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

Percentage Accurate: 90.3% → 93.1%

Time: 10.4s

Alternatives: 8

Speedup: 0.6×

• 49.7% 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: 90.3% 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: 93.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(t \cdot \left(4 \cdot \left(\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{-1}{t}\right)\right) - t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+300)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (+ (* x x) (- (* t (* 4.0 (* (* z (* z y)) (/ -1.0 t)))) (* t (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+300) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x * x) + ((t * (4.0 * ((z * (z * y)) * (-1.0 / t)))) - (t * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+300)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(t * Float64(4.0 * Float64(Float64(z * Float64(z * y)) * Float64(-1.0 / t)))) - Float64(t * Float64(y * 4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+300], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(N[(t * N[(4.0 * N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e300

   1. Initial program 96.8%

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

      1. fma-neg 98.4%

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

      2. distribute-lft-neg-in 98.4%

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

      3. *-commutative 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

      5. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   if 2.0000000000000001e300 < (*.f64 z z)

   1. Initial program 67.3%

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

   3. Taylor expanded in t around inf 67.3%

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

      1. +-commutative 67.3%

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

      2. fma-define 67.3%

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

      3. associate-/l* 67.3%

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

      4. *-commutative 67.3%

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

   5. Simplified 67.3%

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

      1. add-sqr-sqrt 20.4%

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

      2. sqrt-prod 15.7%

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

      3. sqr-neg 15.7%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 7.7%

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

      6. fma-undefine 7.7%

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

      7. distribute-rgt-in 7.7%

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

      8. clear-num 7.7%

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

      9. un-div-inv 7.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 15.7%

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

      12. sqr-neg 15.7%

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

      13. sqrt-prod 20.4%

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

      14. add-sqr-sqrt 64.3%

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

      15. *-commutative 64.3%

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

      16. add-sqr-sqrt 43.8%

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

      17. sqrt-unprod 51.6%

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

      18. sqr-neg 51.6%

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

      19. sqrt-prod 17.4%

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

      20. add-sqr-sqrt 64.3%

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

      21. add-sqr-sqrt 41.9%

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

      22. sqrt-unprod 59.7%

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

   7. Applied egg-rr 64.3%

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

      1. div-inv 64.3%

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

      2. clear-num 64.3%

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

      3. unpow2 64.3%

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

      4. associate-*r/ 71.6%

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

      5. /-rgt-identity 71.6%

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

      6. /-rgt-identity 71.6%

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

      7. associate-*r* 80.2%

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

      8. div-inv 80.2%

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

      9. associate-*r* 81.6%

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

   9. Applied egg-rr 81.6%

      \[\leadsto x \cdot x - \left(\left(4 \cdot \color{blue}{\left(\left(\left(y \cdot z\right) \cdot z\right) \cdot \frac{1}{t}\right)}\right) \cdot t + t \cdot \left(y \cdot 4\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(t \cdot \left(4 \cdot \left(\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{-1}{t}\right)\right) - t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+150}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.45e+150)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (fma x x (* t (* y 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.45e+150) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = fma(x, x, (t * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.45e+150)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = fma(x, x, Float64(t * Float64(y * 4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.45e+150], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45000000000000005e150

   1. Initial program 91.3%

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

   if 1.45000000000000005e150 < x

   1. Initial program 75.8%

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

      1. fma-neg 81.8%

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

      2. distribute-lft-neg-in 81.8%

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

      3. *-commutative 81.8%

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

      4. distribute-rgt-neg-in 81.8%

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

      5. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in z around 0 97.0%

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

      1. neg-mul-1 97.0%

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

   7. Simplified 97.0%

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

      1. add-exp-log 93.2%

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

      2. add-sqr-sqrt 36.0%

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

      3. sqrt-unprod 78.5%

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

      4. sqr-neg 78.5%

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

      5. sqrt-prod 57.2%

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

      6. add-sqr-sqrt 93.2%

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

      7. add-sqr-sqrt 48.1%

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

      8. sqrt-unprod 93.2%

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

      9. swap-sqr 93.2%

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

      10. metadata-eval 93.2%

          \[\leadsto e^{\log \left(\mathsf{fma}\left(x, x, t \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}\right)\right)} \]

