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

Percentage Accurate: 91.1% → 95.1%

Time: 10.9s

Alternatives: 10

Speedup: 0.6×

• 48.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% 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: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y + \frac{\frac{z \cdot y}{t}}{\frac{-1}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+297)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (+ (* x x) (* t (* 4.0 (+ y (/ (/ (* z y) t) (/ -1.0 z))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+297) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} else {
		tmp = (x * x) + (t * (4.0 * (y + (((z * y) / t) / (-1.0 / z)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+297)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(Float64(x * x) + Float64(t * Float64(4.0 * Float64(y + Float64(Float64(Float64(z * y) / t) / Float64(-1.0 / z))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+297], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(t * N[(4.0 * N[(y + N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] / N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e297

   1. Initial program 97.8%

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

      1. cancel-sign-sub-inv 97.8%

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

      2. distribute-lft-neg-out 97.8%

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

      3. +-commutative 97.8%

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

      4. associate-*l* 97.8%

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

      5. distribute-lft-neg-in 97.8%

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

      6. associate-*l* 97.8%

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

      7. distribute-rgt-neg-in 97.8%

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

      8. fma-define 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. remove-double-neg 99.4%

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

      13. sub-neg 99.4%

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

   3. Simplified 99.4%

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

   if 2e297 < (*.f64 z z)

   1. Initial program 69.0%

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

   3. Taylor expanded in t around inf 69.0%

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

      1. +-commutative 69.0%

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

      2. *-commutative 69.0%

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

      3. *-commutative 69.0%

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

      4. metadata-eval 69.0%

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

      5. distribute-rgt-neg-in 69.0%

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

      6. distribute-lft-neg-in 69.0%

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

      7. distribute-rgt-out 69.0%

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

      8. unsub-neg 69.0%

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

      9. associate-/l* 69.0%

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

   5. Simplified 69.0%

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

      1. pow2 69.0%

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

      2. *-un-lft-identity 69.0%

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

      3. times-frac 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. clear-num 71.8%

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

      2. frac-times 71.8%

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

      3. metadata-eval 71.8%

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

      4. div-inv 71.8%

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

      5. /-rgt-identity 71.8%

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

      6. metadata-eval 71.8%

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

      7. times-frac 71.8%

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

      8. *-un-lft-identity 71.8%

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

      9. *-un-lft-identity 71.8%

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

   9. Applied egg-rr 71.8%

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

       1. associate-*r/ 77.4%

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

       2. div-inv 77.4%

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

       3. associate-/r* 81.7%

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

   11. Applied egg-rr 81.7%

       \[\leadsto x \cdot x - t \cdot \left(4 \cdot \left(\color{blue}{\frac{\frac{y \cdot z}{t}}{\frac{1}{z}}} - y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y + \frac{\frac{z \cdot y}{t}}{\frac{-1}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq 10^{-145}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y + \frac{\frac{z \cdot y}{t}}{\frac{-1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y 4.0) 1e-145)
   (+ (* x x) (* t (* 4.0 (+ y (/ (/ (* z y) t) (/ -1.0 z))))))
   (fma x x (* (- (* z z) t) (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 4.0) <= 1e-145) {
		tmp = (x * x) + (t * (4.0 * (y + (((z * y) / t) / (-1.0 / z)))));
	} else {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * 4.0) <= 1e-145)
		tmp = Float64(Float64(x * x) + Float64(t * Float64(4.0 * Float64(y + Float64(Float64(Float64(z * y) / t) / Float64(-1.0 / z))))));
	else
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * 4.0), $MachinePrecision], 1e-145], N[(N[(x * x), $MachinePrecision] + N[(t * N[(4.0 * N[(y + N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] / N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 4 binary64)) < 9.99999999999999915e-146

