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

Percentage Accurate: 89.9% → 95.4%

Time: 13.6s

Alternatives: 9

Speedup: 0.8×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% 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: 95.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{y \cdot -4}\\ \mathbf{if}\;y \cdot 4 \leq -2 \cdot 10^{-297}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(t, y \cdot -4, z \cdot \left(\left(-t_1\right) \cdot \left(z \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;y \cdot 4 \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(y \cdot t\right) + 4 \cdot {\left(z \cdot \sqrt{y}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot \mathsf{fma}\left(z, -z, t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* y -4.0))))
   (if (<= (* y 4.0) -2e-297)
     (- (* x x) (fma t (* y -4.0) (* z (* (- t_1) (* z t_1)))))
     (if (<= (* y 4.0) 5e-154)
       (- (* x x) (+ (* -4.0 (* y t)) (* 4.0 (pow (* z (sqrt y)) 2.0))))
       (fma x x (* 4.0 (* y (fma z (- z) t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y * -4.0));
	double tmp;
	if ((y * 4.0) <= -2e-297) {
		tmp = (x * x) - fma(t, (y * -4.0), (z * (-t_1 * (z * t_1))));
	} else if ((y * 4.0) <= 5e-154) {
		tmp = (x * x) - ((-4.0 * (y * t)) + (4.0 * pow((z * sqrt(y)), 2.0)));
	} else {
		tmp = fma(x, x, (4.0 * (y * fma(z, -z, t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(y * -4.0))
	tmp = 0.0
	if (Float64(y * 4.0) <= -2e-297)
		tmp = Float64(Float64(x * x) - fma(t, Float64(y * -4.0), Float64(z * Float64(Float64(-t_1) * Float64(z * t_1)))));
	elseif (Float64(y * 4.0) <= 5e-154)
		tmp = Float64(Float64(x * x) - Float64(Float64(-4.0 * Float64(y * t)) + Float64(4.0 * (Float64(z * sqrt(y)) ^ 2.0))));
	else
		tmp = fma(x, x, Float64(4.0 * Float64(y * fma(z, Float64(-z), t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(y * 4.0), $MachinePrecision], -2e-297], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision] + N[(z * N[((-t$95$1) * N[(z * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * 4.0), $MachinePrecision], 5e-154], N[(N[(x * x), $MachinePrecision] - N[(N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[Power[N[(z * N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(4.0 * N[(y * N[(z * (-z) + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y 4) < -2.00000000000000008e-297

   1. Initial program 92.3%

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

   2. Taylor expanded in z around 0 87.9%

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

      1. add-sqr-sqrt 30.1%

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

      2. sqrt-unprod 60.2%

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

      3. swap-sqr 60.2%

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

      4. metadata-eval 60.2%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(t \cdot y\right) + \sqrt{\color{blue}{16} \cdot \left(\left(y \cdot {z}^{2}\right) \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

      5. metadata-eval 60.2%

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

      6. swap-sqr 60.2%

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

      7. associate-*l* 60.2%

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

      8. associate-*l* 60.2%

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

      9. sqrt-unprod 61.1%

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

      10. add-sqr-sqrt 61.1%

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

      11. associate-*l* 61.1%

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

      12. metadata-eval 61.1%

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

      13. cancel-sign-sub-inv 61.1%

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

      14. add-sqr-sqrt 29.1%

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

      15. add-sqr-sqrt 12.3%

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

      16. difference-of-squares 12.3%

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

   4. Applied egg-rr 49.3%

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

      1. +-commutative 49.3%

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

      2. *-commutative 49.3%

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

      3. *-commutative 49.3%

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

      4. associate-*r* 49.3%

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

      5. *-commutative 49.3%

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

      6. associate-*r* 49.3%

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

      7. *-commutative 49.3%

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

   6. Simplified 49.3%

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

      1. +-commutative 49.3%

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

      2. difference-of-squares 49.3%

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

      3. cancel-sign-sub-inv 49.3%

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

      4. add-sqr-sqrt 93.6%

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

      5. *-commutative 93.6%

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

      6. associate-*l* 93.6%

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

      7. *-commutative 93.6%

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

      8. distribute-rgt-neg-in 93.6%

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

      9. *-commutative 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. fma-def 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*l* 98.1%