      11. metadata-eval 93.2%

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

      12. swap-sqr 93.2%

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

      13. sqrt-unprod 45.1%

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

      14. add-sqr-sqrt 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. rem-exp-log 97.0%

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

   11. Applied egg-rr 97.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+150}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+284}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(t \cdot \left(4 \cdot \left(\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{-1}{t}\right)\right) - t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+284)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (+ (* x x) (- (* t (* 4.0 (* (* z (* z y)) (/ -1.0 t)))) (* t (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+284) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + ((t * (4.0 * ((z * (z * y)) * (-1.0 / t)))) - (t * (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) <= 5d+284) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) + ((t * (4.0d0 * ((z * (z * y)) * ((-1.0d0) / t)))) - (t * (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) <= 5e+284) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + ((t * (4.0 * ((z * (z * y)) * (-1.0 / t)))) - (t * (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+284:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) + ((t * (4.0 * ((z * (z * y)) * (-1.0 / t)))) - (t * (y * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+284)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(t * Float64(4.0 * Float64(Float64(z * Float64(z * y)) * Float64(-1.0 / t)))) - Float64(t * Float64(y * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+284)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) + ((t * (4.0 * ((z * (z * y)) * (-1.0 / t)))) - (t * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+284], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(N[(t * N[(4.0 * N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e284

   1. Initial program 97.2%

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

   if 4.9999999999999999e284 < (*.f64 z z)

   1. Initial program 67.8%

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

   3. Taylor expanded in t around inf 67.8%

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

      1. +-commutative 67.8%

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

      2. fma-define 67.8%

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

      3. associate-/l* 67.8%

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

      4. *-commutative 67.8%

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

   5. Simplified 67.8%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-prod 14.8%

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

      3. sqr-neg 14.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 8.7%

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

      6. fma-undefine 8.7%

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

      7. distribute-rgt-in 8.7%

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

      8. clear-num 8.7%

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

      9. un-div-inv 8.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 14.8%

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

      12. sqr-neg 14.8%

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

      13. sqrt-prod 19.2%

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

      14. add-sqr-sqrt 64.9%

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

      15. *-commutative 64.9%

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

      16. add-sqr-sqrt 45.6%

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

      17. sqrt-unprod 53.0%

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

      18. sqr-neg 53.0%

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

      19. sqrt-prod 16.4%

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

      20. add-sqr-sqrt 64.9%

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

      21. add-sqr-sqrt 40.9%

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

      22. sqrt-unprod 59.1%

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

   7. Applied egg-rr 64.9%

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

      1. div-inv 64.9%

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

      2. clear-num 64.9%

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

      3. unpow2 64.9%

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

      4. associate-*r/ 71.8%

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

      5. /-rgt-identity 71.8%

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

      6. /-rgt-identity 71.8%

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

      7. associate-*r* 79.9%

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

      8. div-inv 79.8%

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

      9. associate-*r* 81.3%

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

   9. Applied egg-rr 81.3%

      \[\leadsto x \cdot x - \left(\left(4 \cdot \color{blue}{\left(\left(\left(y \cdot z\right) \cdot z\right) \cdot \frac{1}{t}\right)}\right) \cdot t + t \cdot \left(y \cdot 4\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+284}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(t \cdot \left(4 \cdot \left(\left(z \cdot \left(z \cdot y\right)\right) \cdot \frac{-1}{t}\right)\right) - t \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+284}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(t \cdot \left(y \cdot 4\right) + t \cdot \left(4 \cdot \left(\left(z \cdot y\right) \cdot \frac{z}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+284)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (+ (* t (* y 4.0)) (* t (* 4.0 (* (* z y) (/ z t))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+284) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - ((t * (y * 4.0)) + (t * (4.0 * ((z * y) * (z / t)))));
	}
	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) <= 5d+284) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - ((t * (y * 4.0d0)) + (t * (4.0d0 * ((z * y) * (z / t)))))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+284:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - ((t * (y * 4.0)) + (t * (4.0 * ((z * y) * (z / t)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+284)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(t * Float64(y * 4.0)) + Float64(t * Float64(4.0 * Float64(Float64(z * y) * Float64(z / t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+284)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - ((t * (y * 4.0)) + (t * (4.0 * ((z * y) * (z / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+284], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(t * N[(y * 4.0), $MachinePrecision]), $MachinePrecision] + N[(t * N[(4.0 * N[(N[(z * y), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999999e284