   1. Initial program 89.2%

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

   3. Taylor expanded in t around inf 87.4%

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

      1. +-commutative 87.4%

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

      2. *-commutative 87.4%

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

      3. *-commutative 87.4%

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

      4. metadata-eval 87.4%

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

      5. distribute-rgt-neg-in 87.4%

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

      6. distribute-lft-neg-in 87.4%

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

      7. distribute-rgt-out 87.4%

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

      8. unsub-neg 87.4%

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

      9. associate-/l* 84.5%

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

   5. Simplified 84.5%

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

      1. pow2 84.5%

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

      2. *-un-lft-identity 84.5%

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

      3. times-frac 85.7%

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

   7. Applied egg-rr 85.7%

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

      1. clear-num 85.7%

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

      2. frac-times 85.7%

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

      3. metadata-eval 85.7%

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

      4. div-inv 85.7%

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

      5. /-rgt-identity 85.7%

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

      6. metadata-eval 85.7%

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

      7. times-frac 85.7%

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

      8. *-un-lft-identity 85.7%

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

      9. *-un-lft-identity 85.7%

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

   9. Applied egg-rr 85.7%

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

       1. associate-*r/ 90.3%

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

       2. div-inv 90.3%

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

       3. associate-/r* 92.9%

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

   11. Applied egg-rr 92.9%

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

   if 9.99999999999999915e-146 < (*.f64 y #s(literal 4 binary64))

   1. Initial program 91.7%

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

      1. fma-neg 95.8%

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

      2. distribute-lft-neg-in 95.8%

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

      3. *-commutative 95.8%

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

      4. distribute-rgt-neg-in 95.8%

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

      5. metadata-eval 95.8%

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

   3. Simplified 95.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq 10^{-145}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y + \frac{\frac{z \cdot y}{t}}{\frac{-1}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y + \frac{\frac{z \cdot y}{t}}{\frac{-1}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+297)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (+ (* x x) (* t (* 4.0 (+ y (/ (/ (* z y) t) (/ -1.0 z))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 2e+297) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + (t * (4.0 * (y + (((z * y) / t) / (-1.0 / 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) :: tmp
    if ((z * z) <= 2d+297) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) + (t * (4.0d0 * (y + (((z * y) / t) / ((-1.0d0) / z)))))
    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+297) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + (t * (4.0 * (y + (((z * y) / t) / (-1.0 / z)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 2e+297:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) + (t * (4.0 * (y + (((z * y) / t) / (-1.0 / z)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+297)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) + Float64(t * Float64(4.0 * Float64(y + Float64(Float64(Float64(z * y) / t) / Float64(-1.0 / z))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 2e+297)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) + (t * (4.0 * (y + (((z * y) / t) / (-1.0 / z)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+297], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(t * N[(4.0 * N[(y + N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] / N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e297

   1. Initial program 97.8%

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

   if 2e297 < (*.f64 z z)

   1. Initial program 69.0%

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

   3. Taylor expanded in t around inf 69.0%

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

      1. +-commutative 69.0%

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

      2. *-commutative 69.0%

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

      3. *-commutative 69.0%

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

      4. metadata-eval 69.0%

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

      5. distribute-rgt-neg-in 69.0%

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

      6. distribute-lft-neg-in 69.0%

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

      7. distribute-rgt-out 69.0%

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

      8. unsub-neg 69.0%

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

      9. associate-/l* 69.0%

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

   5. Simplified 69.0%

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

      1. pow2 69.0%

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

      2. *-un-lft-identity 69.0%

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

      3. times-frac 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. clear-num 71.8%

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

      2. frac-times 71.8%

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

      3. metadata-eval 71.8%

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

      4. div-inv 71.8%

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

      5. /-rgt-identity 71.8%

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

      6. metadata-eval 71.8%

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

      7. times-frac 71.8%

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

      8. *-un-lft-identity 71.8%

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

      9. *-un-lft-identity 71.8%

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

   9. Applied egg-rr 71.8%

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

       1. associate-*r/ 77.4%

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

       2. div-inv 77.4%

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

       3. associate-/r* 81.7%

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

   11. Applied egg-rr 81.7%

       \[\leadsto x \cdot x - t \cdot \left(4 \cdot \left(\color{blue}{\frac{\frac{y \cdot z}{t}}{\frac{1}{z}}} - y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y + \frac{\frac{z \cdot y}{t}}{\frac{-1}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+304}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+304)
   (+ (* x x) (* (* y 4.0) (- t (* z z))))
   (+ (* x x) (* t (* 4.0 (- y (/ (* z y) (/ t z))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+304) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / 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) :: tmp
    if ((z * z) <= 1d+304) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = (x * x) + (t * (4.0d0 * (y - ((z * y) / (t / z)))))
    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+304) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+304:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+304)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = Float64(Float64(x * x) + Float64(t * Float64(4.0 * Float64(y - Float64(Float64(z * y) / Float64(t / z))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+304)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = (x * x) + (t * (4.0 * (y - ((z * y) / (t / z)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+304], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(t * N[(4.0 * N[(y - N[(N[(z * y), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999994e303

