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

Percentage Accurate: 91.2% → 96.4%

Time: 12.9s

Alternatives: 8

Speedup: 0.6×

• 48.8% 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: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+217)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (- (* x x) (* (* z y) (* z 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+217) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+217)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(z * y) * Float64(z * 4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+217], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e217

   1. Initial program 96.4%

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

      1. cancel-sign-sub-inv 96.4%

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

      2. distribute-lft-neg-out 96.4%

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

      3. +-commutative 96.4%

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

      4. associate-*l* 96.4%

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

      5. distribute-lft-neg-in 96.4%

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

      6. associate-*l* 96.4%

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

      7. distribute-rgt-neg-in 96.4%

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

      8. fma-define 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. remove-double-neg 99.4%

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

      13. sub-neg 99.4%

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

   3. Simplified 99.4%

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

   if 5.00000000000000041e217 < (*.f64 z z)

   1. Initial program 74.5%

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

      1. sub-neg 74.5%

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

      2. flip-+ 6.0%

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

      3. pow2 6.0%

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

      4. pow2 6.0%

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

      5. pow-prod-up 6.0%

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

      6. metadata-eval 6.0%

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

      7. pow2 6.0%

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

   4. Applied egg-rr 6.0%

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

      1. clear-num 6.0%

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

      2. un-div-inv 6.0%

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

      3. clear-num 6.0%

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

      4. metadata-eval 6.0%

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

      5. pow-prod-up 6.0%

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

      6. flip-+ 74.5%

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

      7. unpow2 74.5%

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

      8. fma-define 74.5%

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

      9. add-sqr-sqrt 34.4%

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

      10. sqrt-prod 74.5%

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

      11. sqr-neg 74.5%

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

      12. sqrt-prod 40.1%

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

      13. add-sqr-sqrt 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in z around inf 74.5%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 74.5%

         \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\sqrt{\frac{1}{{z}^{2}}} \cdot \sqrt{\frac{1}{{z}^{2}}}}} \]

      2. times-frac 74.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{y}{\sqrt{\frac{1}{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}}} \]

      3. sqrt-div 74.5%

         \[\leadsto x \cdot x - \frac{y}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      4. metadata-eval 74.5%

         \[\leadsto x \cdot x - \frac{y}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      5. sqrt-pow1 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{{z}^{\color{blue}{1}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      7. pow1 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{\color{blue}{z}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      8. sqrt-div 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \]

      9. metadata-eval 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \]

      10. sqrt-pow1 96.4%

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

      11. metadata-eval 96.4%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{{z}^{\color{blue}{1}}}} \]

      12. pow1 96.4%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{\color{blue}{z}}} \]

   9. Applied egg-rr 96.4%

      \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{z}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 96.4%

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

       2. /-rgt-identity 96.4%

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

       3. associate-/r/ 96.4%

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

       4. metadata-eval 96.4%

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

   11. Simplified 96.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+217)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (- (* x x) (* (* z y) (* z 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+217) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+217)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(z * y) * Float64(z * 4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+217], 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[(z * y), $MachinePrecision] * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e217

   1. Initial program 96.4%

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

      1. fma-neg 97.0%

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

      2. distribute-lft-neg-in 97.0%

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

      3. *-commutative 97.0%

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

      4. distribute-rgt-neg-in 97.0%

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

      5. metadata-eval 97.0%

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

   3. Simplified 97.0%

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

   if 5.00000000000000041e217 < (*.f64 z z)

   1. Initial program 74.5%

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

      1. sub-neg 74.5%

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

      2. flip-+ 6.0%

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

      3. pow2 6.0%

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

      4. pow2 6.0%

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

      5. pow-prod-up 6.0%

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

      6. metadata-eval 6.0%

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

      7. pow2 6.0%

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

   4. Applied egg-rr 6.0%

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

      1. clear-num 6.0%

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

      2. un-div-inv 6.0%

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

      3. clear-num 6.0%

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

      4. metadata-eval 6.0%

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

      5. pow-prod-up 6.0%

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

      6. flip-+ 74.5%

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

      7. unpow2 74.5%

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

      8. fma-define 74.5%

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

      9. add-sqr-sqrt 34.4%

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

      10. sqrt-prod 74.5%

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

      11. sqr-neg 74.5%

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

      12. sqrt-prod 40.1%

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

      13. add-sqr-sqrt 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in z around inf 74.5%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 74.5%