   1. Initial program 97.2%

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

   if 4.9999999999999999e284 < (*.f64 z z)

   1. Initial program 67.8%

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

   3. Taylor expanded in t around inf 67.8%

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

      1. +-commutative 67.8%

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

      2. fma-define 67.8%

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

      3. associate-/l* 67.8%

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

      4. *-commutative 67.8%

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

   5. Simplified 67.8%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-prod 14.8%

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

      3. sqr-neg 14.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 8.7%

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

      6. fma-undefine 8.7%

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

      7. distribute-rgt-in 8.7%

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

      8. clear-num 8.7%

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

      9. un-div-inv 8.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 14.8%

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

      12. sqr-neg 14.8%

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

      13. sqrt-prod 19.2%

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

      14. add-sqr-sqrt 64.9%

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

      15. *-commutative 64.9%

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

      16. add-sqr-sqrt 45.6%

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

      17. sqrt-unprod 53.0%

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

      18. sqr-neg 53.0%

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

      19. sqrt-prod 16.4%

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

      20. add-sqr-sqrt 64.9%

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

      21. add-sqr-sqrt 40.9%

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

      22. sqrt-unprod 59.1%

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

   7. Applied egg-rr 64.9%

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

      1. div-inv 64.9%

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

      2. clear-num 64.9%

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

      3. unpow2 64.9%

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

      4. associate-*r/ 71.8%

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

      5. /-rgt-identity 71.8%

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

      6. /-rgt-identity 71.8%

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

      7. associate-*r* 79.9%

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

   9. Applied egg-rr 79.9%

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

4. Final simplification 92.6%

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

Alternative 5: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5e+294) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5e+294) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * 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 * x) <= 5d+294) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * 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 * x) <= 5e+294) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 5e+294:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5e+294)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 5e+294)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+294], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999999e294

   1. Initial program 93.5%

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

   if 4.9999999999999999e294 < (*.f64 x x)

   1. Initial program 76.9%

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

   3. Taylor expanded in y around 0 76.9%

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

   5. Simplified 89.2%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e+29) {
		tmp = 4.0 * (t * 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 * x) <= 1d+29) then
        tmp = 4.0d0 * (t * 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 * x) <= 1e+29) {
		tmp = 4.0 * (t * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1e+29:
		tmp = 4.0 * (t * y)
	else:
		tmp = x * x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1e+29)
		tmp = 4.0 * (t * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.5%

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

   3. Taylor expanded in t around inf 47.5%

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

      1. *-commutative 47.5%

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

   5. Simplified 47.5%

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

   if 9.99999999999999914e28 < (*.f64 x x)

   1. Initial program 86.9%

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

   3. Taylor expanded in y around 0 86.9%

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

   5. Simplified 72.5%

      \[\leadsto x \cdot x - \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

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

Alternative 7: 67.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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) * (t * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in z around 0 65.6%

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

   1. *-commutative 65.6%

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

5. Simplified 65.6%

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

6. Final simplification 65.6%

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

Alternative 8: 30.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 * (t * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in t around inf 31.3%

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

   1. *-commutative 31.3%

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

5. Simplified 31.3%

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

6. Final simplification 31.3%

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

Developer target: 90.3% 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 2024100 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))