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

Percentage Accurate: 90.9% → 94.8%

Time: 8.1s

Alternatives: 10

Speedup: 0.8×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.9% 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: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+286}:\\ \;\;\;\;x \cdot x + 4 \cdot \left(y \cdot t - z \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+286)
   (+ (* x x) (* 4.0 (- (* y t) (* z (* z y)))))
   (fma x x (* y (* 4.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+286) {
		tmp = (x * x) + (4.0 * ((y * t) - (z * (z * y))));
	} else {
		tmp = fma(x, x, (y * (4.0 * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+286)
		tmp = Float64(Float64(x * x) + Float64(4.0 * Float64(Float64(y * t) - Float64(z * Float64(z * y)))));
	else
		tmp = fma(x, x, Float64(y * Float64(4.0 * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+286], N[(N[(x * x), $MachinePrecision] + N[(4.0 * N[(N[(y * t), $MachinePrecision] - N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(4.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000007e286

   1. Initial program 93.8%

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

      1. sub-neg 93.8%

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

      2. distribute-rgt-in 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. distribute-rgt-out 93.8%

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

      2. unsub-neg 93.8%

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

      3. add-sqr-sqrt 39.5%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-unprod 35.8%

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

      7. add-sqr-sqrt 57.9%

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

      8. distribute-rgt-out-- 55.7%

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

      9. associate-*r* 55.7%

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

      10. associate-*r* 55.7%

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

      11. *-commutative 55.7%

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

      12. distribute-rgt-out-- 55.7%

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

      13. *-commutative 55.7%

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

      14. associate-*l* 61.8%

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

      15. *-commutative 61.8%

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

      16. add-sqr-sqrt 35.1%

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

      17. sqrt-unprod 65.0%

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

      18. sqr-neg 65.0%

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

      19. sqrt-unprod 42.5%

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

      20. add-sqr-sqrt 98.3%

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

   5. Applied egg-rr 98.3%

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

   if 2.00000000000000007e286 < (*.f64 x x)

   1. Initial program 80.8%

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

      1. sub-neg 80.8%

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

      2. distribute-rgt-in 79.5%

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

   3. Applied egg-rr 79.5%

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

   4. Taylor expanded in z around 0 87.7%

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

      1. cancel-sign-sub-inv 87.7%

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

      2. metadata-eval 87.7%

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

      3. unpow2 87.7%

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

      4. associate-*r* 87.7%

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

      5. *-commutative 87.7%

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

      6. associate-*r* 87.7%

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

      7. fma-def 93.2%

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

   6. Simplified 93.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+286}:\\ \;\;\;\;x \cdot x + 4 \cdot \left(y \cdot t - z \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot t\right)\right)\\ \end{array} \]