   1. Initial program 97.8%

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

   if 9.9999999999999994e303 < (*.f64 z z)

   1. Initial program 68.0%

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

   3. Taylor expanded in t around inf 68.0%

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

      1. +-commutative 68.0%

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

      2. *-commutative 68.0%

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

      3. *-commutative 68.0%

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

      4. metadata-eval 68.0%

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

      5. distribute-rgt-neg-in 68.0%

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

      6. distribute-lft-neg-in 68.0%

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

      7. distribute-rgt-out 68.0%

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

      8. unsub-neg 68.0%

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

      9. associate-/l* 68.1%

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

   5. Simplified 68.1%

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

      1. pow2 68.1%

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

      2. *-un-lft-identity 68.1%

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

      3. times-frac 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. clear-num 70.9%

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

      2. frac-times 70.9%

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

      3. metadata-eval 70.9%

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

      4. div-inv 70.9%

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

      5. /-rgt-identity 70.9%

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

      6. metadata-eval 70.9%

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

      7. times-frac 70.9%

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

      8. *-un-lft-identity 70.9%

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

      9. *-un-lft-identity 70.9%

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

   9. Applied egg-rr 70.9%

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

       1. associate-*r/ 76.7%

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

   11. Applied egg-rr 76.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+304}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+249}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+249) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+249) {
		tmp = (x * x) + ((y * 4.0) * (t - (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) <= 2d+249) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (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) <= 2e+249) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2e+249:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+249)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - 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) <= 2e+249)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+249], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.9999999999999998e249

   1. Initial program 91.2%

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

   if 1.9999999999999998e249 < (*.f64 x x)

   1. Initial program 87.3%

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

   3. Taylor expanded in y around 0 87.3%

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

   5. Simplified 94.4%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 94.4%

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

   7. Applied egg-rr 94.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+249}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-240}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-89}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 1e-240)
   (* (* z z) (* y -4.0))
   (if (<= (* x x) 5e-89) (* 4.0 (* y t)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e-240) {
		tmp = (z * z) * (y * -4.0);
	} else if ((x * x) <= 5e-89) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 1d-240) then
        tmp = (z * z) * (y * (-4.0d0))
    else if ((x * x) <= 5d-89) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 1e-240) {
		tmp = (z * z) * (y * -4.0);
	} else if ((x * x) <= 5e-89) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 1e-240:
		tmp = (z * z) * (y * -4.0)
	elif (x * x) <= 5e-89:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 1e-240)
		tmp = Float64(Float64(z * z) * Float64(y * -4.0));
	elseif (Float64(x * x) <= 5e-89)
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 1e-240)
		tmp = (z * z) * (y * -4.0);
	elseif ((x * x) <= 5e-89)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-240], N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e-89], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.9999999999999997e-241

   1. Initial program 93.9%

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

      1. fma-neg 93.9%

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

      2. distribute-lft-neg-in 93.9%

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

      3. *-commutative 93.9%

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

      4. distribute-rgt-neg-in 93.9%

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

      5. metadata-eval 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around inf 55.9%

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

      1. associate-*r* 55.9%

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

      2. *-commutative 55.9%

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

   7. Simplified 55.9%

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

      1. unpow2 55.9%

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

   9. Applied egg-rr 55.9%

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

   if 9.9999999999999997e-241 < (*.f64 x x) < 4.99999999999999967e-89

   1. Initial program 86.3%

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

      1. fma-neg 86.3%

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

      2. distribute-lft-neg-in 86.3%

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

      3. *-commutative 86.3%

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

      4. distribute-rgt-neg-in 86.3%

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

      5. metadata-eval 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in t around inf 45.9%

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

      1. *-commutative 45.9%

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

   7. Simplified 45.9%

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

   if 4.99999999999999967e-89 < (*.f64 x x)

   1. Initial program 89.1%

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

   3. Taylor expanded in y around 0 89.1%

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

   5. Simplified 69.0%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 69.0%

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

   7. Applied egg-rr 69.0%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-240}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-89}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+186}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -4\right) \cdot \frac{z}{\frac{1}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+186)
   (- (* x x) (* y (* t -4.0)))
   (* (* y -4.0) (/ z (/ 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+186) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * (z / (1.0 / 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) :: tmp
    if ((z * z) <= 1d+186) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (y * (-4.0d0)) * (z / (1.0d0 / z))
    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+186) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (y * -4.0) * (z / (1.0 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1e+186:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (y * -4.0) * (z / (1.0 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+186)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(y * -4.0) * Float64(z / Float64(1.0 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1e+186)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (y * -4.0) * (z / (1.0 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+186], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -4.0), $MachinePrecision] * N[(z / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999998e185