         \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\sqrt{\frac{1}{{z}^{2}}} \cdot \sqrt{\frac{1}{{z}^{2}}}}} \]

      2. times-frac 74.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{y}{\sqrt{\frac{1}{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}}} \]

      3. sqrt-div 74.5%

         \[\leadsto x \cdot x - \frac{y}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      4. metadata-eval 74.5%

         \[\leadsto x \cdot x - \frac{y}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      5. sqrt-pow1 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{{z}^{\color{blue}{1}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      7. pow1 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{\color{blue}{z}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      8. sqrt-div 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \]

      9. metadata-eval 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \]

      10. sqrt-pow1 96.4%

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

      11. metadata-eval 96.4%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{{z}^{\color{blue}{1}}}} \]

      12. pow1 96.4%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{\color{blue}{z}}} \]

   9. Applied egg-rr 96.4%

      \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{z}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 96.4%

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

       2. /-rgt-identity 96.4%

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

       3. associate-/r/ 96.4%

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

       4. metadata-eval 96.4%

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

   11. Simplified 96.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+217}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e+217)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* (* z y) (* z 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e+217) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - ((z * y) * (z * 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+217) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - ((z * y) * (z * 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+217) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 5e+217:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - ((z * y) * (z * 4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e+217)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(z * y) * Float64(z * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 5e+217)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - ((z * y) * (z * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+217], 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[(z * y), $MachinePrecision] * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000041e217

   1. Initial program 96.4%

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

   if 5.00000000000000041e217 < (*.f64 z z)

   1. Initial program 74.5%

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

      1. sub-neg 74.5%

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

      2. flip-+ 6.0%

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

      3. pow2 6.0%

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

      4. pow2 6.0%

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

      5. pow-prod-up 6.0%

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

      6. metadata-eval 6.0%

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

      7. pow2 6.0%

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

   4. Applied egg-rr 6.0%

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

      1. clear-num 6.0%

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

      2. un-div-inv 6.0%

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

      3. clear-num 6.0%

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

      4. metadata-eval 6.0%

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

      5. pow-prod-up 6.0%

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

      6. flip-+ 74.5%

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

      7. unpow2 74.5%

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

      8. fma-define 74.5%

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

      9. add-sqr-sqrt 34.4%

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

      10. sqrt-prod 74.5%

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

      11. sqr-neg 74.5%

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

      12. sqrt-prod 40.1%

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

      13. add-sqr-sqrt 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in z around inf 74.5%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 74.5%

         \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\sqrt{\frac{1}{{z}^{2}}} \cdot \sqrt{\frac{1}{{z}^{2}}}}} \]

      2. times-frac 74.5%

         \[\leadsto x \cdot x - \color{blue}{\frac{y}{\sqrt{\frac{1}{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}}} \]

      3. sqrt-div 74.5%

         \[\leadsto x \cdot x - \frac{y}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      4. metadata-eval 74.5%

         \[\leadsto x \cdot x - \frac{y}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      5. sqrt-pow1 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{{z}^{\color{blue}{1}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      7. pow1 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{\color{blue}{z}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      8. sqrt-div 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \]

      9. metadata-eval 44.7%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \]

      10. sqrt-pow1 96.4%

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

      11. metadata-eval 96.4%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{{z}^{\color{blue}{1}}}} \]

      12. pow1 96.4%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{\color{blue}{z}}} \]

   9. Applied egg-rr 96.4%

      \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{z}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 96.4%

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

       2. /-rgt-identity 96.4%

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

       3. associate-/r/ 96.4%

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

       4. metadata-eval 96.4%

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

   11. Simplified 96.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+217}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2e+62)
   (- (* x x) (* -4.0 (* y t)))
   (- (* x x) (* (* z y) (* z 4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2e+62) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = (x * x) - ((z * y) * (z * 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 <= 2d+62) then
        tmp = (x * x) - ((-4.0d0) * (y * t))
    else
        tmp = (x * x) - ((z * y) * (z * 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 <= 2e+62) {
		tmp = (x * x) - (-4.0 * (y * t));
	} else {
		tmp = (x * x) - ((z * y) * (z * 4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 2e+62:
		tmp = (x * x) - (-4.0 * (y * t))
	else:
		tmp = (x * x) - ((z * y) * (z * 4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2e+62)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(y * t)));
	else
		tmp = Float64(Float64(x * x) - Float64(Float64(z * y) * Float64(z * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2e+62)
		tmp = (x * x) - (-4.0 * (y * t));
	else
		tmp = (x * x) - ((z * y) * (z * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2e+62], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] * N[(z * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000007e62