Alternative 2: 58.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -4 \cdot \left(y \cdot \left(z \cdot z\right)\right)\\ t_2 := t \cdot \left(4 \cdot y\right)\\ \mathbf{if}\;x \cdot x \leq 1.58 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 1.65 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 33000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 9.6 \cdot 10^{+50} \lor \neg \left(x \cdot x \leq 1.3 \cdot 10^{+132}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* -4.0 (* y (* z z)))) (t_2 (* t (* 4.0 y))))
   (if (<= (* x x) 1.58e-164)
     t_2
     (if (<= (* x x) 1.65e-82)
       t_1
       (if (<= (* x x) 33000000.0)
         t_2
         (if (or (<= (* x x) 9.6e+50) (not (<= (* x x) 1.3e+132)))
           (* x x)
           t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * (y * (z * z));
	double t_2 = t * (4.0 * y);
	double tmp;
	if ((x * x) <= 1.58e-164) {
		tmp = t_2;
	} else if ((x * x) <= 1.65e-82) {
		tmp = t_1;
	} else if ((x * x) <= 33000000.0) {
		tmp = t_2;
	} else if (((x * x) <= 9.6e+50) || !((x * x) <= 1.3e+132)) {
		tmp = x * x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-4.0d0) * (y * (z * z))
    t_2 = t * (4.0d0 * y)
    if ((x * x) <= 1.58d-164) then
        tmp = t_2
    else if ((x * x) <= 1.65d-82) then
        tmp = t_1
    else if ((x * x) <= 33000000.0d0) then
        tmp = t_2
    else if (((x * x) <= 9.6d+50) .or. (.not. ((x * x) <= 1.3d+132))) then
        tmp = x * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -4.0 * (y * (z * z));
	double t_2 = t * (4.0 * y);
	double tmp;
	if ((x * x) <= 1.58e-164) {
		tmp = t_2;
	} else if ((x * x) <= 1.65e-82) {
		tmp = t_1;
	} else if ((x * x) <= 33000000.0) {
		tmp = t_2;
	} else if (((x * x) <= 9.6e+50) || !((x * x) <= 1.3e+132)) {
		tmp = x * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -4.0 * (y * (z * z))
	t_2 = t * (4.0 * y)
	tmp = 0
	if (x * x) <= 1.58e-164:
		tmp = t_2
	elif (x * x) <= 1.65e-82:
		tmp = t_1
	elif (x * x) <= 33000000.0:
		tmp = t_2
	elif ((x * x) <= 9.6e+50) or not ((x * x) <= 1.3e+132):
		tmp = x * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-4.0 * Float64(y * Float64(z * z)))
	t_2 = Float64(t * Float64(4.0 * y))
	tmp = 0.0
	if (Float64(x * x) <= 1.58e-164)
		tmp = t_2;
	elseif (Float64(x * x) <= 1.65e-82)
		tmp = t_1;
	elseif (Float64(x * x) <= 33000000.0)
		tmp = t_2;
	elseif ((Float64(x * x) <= 9.6e+50) || !(Float64(x * x) <= 1.3e+132))
		tmp = Float64(x * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -4.0 * (y * (z * z));
	t_2 = t * (4.0 * y);
	tmp = 0.0;
	if ((x * x) <= 1.58e-164)
		tmp = t_2;
	elseif ((x * x) <= 1.65e-82)
		tmp = t_1;
	elseif ((x * x) <= 33000000.0)
		tmp = t_2;
	elseif (((x * x) <= 9.6e+50) || ~(((x * x) <= 1.3e+132)))
		tmp = x * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-4.0 * N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1.58e-164], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1.65e-82], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 33000000.0], t$95$2, If[Or[LessEqual[N[(x * x), $MachinePrecision], 9.6e+50], N[Not[LessEqual[N[(x * x), $MachinePrecision], 1.3e+132]], $MachinePrecision]], N[(x * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.57999999999999998e-164 or 1.65000000000000011e-82 < (*.f64 x x) < 3.3e7

   1. Initial program 91.5%

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

   2. Taylor expanded in t around inf 57.5%

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

      1. associate-*r* 57.6%

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

   4. Simplified 57.6%

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

   if 1.57999999999999998e-164 < (*.f64 x x) < 1.65000000000000011e-82 or 9.6000000000000007e50 < (*.f64 x x) < 1.3e132

   1. Initial program 94.3%

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

   2. Taylor expanded in z around inf 53.9%

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

      1. unpow2 53.9%

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

   4. Simplified 53.9%

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

   if 3.3e7 < (*.f64 x x) < 9.6000000000000007e50 or 1.3e132 < (*.f64 x x)

   1. Initial program 87.9%

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

   2. Taylor expanded in x around inf 81.1%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 81.1%

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

   4. Simplified 81.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.58 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 1.65 \cdot 10^{-82}:\\ \;\;\;\;-4 \cdot \left(y \cdot \left(z \cdot z\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 33000000:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 9.6 \cdot 10^{+50} \lor \neg \left(x \cdot x \leq 1.3 \cdot 10^{+132}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(y \cdot \left(z \cdot z\right)\right)\\ \end{array} \]

