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

Percentage Accurate: 90.3% → 95.8%

Time: 8.1s

Alternatives: 10

Speedup: 1.1×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 4e+296)
   (fma (- (* z z) t) (* y -4.0) (* x x))
   (* (* -4.0 z) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 4e+296) {
		tmp = fma(((z * z) - t), (y * -4.0), (x * x));
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 4e+296)
		tmp = fma(Float64(Float64(z * z) - t), Float64(y * -4.0), Float64(x * x));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+296], N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.99999999999999993e296

   1. Initial program 95.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.9

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

   4. Applied rewrites 98.9%

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

   if 3.99999999999999993e296 < (*.f64 z z)

   1. Initial program 80.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 93.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z - t, y \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ t_2 := \left(4 \cdot y\right) \cdot t\_1\\ t_3 := \left(-4 \cdot t\_1\right) \cdot y\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+186}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot t\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)) (t_2 (* (* 4.0 y) t_1)) (t_3 (* (* -4.0 t_1) y)))
   (if (<= t_2 -1e+186)
     t_3
     (if (<= t_2 5e+83) (fma x x (* (* y t) 4.0)) t_3))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double t_2 = (4.0 * y) * t_1;
	double t_3 = (-4.0 * t_1) * y;
	double tmp;
	if (t_2 <= -1e+186) {
		tmp = t_3;
	} else if (t_2 <= 5e+83) {
		tmp = fma(x, x, ((y * t) * 4.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	t_2 = Float64(Float64(4.0 * y) * t_1)
	t_3 = Float64(Float64(-4.0 * t_1) * y)
	tmp = 0.0
	if (t_2 <= -1e+186)
		tmp = t_3;
	elseif (t_2 <= 5e+83)
		tmp = fma(x, x, Float64(Float64(y * t) * 4.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * y), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * t$95$1), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+186], t$95$3, If[LessEqual[t$95$2, 5e+83], N[(x * x + N[(N[(y * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < -9.9999999999999998e185 or 5.00000000000000029e83 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t))

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 86.8%

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

   if -9.9999999999999998e185 < (*.f64 (*.f64 y #s(literal 4 binary64)) (-.f64 (*.f64 z z) t)) < 5.00000000000000029e83

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 87.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot \left(z \cdot z - t\right) \leq -1 \cdot 10^{+186}:\\ \;\;\;\;\left(-4 \cdot \left(z \cdot z - t\right)\right) \cdot y\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot \left(z \cdot z - t\right) \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot t\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot \left(z \cdot z - t\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(z \cdot z\right), -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-49)
   (fma (* 4.0 t) y (* x x))
   (if (<= (* z z) 1e+283)
     (fma (* y (* z z)) -4.0 (* x x))
     (* (* -4.0 z) (* y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-49) {
		tmp = fma((4.0 * t), y, (x * x));
	} else if ((z * z) <= 1e+283) {
		tmp = fma((y * (z * z)), -4.0, (x * x));
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-49)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	elseif (Float64(z * z) <= 1e+283)
		tmp = fma(Float64(y * Float64(z * z)), -4.0, Float64(x * x));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-49], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+283], N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.9999999999999999e-49

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.1

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

   5. Applied rewrites 93.1%

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

   7. Applied rewrites 93.9%

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

   if 4.9999999999999999e-49 < (*.f64 z z) < 9.99999999999999955e282

   1. Initial program 92.8%

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

   3. Taylor expanded in t around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if 9.99999999999999955e282 < (*.f64 z z)

   1. Initial program 79.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 91.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(z \cdot z\right), -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot z, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 5e-49)
   (fma (* 4.0 t) y (* x x))
   (if (<= (* z z) 1e+283)
     (fma (* (* y z) z) -4.0 (* x x))
     (* (* -4.0 z) (* y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 5e-49) {
		tmp = fma((4.0 * t), y, (x * x));
	} else if ((z * z) <= 1e+283) {
		tmp = fma(((y * z) * z), -4.0, (x * x));
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 5e-49)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	elseif (Float64(z * z) <= 1e+283)
		tmp = fma(Float64(Float64(y * z) * z), -4.0, Float64(x * x));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-49], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+283], N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.9999999999999999e-49

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 93.1

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

   5. Applied rewrites 93.1%

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

   7. Applied rewrites 93.9%

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

   if 4.9999999999999999e-49 < (*.f64 z z) < 9.99999999999999955e282

   1. Initial program 92.8%

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

   3. Taylor expanded in t around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 79.7%

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

   if 9.99999999999999955e282 < (*.f64 z z)