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

      4. *-commutative 98.1%

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

      5. *-commutative 98.1%

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

   10. Simplified 98.1%

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

   if -2.00000000000000008e-297 < (*.f64 y 4) < 5.0000000000000002e-154

   1. Initial program 79.8%

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

   2. Taylor expanded in z around 0 79.8%

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

      1. add-sqr-sqrt 79.8%

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

      2. pow2 79.8%

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

      3. *-commutative 79.8%

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

      4. sqrt-prod 79.8%

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

      5. unpow2 79.8%

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

      6. sqrt-prod 39.3%

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

      7. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 5.0000000000000002e-154 < (*.f64 y 4)

   1. Initial program 86.7%

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

      1. fma-neg 95.2%

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

      2. distribute-rgt-neg-in 95.2%

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

      3. *-commutative 95.2%

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

      4. associate-*l* 96.1%

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

      5. sub-neg 96.1%

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

      6. distribute-neg-in 96.1%

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

      7. distribute-rgt-neg-out 96.1%

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

      8. remove-double-neg 96.1%

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

      9. fma-def 96.1%

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

   3. Simplified 96.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq -2 \cdot 10^{-297}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(t, y \cdot -4, z \cdot \left(\left(-\sqrt{y \cdot -4}\right) \cdot \left(z \cdot \sqrt{y \cdot -4}\right)\right)\right)\\ \mathbf{elif}\;y \cdot 4 \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x \cdot x - \left(-4 \cdot \left(y \cdot t\right) + 4 \cdot {\left(z \cdot \sqrt{y}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot \mathsf{fma}\left(z, -z, t\right)\right)\right)\\ \end{array} \]

Alternative 2: 96.1% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+293)
   (fma y (* -4.0 (- (pow z 2.0) t)) (pow x 2.0))
   (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+293) {
		tmp = fma(y, (-4.0 * (pow(z, 2.0) - t)), pow(x, 2.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+293)
		tmp = fma(y, Float64(-4.0 * Float64((z ^ 2.0) - t)), (x ^ 2.0));
	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+293], N[(y * N[(-4.0 * N[(N[Power[z, 2.0], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[Power[x, 2.0], $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) < 1.9999999999999998e293

   1. Initial program 95.1%

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

   2. Taylor expanded in z around 0 95.1%

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

   3. Taylor expanded in x around 0 95.1%

      \[\leadsto \color{blue}{-1 \cdot \left(-4 \cdot \left(t \cdot y\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right) + {x}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 95.1%

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

      2. associate-*r* 95.1%

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

      3. *-commutative 95.1%

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

      4. distribute-neg-in 95.1%

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

      5. distribute-lft-neg-in 95.1%

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

      6. metadata-eval 95.1%

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

      7. associate-*r* 94.5%

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

      8. *-commutative 94.5%

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

      9. associate-*r* 95.1%

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

      10. +-commutative 95.1%

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

      11. sub-neg 95.1%

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

      12. distribute-lft-out-- 95.6%

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

      13. cancel-sign-sub-inv 95.6%

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

      14. metadata-eval 95.6%

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

      15. fma-def 97.8%

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

   5. Simplified 97.8%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 69.7%

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

   2. Taylor expanded in z around inf 76.8%

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

      1. associate-*r* 76.8%

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

   4. Simplified 76.8%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. *-commutative 30.7%

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

      4. sqrt-prod 30.7%

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

      5. unpow2 30.7%

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

      6. sqrt-prod 22.9%

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

      7. add-sqr-sqrt 37.2%

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

      8. *-commutative 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. unpow-prod-down 30.7%