Alternative 3: 59.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.2 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 44000000:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 2.4 \cdot 10^{+56}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 3.35 \cdot 10^{+127}:\\ \;\;\;\;-4 \cdot \left(y \cdot \left(z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 7.2e-163)
   (* t (* 4.0 y))
   (if (<= (* x x) 44000000.0)
     (* z (* z (* y -4.0)))
     (if (<= (* x x) 2.4e+56)
       (* x x)
       (if (<= (* x x) 3.35e+127) (* -4.0 (* y (* z z))) (* x x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 7.2e-163) {
		tmp = t * (4.0 * y);
	} else if ((x * x) <= 44000000.0) {
		tmp = z * (z * (y * -4.0));
	} else if ((x * x) <= 2.4e+56) {
		tmp = x * x;
	} else if ((x * x) <= 3.35e+127) {
		tmp = -4.0 * (y * (z * z));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 7.2d-163) then
        tmp = t * (4.0d0 * y)
    else if ((x * x) <= 44000000.0d0) then
        tmp = z * (z * (y * (-4.0d0)))
    else if ((x * x) <= 2.4d+56) then
        tmp = x * x
    else if ((x * x) <= 3.35d+127) then
        tmp = (-4.0d0) * (y * (z * z))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 7.2e-163) {
		tmp = t * (4.0 * y);
	} else if ((x * x) <= 44000000.0) {
		tmp = z * (z * (y * -4.0));
	} else if ((x * x) <= 2.4e+56) {
		tmp = x * x;
	} else if ((x * x) <= 3.35e+127) {
		tmp = -4.0 * (y * (z * z));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 7.2e-163:
		tmp = t * (4.0 * y)
	elif (x * x) <= 44000000.0:
		tmp = z * (z * (y * -4.0))
	elif (x * x) <= 2.4e+56:
		tmp = x * x
	elif (x * x) <= 3.35e+127:
		tmp = -4.0 * (y * (z * z))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 7.2e-163)
		tmp = Float64(t * Float64(4.0 * y));
	elseif (Float64(x * x) <= 44000000.0)
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	elseif (Float64(x * x) <= 2.4e+56)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 3.35e+127)
		tmp = Float64(-4.0 * Float64(y * Float64(z * z)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 7.2e-163)
		tmp = t * (4.0 * y);
	elseif ((x * x) <= 44000000.0)
		tmp = z * (z * (y * -4.0));
	elseif ((x * x) <= 2.4e+56)
		tmp = x * x;
	elseif ((x * x) <= 3.35e+127)
		tmp = -4.0 * (y * (z * z));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 7.2e-163], N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 44000000.0], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2.4e+56], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 3.35e+127], N[(-4.0 * N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 7.1999999999999996e-163

   1. Initial program 94.5%

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

   2. Taylor expanded in t around inf 60.6%

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

      1. associate-*r* 60.6%

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

   4. Simplified 60.6%

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

   if 7.1999999999999996e-163 < (*.f64 x x) < 4.4e7

   1. Initial program 84.2%

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

   2. Taylor expanded in z around inf 39.0%

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

      1. metadata-eval 39.0%

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

      2. distribute-lft-neg-in 39.0%

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

      3. *-commutative 39.0%

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

      4. unpow2 39.0%

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

      5. *-commutative 39.0%

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

      6. associate-*r* 39.0%

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

      7. associate-*l* 54.7%

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

      8. distribute-rgt-neg-in 54.7%

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

      9. distribute-rgt-neg-in 54.7%

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

      10. distribute-rgt-neg-in 54.7%

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

      11. metadata-eval 54.7%

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

   4. Simplified 54.7%

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

   if 4.4e7 < (*.f64 x x) < 2.40000000000000013e56 or 3.3499999999999998e127 < (*.f64 x x)

   1. Initial program 87.9%

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

   2. Taylor expanded in x around inf 81.1%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 81.1%

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

   4. Simplified 81.1%

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

   if 2.40000000000000013e56 < (*.f64 x x) < 3.3499999999999998e127

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 65.1%

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

      1. unpow2 65.1%

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

   4. Simplified 65.1%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.2 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{elif}\;x \cdot x \leq 44000000:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 2.4 \cdot 10^{+56}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 3.35 \cdot 10^{+127}:\\ \;\;\;\;-4 \cdot \left(y \cdot \left(z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{+289}:\\ \;\;\;\;x \cdot x + 4 \cdot \left(y \cdot t - z \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 5.5e+289)
   (+ (* x x) (* 4.0 (- (* y t) (* z (* z y)))))
   (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5.5e+289) {
		tmp = (x * x) + (4.0 * ((y * t) - (z * (z * y))));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 5.5d+289) then
        tmp = (x * x) + (4.0d0 * ((y * t) - (z * (z * y))))
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 5.5e+289) {
		tmp = (x * x) + (4.0 * ((y * t) - (z * (z * y))));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 5.5e+289:
		tmp = (x * x) + (4.0 * ((y * t) - (z * (z * y))))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 5.5e+289)
		tmp = Float64(Float64(x * x) + Float64(4.0 * Float64(Float64(y * t) - Float64(z * Float64(z * y)))));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 5.5e+289)
		tmp = (x * x) + (4.0 * ((y * t) - (z * (z * y))));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.5e+289], N[(N[(x * x), $MachinePrecision] + N[(4.0 * N[(N[(y * t), $MachinePrecision] - N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.4999999999999999e289