   1. Initial program 90.6%

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

   3. Taylor expanded in z around 0 76.9%

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

      1. *-commutative 76.9%

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

   5. Simplified 76.9%

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

   if 2.00000000000000007e62 < z

   1. Initial program 82.7%

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

      1. sub-neg 82.7%

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

      2. flip-+ 12.3%

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

      3. pow2 12.3%

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

      4. pow2 12.3%

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

      5. pow-prod-up 12.4%

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

      6. metadata-eval 12.4%

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

      7. pow2 12.4%

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

   4. Applied egg-rr 12.4%

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

      1. clear-num 12.4%

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

      2. un-div-inv 12.4%

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

      3. clear-num 12.4%

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

      4. metadata-eval 12.4%

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

      5. pow-prod-up 12.3%

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

      6. flip-+ 82.7%

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

      7. unpow2 82.7%

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

      8. fma-define 82.7%

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

      9. add-sqr-sqrt 32.3%

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

      10. sqrt-prod 80.8%

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

      11. sqr-neg 80.8%

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

      12. sqrt-prod 49.8%

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

      13. add-sqr-sqrt 82.2%

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

   6. Applied egg-rr 82.2%

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

   7. Taylor expanded in z around inf 82.2%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{{z}^{2}}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 82.2%

         \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\sqrt{\frac{1}{{z}^{2}}} \cdot \sqrt{\frac{1}{{z}^{2}}}}} \]

      2. times-frac 82.2%

         \[\leadsto x \cdot x - \color{blue}{\frac{y}{\sqrt{\frac{1}{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}}} \]

      3. sqrt-div 82.2%

         \[\leadsto x \cdot x - \frac{y}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      4. metadata-eval 82.2%

         \[\leadsto x \cdot x - \frac{y}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      5. sqrt-pow1 82.2%

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

      6. metadata-eval 82.2%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{{z}^{\color{blue}{1}}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      7. pow1 82.2%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{\color{blue}{z}}} \cdot \frac{4}{\sqrt{\frac{1}{{z}^{2}}}} \]

      8. sqrt-div 82.2%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\color{blue}{\frac{\sqrt{1}}{\sqrt{{z}^{2}}}}} \]

      9. metadata-eval 82.2%

         \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{\color{blue}{1}}{\sqrt{{z}^{2}}}} \]

      10. sqrt-pow1 97.3%

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

      11. metadata-eval 97.3%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{{z}^{\color{blue}{1}}}} \]

      12. pow1 97.3%

          \[\leadsto x \cdot x - \frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{\color{blue}{z}}} \]

   9. Applied egg-rr 97.3%

      \[\leadsto x \cdot x - \color{blue}{\frac{y}{\frac{1}{z}} \cdot \frac{4}{\frac{1}{z}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 97.3%

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

       2. /-rgt-identity 97.3%

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

       3. associate-/r/ 97.3%

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

       4. metadata-eval 97.3%

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

   11. Simplified 97.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(z \cdot y\right) \cdot \left(z \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.45 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1.45e+98) (* y (* 4.0 t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1.45e+98) {
		tmp = y * (4.0 * t);
	} 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) <= 1.45d+98) then
        tmp = y * (4.0d0 * t)
    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) <= 1.45e+98) {
		tmp = y * (4.0 * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1.45e+98:
		tmp = y * (4.0 * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1.45e+98)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1.45e+98)
		tmp = y * (4.0 * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.45e+98], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.45000000000000005e98

   1. Initial program 91.8%

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

   3. Taylor expanded in t around inf 44.4%

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

      1. associate-*r* 44.4%

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

   5. Simplified 44.4%

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

   if 1.45000000000000005e98 < (*.f64 x x)

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0 85.8%

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

   5. Simplified 70.7%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.45 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.75e+153) (* y (* 4.0 t)) (* y (* t -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.75e+153) {
		tmp = y * (4.0 * t);
	} else {
		tmp = y * (t * -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 <= 1.75d+153) then
        tmp = y * (4.0d0 * t)
    else
        tmp = y * (t * (-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 <= 1.75e+153) {
		tmp = y * (4.0 * t);
	} else {
		tmp = y * (t * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.75e+153:
		tmp = y * (4.0 * t)
	else:
		tmp = y * (t * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.75e+153)
		tmp = Float64(y * Float64(4.0 * t));
	else
		tmp = Float64(y * Float64(t * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.75e+153)
		tmp = y * (4.0 * t);
	else
		tmp = y * (t * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.75e+153], N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.75e153