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

      2. unpow2 30.7%

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

      3. associate-*l* 37.3%

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

      4. unpow2 37.3%

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

      5. add-sqr-sqrt 90.1%

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

   8. Applied egg-rr 90.1%

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

4. Final simplification 95.7%

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

Alternative 3: 95.4% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+293)
   (fma x x (* 4.0 (* y (fma 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+293) {
		tmp = fma(x, x, (4.0 * (y * fma(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+293)
		tmp = fma(x, x, Float64(4.0 * Float64(y * fma(z, Float64(-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+293], N[(x * x + N[(4.0 * N[(y * N[(z * (-z) + t), $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) < 1.9999999999999998e293

   1. Initial program 95.1%

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

      1. fma-neg 97.3%

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

      2. distribute-rgt-neg-in 97.3%

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

      3. *-commutative 97.3%

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

      4. associate-*l* 97.8%

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

      5. sub-neg 97.8%

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

      6. distribute-neg-in 97.8%

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

      7. distribute-rgt-neg-out 97.8%

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

      8. remove-double-neg 97.8%

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

      9. fma-def 97.8%

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

   3. Simplified 97.8%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 69.7%

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

   2. Taylor expanded in z around inf 76.8%

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

      1. associate-*r* 76.8%

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

   4. Simplified 76.8%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. *-commutative 30.7%

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

      4. sqrt-prod 30.7%

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

      5. unpow2 30.7%

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

      6. sqrt-prod 22.9%

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

      7. add-sqr-sqrt 37.2%

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

      8. *-commutative 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. unpow-prod-down 30.7%

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

      2. unpow2 30.7%

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

      3. associate-*l* 37.3%

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

      4. unpow2 37.3%

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

      5. add-sqr-sqrt 90.1%

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

   8. Applied egg-rr 90.1%

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

4. Final simplification 95.7%

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

Alternative 4: 95.4% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+293)
   (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+293) {
		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+293)
		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+293], 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) < 1.9999999999999998e293

   1. Initial program 95.1%

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

      1. fma-neg 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-rgt-neg-in 97.3%

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

      4. distribute-lft-neg-out 97.3%

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

      5. distribute-lft-neg-out 97.3%

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

      6. distribute-rgt-neg-in 97.3%

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

      7. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 69.7%

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

   2. Taylor expanded in z around inf 76.8%

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

      1. associate-*r* 76.8%

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

   4. Simplified 76.8%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. *-commutative 30.7%

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

      4. sqrt-prod 30.7%

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

      5. unpow2 30.7%

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

      6. sqrt-prod 22.9%

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

      7. add-sqr-sqrt 37.2%

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

      8. *-commutative 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. unpow-prod-down 30.7%

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

      2. unpow2 30.7%

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

      3. associate-*l* 37.3%

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

      4. unpow2 37.3%

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

      5. add-sqr-sqrt 90.1%

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

   8. Applied egg-rr 90.1%

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

4. Final simplification 95.3%

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

Alternative 5: 94.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+293)
   (+ (* 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) <= 2e+293) {
		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) <= 2d+293) 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) <= 2e+293) {
		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) <= 2e+293:
		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) <= 2e+293)
		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) <= 2e+293)
		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], 2e+293], 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) < 1.9999999999999998e293

   1. Initial program 95.1%

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

   if 1.9999999999999998e293 < (*.f64 z z)

   1. Initial program 69.7%

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

   2. Taylor expanded in z around inf 76.8%

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

      1. associate-*r* 76.8%

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

   4. Simplified 76.8%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. *-commutative 30.7%

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

      4. sqrt-prod 30.7%

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

      5. unpow2 30.7%

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

      6. sqrt-prod 22.9%

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

      7. add-sqr-sqrt 37.2%

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

      8. *-commutative 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. unpow-prod-down 30.7%