   1. Initial program 79.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 91.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot z, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot z - t\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{+206}:\\ \;\;\;\;1 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z z) t)))
   (if (<= t_1 -5000000000000.0)
     (* (* 4.0 t) y)
     (if (<= t_1 3e+206) (* 1.0 (* x x)) (* (* -4.0 z) (* y z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = (4.0 * t) * y;
	} else if (t_1 <= 3e+206) {
		tmp = 1.0 * (x * x);
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	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) :: tmp
    t_1 = (z * z) - t
    if (t_1 <= (-5000000000000.0d0)) then
        tmp = (4.0d0 * t) * y
    else if (t_1 <= 3d+206) then
        tmp = 1.0d0 * (x * x)
    else
        tmp = ((-4.0d0) * z) * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) - t;
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = (4.0 * t) * y;
	} else if (t_1 <= 3e+206) {
		tmp = 1.0 * (x * x);
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) - t
	tmp = 0
	if t_1 <= -5000000000000.0:
		tmp = (4.0 * t) * y
	elif t_1 <= 3e+206:
		tmp = 1.0 * (x * x)
	else:
		tmp = (-4.0 * z) * (y * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) - t)
	tmp = 0.0
	if (t_1 <= -5000000000000.0)
		tmp = Float64(Float64(4.0 * t) * y);
	elseif (t_1 <= 3e+206)
		tmp = Float64(1.0 * Float64(x * x));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) - t;
	tmp = 0.0;
	if (t_1 <= -5000000000000.0)
		tmp = (4.0 * t) * y;
	elseif (t_1 <= 3e+206)
		tmp = 1.0 * (x * x);
	else
		tmp = (-4.0 * z) * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000000.0], N[(N[(4.0 * t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 3e+206], N[(1.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 z z) t) < -5e12

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

   7. Applied rewrites 78.9%

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

   if -5e12 < (-.f64 (*.f64 z z) t) < 3.0000000000000001e206

   1. Initial program 96.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 96.5

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

   4. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 71.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 55.1%

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]

   if 3.0000000000000001e206 < (-.f64 (*.f64 z z) t)

   1. Initial program 83.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   7. Applied rewrites 73.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq -5000000000000:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{elif}\;z \cdot z - t \leq 3 \cdot 10^{+206}:\\ \;\;\;\;1 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+213) (fma (* 4.0 t) y (* x x)) (* (* -4.0 z) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+213) {
		tmp = fma((4.0 * t), y, (x * x));
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+213)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+213], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999984e212

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   7. Applied rewrites 84.1%

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

   if 9.99999999999999984e212 < (*.f64 z z)

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   7. Applied rewrites 88.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot t\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+213) (fma x x (* (* y t) 4.0)) (* (* -4.0 z) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+213) {
		tmp = fma(x, x, ((y * t) * 4.0));
	} else {
		tmp = (-4.0 * z) * (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+213)
		tmp = fma(x, x, Float64(Float64(y * t) * 4.0));
	else
		tmp = Float64(Float64(-4.0 * z) * Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+213], N[(x * x + N[(N[(y * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999984e212

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   7. Applied rewrites 83.0%

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

   if 9.99999999999999984e212 < (*.f64 z z)

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   7. Applied rewrites 88.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot t\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot z\right) \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 95.3

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

4. Applied rewrites 95.3%

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

5. Final simplification 95.3%

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

Alternative 9: 45.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-34}:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.8e-34) (* (* 4.0 t) y) (* 1.0 (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.8e-34) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = 1.0 * (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 <= 2.8d-34) then
        tmp = (4.0d0 * t) * y
    else
        tmp = 1.0d0 * (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 <= 2.8e-34) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = 1.0 * (x * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.8e-34:
		tmp = (4.0 * t) * y
	else:
		tmp = 1.0 * (x * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.8e-34)
		tmp = Float64(Float64(4.0 * t) * y);
	else
		tmp = Float64(1.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.8e-34)
		tmp = (4.0 * t) * y;
	else
		tmp = 1.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.8e-34], N[(N[(4.0 * t), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999997e-34

   1. Initial program 94.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 37.8

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

   5. Applied rewrites 37.8%

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

   7. Applied rewrites 37.8%

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

   if 2.79999999999999997e-34 < x

   1. Initial program 87.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 94.5

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

   6. Applied rewrites 94.5%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 94.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 64.5%

       \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-34}:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 1.0 * (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 = 1.0d0 * (x * x)
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return Float64(1.0 * Float64(x * x))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 95.3

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

4. Applied rewrites 95.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. flip-- N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. clear-num N/A

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

   13. flip-- N/A

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

   14. lift--.f64 N/A

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

   15. lower-/.f64 95.2

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

6. Applied rewrites 95.2%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 70.5%

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

10. Taylor expanded in x around inf

    \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
11. Step-by-step derivation

12. Applied rewrites 39.2%

    \[\leadsto 1 \cdot \left(\color{blue}{x} \cdot x\right) \]
13. Add Preprocessing

Developer Target 1: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024295 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

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