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

Percentage Accurate: 90.9% → 96.4%

Time: 11.4s

Alternatives: 7

Speedup: 0.6×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.9% 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: 96.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+302) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+302)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(y * 4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+302], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e302

   1. Initial program 96.7%

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

      1. fma-neg 98.9%

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

      2. distribute-lft-neg-in 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   if 2.0000000000000002e302 < (*.f64 z z)

   1. Initial program 62.7%

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

      1. sub-neg 62.7%

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

      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%

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

      5. pow-sqr 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-+ 62.7%

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

      7. add-sqr-sqrt 32.1%

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

      8. sqrt-prod 48.3%

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

      9. sqr-neg 48.3%

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

      10. sqrt-prod 30.6%

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

      11. add-sqr-sqrt 62.7%

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

      12. sub-neg 62.7%

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

   6. Applied egg-rr 62.7%

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

   7. Taylor expanded in z around inf 62.7%

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

      1. associate-/r/ 62.7%

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

      2. /-rgt-identity 62.7%

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

      3. unpow2 62.7%

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

      4. associate-*r* 91.4%

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

   9. Applied egg-rr 91.4%

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

4. Final simplification 96.8%

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

Alternative 2: 56.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.06 \cdot 10^{-134} \lor \neg \left(x \cdot x \leq 7 \cdot 10^{+26}\right) \land x \cdot x \leq 4.5 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* x x) 1.06e-134)
         (and (not (<= (* x x) 7e+26)) (<= (* x x) 4.5e+140)))
   (* y (* t 4.0))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 1.06e-134) || (!((x * x) <= 7e+26) && ((x * x) <= 4.5e+140))) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) <= 1.06d-134) .or. (.not. ((x * x) <= 7d+26)) .and. ((x * x) <= 4.5d+140)) then
        tmp = y * (t * 4.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) <= 1.06e-134) || (!((x * x) <= 7e+26) && ((x * x) <= 4.5e+140))) {
		tmp = y * (t * 4.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) <= 1.06e-134) or (not ((x * x) <= 7e+26) and ((x * x) <= 4.5e+140)):
		tmp = y * (t * 4.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x * x) <= 1.06e-134) || (!(Float64(x * x) <= 7e+26) && (Float64(x * x) <= 4.5e+140)))
		tmp = Float64(y * Float64(t * 4.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) <= 1.06e-134) || (~(((x * x) <= 7e+26)) && ((x * x) <= 4.5e+140)))
		tmp = y * (t * 4.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 1.06e-134], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 7e+26]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 4.5e+140]]], N[(y * N[(t * 4.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.06e-134 or 6.9999999999999998e26 < (*.f64 x x) < 4.5000000000000002e140

   1. Initial program 88.9%

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

   3. Taylor expanded in t around inf 44.6%

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

      1. associate-*r* 44.6%

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

   5. Simplified 44.6%

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

   if 1.06e-134 < (*.f64 x x) < 6.9999999999999998e26 or 4.5000000000000002e140 < (*.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. Step-by-step derivation

      1. sub-neg 85.8%

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

      2. distribute-lft-neg-in 85.8%

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

      3. *-commutative 85.8%

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

      4. distribute-lft-neg-in 85.8%

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

      5. metadata-eval 85.8%

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

      6. *-commutative 85.8%

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

      7. add-sqr-sqrt 35.0%

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

      8. sqrt-unprod 66.2%

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

      9. swap-sqr 66.2%

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

      10. metadata-eval 66.2%

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

      11. metadata-eval 66.2%

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

      12. swap-sqr 66.2%

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

      13. sqrt-unprod 37.0%

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

      14. add-sqr-sqrt 58.0%

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

      15. expm1-log1p-u 56.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

      16. add-sqr-sqrt 36.3%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

   4. Applied egg-rr 70.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \color{blue}{{x}^{2}} \]

      2. pow2 68.9%

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

   8. Applied egg-rr 68.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.06 \cdot 10^{-134} \lor \neg \left(x \cdot x \leq 7 \cdot 10^{+26}\right) \land x \cdot x \leq 4.5 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(t \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+302) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (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+302) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) - (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+302) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+302:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+302)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - 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+302)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+302], 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[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e302

   1. Initial program 96.7%

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

   if 2.0000000000000002e302 < (*.f64 z z)

   1. Initial program 62.7%

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

      1. sub-neg 62.7%

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

      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%

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

      5. pow-sqr 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-+ 62.7%

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

      7. add-sqr-sqrt 32.1%

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

      8. sqrt-prod 48.3%

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

      9. sqr-neg 48.3%

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

      10. sqrt-prod 30.6%

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

      11. add-sqr-sqrt 62.7%

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

      12. sub-neg 62.7%

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

   6. Applied egg-rr 62.7%

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

   7. Taylor expanded in z around inf 62.7%

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

      1. associate-/r/ 62.7%

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

      2. /-rgt-identity 62.7%

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

      3. unpow2 62.7%

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

      4. associate-*r* 91.4%

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

   9. Applied egg-rr 91.4%

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

4. Final simplification 95.2%

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

Alternative 4: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+41}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.3e+41)
   (- (* x x) (* -4.0 (* t y)))
   (- (* x x) (* z (* z (* y 4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.3e+41) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - (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 <= 3.3d+41) then
        tmp = (x * x) - ((-4.0d0) * (t * y))
    else
        tmp = (x * x) - (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 <= 3.3e+41) {
		tmp = (x * x) - (-4.0 * (t * y));
	} else {
		tmp = (x * x) - (z * (z * (y * 4.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.3e+41:
		tmp = (x * x) - (-4.0 * (t * y))
	else:
		tmp = (x * x) - (z * (z * (y * 4.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.3e+41)
		tmp = Float64(Float64(x * x) - Float64(-4.0 * Float64(t * y)));
	else
		tmp = Float64(Float64(x * x) - 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 <= 3.3e+41)
		tmp = (x * x) - (-4.0 * (t * y));
	else
		tmp = (x * x) - (z * (z * (y * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.3e+41], N[(N[(x * x), $MachinePrecision] - N[(-4.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.3e41