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

      2. unpow2 30.7%

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

      3. associate-*l* 37.3%

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

      4. unpow2 37.3%

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

      5. add-sqr-sqrt 90.1%

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

   8. Applied egg-rr 90.1%

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

4. Final simplification 93.7%

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

Alternative 6: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+90}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 7.2e+90) (- (* 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 <= 7.2e+90) {
		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 <= 7.2d+90) 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 <= 7.2e+90) {
		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 <= 7.2e+90:
		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 (z <= 7.2e+90)
		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 <= 7.2e+90)
		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[z, 7.2e+90], 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 z < 7.2e90

   1. Initial program 91.3%

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

   2. Taylor expanded in z around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. *-commutative 76.8%

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

      3. associate-*l* 76.8%

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

   4. Simplified 76.8%

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

   if 7.2e90 < z

   1. Initial program 75.4%

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

   2. Taylor expanded in z around inf 75.9%

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

      1. associate-*r* 75.9%

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

   4. Simplified 75.9%

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

      1. add-sqr-sqrt 28.0%

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

      2. pow2 28.0%

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

      3. *-commutative 28.0%

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

      4. sqrt-prod 28.0%

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

      5. unpow2 28.0%

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

      6. sqrt-prod 33.4%

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

      7. add-sqr-sqrt 33.4%

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

      8. *-commutative 33.4%

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

   6. Applied egg-rr 33.4%

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

      1. unpow-prod-down 28.0%

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

      2. unpow2 28.0%

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

      3. associate-*l* 33.5%

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

      4. unpow2 33.5%

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

      5. add-sqr-sqrt 86.9%

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

   8. Applied egg-rr 86.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+90}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 7: 45.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 5e+40) (* y (* 4.0 t)) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5e+40) {
		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 <= 5d+40) 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 <= 5e+40) {
		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 <= 5e+40:
		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 (z <= 5e+40)
		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 <= 5e+40)
		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[z, 5e+40], 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 z < 5.00000000000000003e40

   1. Initial program 91.5%

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

   2. Taylor expanded in z around 0 92.0%

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

   3. Taylor expanded in t around inf 32.2%

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

      1. associate-*r* 32.2%

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

      2. *-commutative 32.2%

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

      3. *-commutative 32.2%

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

   5. Simplified 32.2%

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

   if 5.00000000000000003e40 < z

   1. Initial program 77.0%

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

   2. Taylor expanded in z around inf 69.1%

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

      1. associate-*r* 69.1%

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

   4. Simplified 69.1%

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

      1. add-sqr-sqrt 27.7%

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

      2. pow2 27.7%

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

      3. *-commutative 27.7%

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

      4. sqrt-prod 27.7%

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

      5. unpow2 27.7%

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

      6. sqrt-prod 32.4%

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

      7. add-sqr-sqrt 32.4%

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

      8. *-commutative 32.4%

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

   6. Applied egg-rr 32.4%

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

      1. unpow-prod-down 27.7%

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

      2. unpow2 27.7%

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

      3. associate-*l* 32.5%

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

      4. unpow2 32.5%

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

      5. add-sqr-sqrt 78.6%

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

   8. Applied egg-rr 78.6%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 8: 31.0% 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 88.2%

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

2. Taylor expanded in t around inf 26.7%

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

   1. *-commutative 26.7%

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

4. Simplified 26.7%

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

5. Final simplification 26.7%

   \[\leadsto 4 \cdot \left(y \cdot t\right) \]

Alternative 9: 31.0% 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 88.2%

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

2. Taylor expanded in z around 0 85.0%

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

3. Taylor expanded in t around inf 26.7%

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

   1. associate-*r* 26.7%

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

   2. *-commutative 26.7%

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

   3. *-commutative 26.7%

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

5. Simplified 26.7%

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

6. Final simplification 26.7%

   \[\leadsto y \cdot \left(4 \cdot t\right) \]

Developer target: 89.9% 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 2023305 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

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