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0 87.4%

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

      1. *-commutative 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l* 87.4%

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

   5. Simplified 87.4%

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

   if 9.9999999999999998e185 < (*.f64 z z)

   1. Initial program 73.1%

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

      1. fma-neg 77.8%

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

      2. distribute-lft-neg-in 77.8%

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

      3. *-commutative 77.8%

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

      4. distribute-rgt-neg-in 77.8%

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

      5. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in z around inf 74.5%

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

      1. associate-*r* 74.5%

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

      2. *-commutative 74.5%

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

   7. Simplified 74.5%

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

      1. unpow2 74.5%

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

      2. /-rgt-identity 74.5%

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

      3. clear-num 74.5%

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

      4. /-rgt-identity 74.5%

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

      5. frac-times 74.5%

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

      6. *-un-lft-identity 74.5%

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

   9. Applied egg-rr 74.5%

      \[\leadsto \left(y \cdot -4\right) \cdot \color{blue}{\frac{z}{\frac{1}{z} \cdot 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

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

Alternative 8: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.02 \cdot 10^{+186}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1.02e+186)
   (- (* x x) (* y (* t -4.0)))
   (* (* z z) (* y -4.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1.02e+186) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * z) <= 1.02d+186) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (z * z) * (y * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1.02e+186) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = (z * z) * (y * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * z) <= 1.02e+186:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = (z * z) * (y * -4.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1.02e+186)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(Float64(z * z) * Float64(y * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * z) <= 1.02e+186)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = (z * z) * (y * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.02e+186], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.01999999999999999e186

   1. Initial program 98.7%

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

   3. Taylor expanded in z around 0 87.4%

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

      1. *-commutative 87.4%

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

      2. *-commutative 87.4%

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

      3. associate-*l* 87.4%

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

   5. Simplified 87.4%

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

   if 1.01999999999999999e186 < (*.f64 z z)

   1. Initial program 73.1%

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

      1. fma-neg 77.8%

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

      2. distribute-lft-neg-in 77.8%

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

      3. *-commutative 77.8%

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

      4. distribute-rgt-neg-in 77.8%

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

      5. metadata-eval 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in z around inf 74.5%

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

      1. associate-*r* 74.5%

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

      2. *-commutative 74.5%

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

   7. Simplified 74.5%

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

      1. unpow2 74.5%

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

   9. Applied egg-rr 74.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.02 \cdot 10^{+186}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \left(y \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 9 \cdot 10^{-84}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 9e-84) (* 4.0 (* y t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 9e-84) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 9d-84) then
        tmp = 4.0d0 * (y * t)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 9e-84) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 9e-84:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 9e-84)
		tmp = Float64(4.0 * Float64(y * t));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 9e-84)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 9e-84], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.00000000000000031e-84

   1. Initial program 91.3%

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

      1. fma-neg 91.3%

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

      2. distribute-lft-neg-in 91.3%

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

      3. *-commutative 91.3%

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

      4. distribute-rgt-neg-in 91.3%

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

      5. metadata-eval 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in t around inf 47.4%

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

      1. *-commutative 47.4%

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

   7. Simplified 47.4%

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

   if 9.00000000000000031e-84 < (*.f64 x x)

   1. Initial program 89.1%

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

   3. Taylor expanded in y around 0 89.1%

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

   5. Simplified 69.0%

      \[\leadsto x \cdot x - \color{blue}{0} \]
   6. Step-by-step derivation

      1. --rgt-identity 69.0%

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

   7. Applied egg-rr 69.0%

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

Alternative 10: 41.2% 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. Add Preprocessing

3. Taylor expanded in y around 0 90.1%

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

5. Simplified 40.8%

   \[\leadsto x \cdot x - \color{blue}{0} \]
6. Step-by-step derivation

   1. --rgt-identity 40.8%

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

7. Applied egg-rr 40.8%

   \[\leadsto \color{blue}{x \cdot x} \]
8. Add Preprocessing

Developer Target 1: 91.1% 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 2024139 
(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))))