   1. Initial program 93.8%

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

      1. sub-neg 93.8%

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

      2. distribute-rgt-in 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. distribute-rgt-out 93.8%

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

      2. unsub-neg 93.8%

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

      3. add-sqr-sqrt 39.5%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-unprod 35.8%

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

      7. add-sqr-sqrt 57.9%

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

      8. distribute-rgt-out-- 55.7%

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

      9. associate-*r* 55.7%

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

      10. associate-*r* 55.7%

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

      11. *-commutative 55.7%

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

      12. distribute-rgt-out-- 55.7%

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

      13. *-commutative 55.7%

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

      14. associate-*l* 61.8%

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

      15. *-commutative 61.8%

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

      16. add-sqr-sqrt 35.1%

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

      17. sqrt-unprod 65.0%

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

      18. sqr-neg 65.0%

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

      19. sqrt-unprod 42.5%

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

      20. add-sqr-sqrt 98.3%

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

   5. Applied egg-rr 98.3%

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

   if 5.4999999999999999e289 < (*.f64 x x)

   1. Initial program 80.8%

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

   2. Taylor expanded in x around inf 91.8%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 91.8%

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

   4. Simplified 91.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{+289}:\\ \;\;\;\;x \cdot x + 4 \cdot \left(y \cdot t - z \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+129}:\\ \;\;\;\;x \cdot x + \left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(4 \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+129)
   (+ (* x x) (* (* 4.0 y) (- t (* z z))))
   (- (* x x) (* z (* z (* 4.0 y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+129) {
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (4.0 * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1d+129) then
        tmp = (x * x) + ((4.0d0 * y) * (t - (z * z)))
    else
        tmp = (x * x) - (z * (z * (4.0d0 * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+129) {
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)));
	} else {
		tmp = (x * x) - (z * (z * (4.0 * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+129:
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)))
	else:
		tmp = (x * x) - (z * (z * (4.0 * y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+129)
		tmp = Float64(Float64(x * x) + Float64(Float64(4.0 * y) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(4.0 * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+129)
		tmp = (x * x) + ((4.0 * y) * (t - (z * z)));
	else
		tmp = (x * x) - (z * (z * (4.0 * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+129], N[(N[(x * x), $MachinePrecision] + N[(N[(4.0 * y), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e129

   1. Initial program 98.0%

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

   if 1e129 < (*.f64 z z)

   1. Initial program 78.3%

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

   2. Taylor expanded in z around inf 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*r* 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-*r* 91.2%

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

      5. *-commutative 91.2%

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

   4. Simplified 91.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+129}:\\ \;\;\;\;x \cdot x + \left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(4 \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 88.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+110}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(4 \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+110)
   (- (* x x) (* t (* y -4.0)))
   (- (* x x) (* z (* z (* 4.0 y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+110) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = (x * x) - (z * (z * (4.0 * y)));
	}
	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+110) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else
        tmp = (x * x) - (z * (z * (4.0d0 * y)))
    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+110) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = (x * x) - (z * (z * (4.0 * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+110:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = (x * x) - (z * (z * (4.0 * y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+110)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) - Float64(z * Float64(z * Float64(4.0 * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+110)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = (x * x) - (z * (z * (4.0 * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+110], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] - N[(z * N[(z * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e110

   1. Initial program 97.9%

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

   2. Taylor expanded in z around 0 93.9%

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

      1. *-commutative 93.9%

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

      2. *-commutative 93.9%

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

      3. associate-*l* 93.9%

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

   4. Simplified 93.9%

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

   if 2e110 < (*.f64 z z)

   1. Initial program 79.5%

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

   2. Taylor expanded in z around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r* 91.0%

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

      5. *-commutative 91.0%

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

   4. Simplified 91.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+110}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - z \cdot \left(z \cdot \left(4 \cdot y\right)\right)\\ \end{array} \]