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0 76.3%

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

      1. *-commutative 76.3%

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

   5. Simplified 76.3%

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

   if 3.3e41 < z

   1. Initial program 70.1%

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

      1. sub-neg 70.1%

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

      2. flip-+ 10.8%

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

      3. pow2 10.8%

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

      4. pow2 10.8%

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

      5. pow-prod-up 10.8%

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

      6. metadata-eval 10.8%

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

      7. pow2 10.8%

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

   4. Applied egg-rr 10.8%

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

      1. clear-num 10.8%

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

      2. un-div-inv 10.8%

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

      3. clear-num 10.8%

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

      4. metadata-eval 10.8%

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

      5. pow-sqr 10.7%

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

      6. flip-+ 70.0%

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

      7. add-sqr-sqrt 34.3%

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

      8. sqrt-prod 54.1%

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

      9. sqr-neg 54.1%

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

      10. sqrt-prod 35.7%

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

      11. add-sqr-sqrt 70.0%

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

      12. sub-neg 70.0%

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

   6. Applied egg-rr 70.0%

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

   7. Taylor expanded in z around inf 70.0%

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

      1. associate-/r/ 70.1%

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

      2. /-rgt-identity 70.1%

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

      3. unpow2 70.1%

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

      4. associate-*r* 92.8%

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

   9. Applied egg-rr 92.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+41}:\\ \;\;\;\;x \cdot x - -4 \cdot \left(t \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around 0 64.1%

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

   1. *-commutative 64.1%

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

5. Simplified 64.1%

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

6. Final simplification 64.1%

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

Alternative 6: 41.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. sub-neg 87.3%

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

   2. distribute-lft-neg-in 87.3%

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

   3. *-commutative 87.3%

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

   4. distribute-lft-neg-in 87.3%

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

   5. metadata-eval 87.3%

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

   6. *-commutative 87.3%

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

   7. add-sqr-sqrt 43.5%

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

   8. sqrt-unprod 57.0%

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

   9. swap-sqr 57.0%

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

   10. metadata-eval 57.0%

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

   11. metadata-eval 57.0%

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

   12. swap-sqr 57.0%

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

   13. sqrt-unprod 22.8%

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

   14. add-sqr-sqrt 36.2%

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

   15. expm1-log1p-u 35.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

   16. add-sqr-sqrt 22.4%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

4. Applied egg-rr 65.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

6. Simplified 42.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 42.9%

      \[\leadsto \color{blue}{{x}^{2}} \]

   2. pow2 42.9%

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

8. Applied egg-rr 42.9%

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

9. Final simplification 42.9%

   \[\leadsto x \cdot x \]
10. Add Preprocessing

Alternative 7: 2.0% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 -1.0)

C

double code(double x, double y, double z, double t) {
	return -1.0;
}

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 = -1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return -1.0;
}

Python

def code(x, y, z, t):
	return -1.0

Julia

function code(x, y, z, t)
	return -1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -1.0;
end

Wolfram

code[x_, y_, z_, t_] := -1.0

Derivation

1. Initial program 87.3%

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

   1. sub-neg 87.3%

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

   2. distribute-lft-neg-in 87.3%

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

   3. *-commutative 87.3%

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

   4. distribute-lft-neg-in 87.3%

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

   5. metadata-eval 87.3%

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

   6. *-commutative 87.3%

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

   7. add-sqr-sqrt 43.5%

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

   8. sqrt-unprod 57.0%

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

   9. swap-sqr 57.0%

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

   10. metadata-eval 57.0%

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

   11. metadata-eval 57.0%

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

   12. swap-sqr 57.0%

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

   13. sqrt-unprod 22.8%

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

   14. add-sqr-sqrt 36.2%

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

   15. expm1-log1p-u 35.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

   16. add-sqr-sqrt 22.4%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot x + \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)} \cdot \left(z \cdot z - t\right)\right)\right) \]

4. Applied egg-rr 65.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(-4 \cdot \left({z}^{2} - t\right)\right)\right)\right)\right)} \]

6. Simplified 42.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{2}\right)\right)} \]

7. Taylor expanded in x around inf 17.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-2 \cdot \log \left(\frac{1}{x}\right)}\right) \]
8. Step-by-step derivation

   1. log-rec 17.0%

      \[\leadsto \mathsf{expm1}\left(-2 \cdot \color{blue}{\left(-\log x\right)}\right) \]

9. Simplified 17.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-2 \cdot \left(-\log x\right)}\right) \]

10. Taylor expanded in x around 0 1.9%

    \[\leadsto \color{blue}{-1} \]

11. Final simplification 1.9%

    \[\leadsto -1 \]
12. Add Preprocessing

Developer target: 91.0% 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 2024050 
(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))))