * t);
	} 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) :: tmp
    if ((z * z) <= 6.5d+81) then
        tmp = y * (4.0d0 * t)
    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 tmp;
	if ((z * z) <= 6.5e+81) {
		tmp = y * (4.0 * t);
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 6.5e+81:
		tmp = y * (4.0 * t)
	else:
		tmp = (z * z) * (y * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 6.5e+81)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(Float64(z * z) * Float64(y * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 6.5e+81)
		tmp = y * (4.0 * t);
	else
		tmp = (z * z) * (y * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 6.5e+81], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 6.4999999999999996e81

   1. Initial program 100.0%

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

      1. fmm-def 100.0%

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

      2. distribute-lft-neg-in 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 48.2%

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

      1. *-commutative 48.2%

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

   7. Simplified 48.2%

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

      1. *-commutative 48.2%

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

      2. metadata-eval 48.2%

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

      3. distribute-rgt-neg-in 48.2%

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

      4. associate-*r* 48.9%

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

      5. add-sqr-sqrt 25.4%

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

      6. sqrt-unprod 19.3%

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

      7. associate-*r* 19.3%

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

      8. associate-*r* 19.3%

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

      9. swap-sqr 19.3%

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

      10. metadata-eval 19.3%

          \[\leadsto -\sqrt{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) \cdot \color{blue}{16}} \]

      11. metadata-eval 19.3%

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

      12. swap-sqr 19.3%

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

      13. *-commutative 19.3%

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

      14. *-commutative 19.3%

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

      15. sqrt-unprod 6.5%

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

      16. add-sqr-sqrt 7.8%

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

      17. *-commutative 7.8%

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

      18. metadata-eval 7.8%

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

      19. distribute-rgt-neg-in 7.8%

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

      20. associate-*r* 7.7%

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

      21. distribute-rgt-neg-in 7.7%

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

      22. *-commutative 7.7%

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

      23. distribute-lft-neg-in 7.7%

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

   9. Applied egg-rr 7.7%

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

       1. add-sqr-sqrt 7.2%

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

       2. sqrt-unprod 26.3%

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

       3. sqr-neg 26.3%

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

       4. sqrt-unprod 29.2%

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

       5. add-sqr-sqrt 48.9%

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

       6. *-commutative 48.9%

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

   11. Applied egg-rr 48.9%

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

   if 6.4999999999999996e81 < (*.f64 z z)

   1. Initial program 82.3%

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

      1. fmm-def 88.0%

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

      2. distribute-lft-neg-in 88.0%

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

      3. *-commutative 88.0%

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

      4. distribute-rgt-neg-in 88.0%

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

      5. metadata-eval 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in z around inf 73.1%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

   7. Simplified 73.1%

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

      1. unpow2 73.1%

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

   9. Applied egg-rr 73.1%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.5 \cdot 10^{+81}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(4 \cdot t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y (* 4.0 t)))

C

double code(double x, double y, double z, double t) {
	return y * (4.0 * 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 = y * (4.0d0 * t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * (4.0 * t);
}

Python

def code(x, y, z, t):
	return y * (4.0 * t)

Julia

function code(x, y, z, t)
	return Float64(y * Float64(4.0 * t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * (4.0 * t);
end

Wolfram

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

Derivation

1. Initial program 92.7%

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

   1. fmm-def 95.1%

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

   2. distribute-lft-neg-in 95.1%

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

   3. *-commutative 95.1%

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

   4. distribute-rgt-neg-in 95.1%

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

   5. metadata-eval 95.1%

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

3. Simplified 95.1%

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

5. Taylor expanded in t around inf 32.7%

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

   1. *-commutative 32.7%

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

7. Simplified 32.7%

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

   1. *-commutative 32.7%

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

   2. metadata-eval 32.7%

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

   3. distribute-rgt-neg-in 32.7%

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

   4. associate-*r* 33.1%

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

   5. add-sqr-sqrt 17.7%

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

   6. sqrt-unprod 17.9%

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

   7. associate-*r* 17.9%

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

   8. associate-*r* 17.9%

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

   9. swap-sqr 17.9%

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

   10. metadata-eval 17.9%

       \[\leadsto -\sqrt{\left(\left(y \cdot t\right) \cdot \left(y \cdot t\right)\right) \cdot \color{blue}{16}} \]

   11. metadata-eval 17.9%

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

   12. swap-sqr 17.9%

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

   13. *-commutative 17.9%

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

   14. *-commutative 17.9%

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

   15. sqrt-unprod 5.4%

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

   16. add-sqr-sqrt 6.7%

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

   17. *-commutative 6.7%

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

   18. metadata-eval 6.7%

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

   19. distribute-rgt-neg-in 6.7%

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

   20. associate-*r* 6.7%

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

   21. distribute-rgt-neg-in 6.7%

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

   22. *-commutative 6.7%

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

   23. distribute-lft-neg-in 6.7%

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

9. Applied egg-rr 6.7%

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

    1. add-sqr-sqrt 4.8%

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

    2. sqrt-unprod 20.6%

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

    3. sqr-neg 20.6%

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

    4. sqrt-unprod 18.8%

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

    5. add-sqr-sqrt 33.1%

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

    6. *-commutative 33.1%

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

11. Applied egg-rr 33.1%

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

12. Final simplification 33.1%

    \[\leadsto y \cdot \left(4 \cdot t\right) \]
13. Add Preprocessing

Alternative 9: 32.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(y \cdot t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 4.0 (* y t)))

C

double code(double x, double y, double z, double t) {
	return 4.0 * (y * 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 = 4.0d0 * (y * t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 4.0 * (y * t);
}

Python

def code(x, y, z, t):
	return 4.0 * (y * t)

Julia

function code(x, y, z, t)
	return Float64(4.0 * Float64(y * t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 4.0 * (y * t);
end

Wolfram

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

Derivation

1. Initial program 92.7%

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

   1. fmm-def 95.1%

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

   2. distribute-lft-neg-in 95.1%

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

   3. *-commutative 95.1%

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

   4. distribute-rgt-neg-in 95.1%

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

   5. metadata-eval 95.1%

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

3. Simplified 95.1%

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

5. Taylor expanded in t around inf 32.7%

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

   1. *-commutative 32.7%

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

7. Simplified 32.7%

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

Developer Target 1: 90.4% 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 2024179 
(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))))