   1. Initial program 91.1%

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

   3. Taylor expanded in t around inf 34.6%

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

      1. associate-*r* 34.6%

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

   5. Simplified 34.6%

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

   if 1.75e153 < z

   1. Initial program 77.4%

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

      1. sub-neg 77.4%

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

      2. flip-+ 0.0%

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

      3. pow2 0.0%

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

      4. pow2 0.0%

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

      5. pow-prod-up 0.0%

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

      6. metadata-eval 0.0%

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

      7. pow2 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. clear-num 0.0%

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

      2. un-div-inv 0.0%

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

      3. clear-num 0.0%

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

      4. metadata-eval 0.0%

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

      5. pow-prod-up 0.0%

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

      6. flip-+ 77.4%

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

      7. unpow2 77.4%

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

      8. fma-define 77.4%

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

      9. add-sqr-sqrt 26.8%

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

      10. sqrt-prod 77.4%

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

      11. sqr-neg 77.4%

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

      12. sqrt-prod 50.5%

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

      13. add-sqr-sqrt 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in z around 0 33.0%

      \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{t}}} \]

   8. Taylor expanded in x around 0 18.2%

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

      1. associate-*r* 18.2%

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

      2. metadata-eval 18.2%

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

      3. distribute-lft-neg-in 18.2%

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

      4. metadata-eval 18.2%

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

      5. associate-/r/ 18.2%

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

      6. distribute-lft-neg-in 18.2%

         \[\leadsto \color{blue}{-\frac{4}{\frac{1}{t}} \cdot y} \]

      7. *-commutative 18.2%

         \[\leadsto -\color{blue}{y \cdot \frac{4}{\frac{1}{t}}} \]

      8. distribute-rgt-neg-in 18.2%

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

      9. associate-/r/ 18.2%

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

      10. metadata-eval 18.2%

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

      11. distribute-lft-neg-in 18.2%

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

      12. metadata-eval 18.2%

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

   10. Simplified 18.2%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(4 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

3. Taylor expanded in z around 0 66.0%

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

   1. *-commutative 66.0%

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

5. Simplified 66.0%

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

6. Final simplification 66.0%

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

Alternative 8: 5.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 * (t * (-4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

   1. sub-neg 89.1%

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

   2. flip-+ 54.4%

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

   3. pow2 54.4%

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

   4. pow2 54.4%

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

   5. pow-prod-up 54.4%

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

   6. metadata-eval 54.4%

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

   7. pow2 54.4%

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

4. Applied egg-rr 54.4%

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

   1. clear-num 54.4%

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

   2. un-div-inv 54.4%

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

   3. clear-num 54.4%

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

   4. metadata-eval 54.4%

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

   5. pow-prod-up 54.4%

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

   6. flip-+ 89.0%

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

   7. unpow2 89.0%

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

   8. fma-define 89.0%

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

   9. add-sqr-sqrt 42.7%

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

   10. sqrt-prod 69.0%

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

   11. sqr-neg 69.0%

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

   12. sqrt-prod 32.2%

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

   13. add-sqr-sqrt 63.3%

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

6. Applied egg-rr 63.3%

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

7. Taylor expanded in z around 0 42.8%

   \[\leadsto x \cdot x - \frac{y \cdot 4}{\color{blue}{\frac{1}{t}}} \]

8. Taylor expanded in x around 0 6.7%

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

   1. associate-*r* 6.7%

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

   2. metadata-eval 6.7%

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

   3. distribute-lft-neg-in 6.7%

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

   4. metadata-eval 6.7%

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

   5. associate-/r/ 6.7%

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

   6. distribute-lft-neg-in 6.7%

      \[\leadsto \color{blue}{-\frac{4}{\frac{1}{t}} \cdot y} \]

   7. *-commutative 6.7%

      \[\leadsto -\color{blue}{y \cdot \frac{4}{\frac{1}{t}}} \]

   8. distribute-rgt-neg-in 6.7%

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

   9. associate-/r/ 6.7%

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

   10. metadata-eval 6.7%

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

   11. distribute-lft-neg-in 6.7%

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

   12. metadata-eval 6.7%

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

10. Simplified 6.7%

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

11. Final simplification 6.7%

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

Developer target: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024067 
(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))))