Alternative 7: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+68} \lor \neg \left(x \leq 5.6 \cdot 10^{+94}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.05e+68) (not (<= x 5.6e+94)))
   (* x x)
   (* (* y -4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.05e+68) || !(x <= 5.6e+94)) {
		tmp = x * x;
	} else {
		tmp = (y * -4.0) * ((z * z) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-3.05d+68)) .or. (.not. (x <= 5.6d+94))) then
        tmp = x * x
    else
        tmp = (y * (-4.0d0)) * ((z * z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.05e+68) || !(x <= 5.6e+94)) {
		tmp = x * x;
	} else {
		tmp = (y * -4.0) * ((z * z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.05e+68) or not (x <= 5.6e+94):
		tmp = x * x
	else:
		tmp = (y * -4.0) * ((z * z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.05e+68) || !(x <= 5.6e+94))
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(y * -4.0) * Float64(Float64(z * z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.05e+68) || ~((x <= 5.6e+94)))
		tmp = x * x;
	else
		tmp = (y * -4.0) * ((z * z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.05e+68], N[Not[LessEqual[x, 5.6e+94]], $MachinePrecision]], N[(x * x), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05e68 or 5.59999999999999997e94 < x

   1. Initial program 86.3%

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

   2. Taylor expanded in x around inf 83.1%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 83.1%

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

   4. Simplified 83.1%

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

   if -3.05e68 < x < 5.59999999999999997e94

   1. Initial program 92.9%

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

   2. Taylor expanded in x around 0 81.0%

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

      1. *-commutative 81.0%

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

      2. *-commutative 81.0%

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

      3. unpow2 81.0%

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

      4. *-commutative 81.0%

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

      5. associate-*l* 81.0%

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

   4. Simplified 81.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+68} \lor \neg \left(x \leq 5.6 \cdot 10^{+94}\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \left(z \cdot z - t\right)\\ \end{array} \]

Alternative 8: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+197}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+197) (- (* x x) (* t (* y -4.0))) (* z (* z (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+197) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 4d+197) then
        tmp = (x * x) - (t * (y * (-4.0d0)))
    else
        tmp = z * (z * (y * (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+197) {
		tmp = (x * x) - (t * (y * -4.0));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 4e+197:
		tmp = (x * x) - (t * (y * -4.0))
	else:
		tmp = z * (z * (y * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+197)
		tmp = Float64(Float64(x * x) - Float64(t * Float64(y * -4.0)));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 4e+197)
		tmp = (x * x) - (t * (y * -4.0));
	else
		tmp = z * (z * (y * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+197], N[(N[(x * x), $MachinePrecision] - N[(t * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.9999999999999998e197

   1. Initial program 97.5%

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

   2. Taylor expanded in z around 0 90.4%

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

      1. *-commutative 90.4%

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

      2. *-commutative 90.4%

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

      3. associate-*l* 90.4%

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

   4. Simplified 90.4%

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

   if 3.9999999999999998e197 < (*.f64 z z)

   1. Initial program 76.6%

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

   2. Taylor expanded in z around inf 74.6%

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

      1. metadata-eval 74.6%

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

      2. distribute-lft-neg-in 74.6%

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

      3. *-commutative 74.6%

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

      4. unpow2 74.6%

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

      5. *-commutative 74.6%

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

      6. associate-*r* 74.6%

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

      7. associate-*l* 83.8%

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

      8. distribute-rgt-neg-in 83.8%

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

      9. distribute-rgt-neg-in 83.8%

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

      10. distribute-rgt-neg-in 83.8%

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

      11. metadata-eval 83.8%

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

   4. Simplified 83.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+197}:\\ \;\;\;\;x \cdot x - t \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]

Alternative 9: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 44000000:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 44000000.0) (* t (* 4.0 y)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 44000000.0) {
		tmp = t * (4.0 * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 44000000.0d0) then
        tmp = t * (4.0d0 * y)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 44000000.0) {
		tmp = t * (4.0 * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 44000000.0:
		tmp = t * (4.0 * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 44000000.0)
		tmp = Float64(t * Float64(4.0 * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 44000000.0)
		tmp = t * (4.0 * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 44000000.0], N[(t * N[(4.0 * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.4e7

   1. Initial program 91.5%

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

   2. Taylor expanded in t around inf 51.7%

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

      1. associate-*r* 51.7%

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

   4. Simplified 51.7%

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

   if 4.4e7 < (*.f64 x x)

   1. Initial program 88.9%

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

   2. Taylor expanded in x around inf 76.0%

      \[\leadsto \color{blue}{{x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 76.0%

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

   4. Simplified 76.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 44000000:\\ \;\;\;\;t \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 10: 42.0% 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 90.1%

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

2. Taylor expanded in x around inf 45.3%

   \[\leadsto \color{blue}{{x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 45.3%

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

4. Simplified 45.3%

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

5. Final simplification 45.3%

   \[\leadsto x \cdot x \]

Developer target: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023171 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

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

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