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

Percentage Accurate: 91.0% → 94.9%

Time: 11.0s

Alternatives: 9

Speedup: 0.6×

• 48.9% 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: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\mathsf{fma}\left(x, x, t \cdot \left(\mathsf{fma}\left(y, z \cdot \frac{z}{t}, -y\right) \cdot -4\right)\right)}\\ \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {t\_1}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (cbrt (fma x x (* t (* (fma y (* z (/ z t)) (- y)) -4.0))))))
   (if (<= (* z z) 1e+308)
     (fma (* y 4.0) (- t (* z z)) (* x x))
     (* t_1 (pow t_1 2.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = cbrt(fma(x, x, (t * (fma(y, (z * (z / t)), -y) * -4.0))));
	double tmp;
	if ((z * z) <= 1e+308) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} else {
		tmp = t_1 * pow(t_1, 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = cbrt(fma(x, x, Float64(t * Float64(fma(y, Float64(z * Float64(z / t)), Float64(-y)) * -4.0))))
	tmp = 0.0
	if (Float64(z * z) <= 1e+308)
		tmp = fma(Float64(y * 4.0), Float64(t - Float64(z * z)), Float64(x * x));
	else
		tmp = Float64(t_1 * (t_1 ^ 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Power[N[(x * x + N[(t * N[(N[(y * N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 1e+308], N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e308

   1. Initial program 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. distribute-lft-neg-out 97.9%

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

      3. +-commutative 97.9%

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

      4. distribute-lft-neg-out 97.9%

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

      5. distribute-lft-neg-in 97.9%

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

      6. distribute-rgt-neg-in 97.9%

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

      7. fma-define 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. remove-double-neg 99.4%

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

      12. 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 1e308 < (*.f64 z z)

   1. Initial program 67.7%

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

   3. Taylor expanded in t around inf 67.7%

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

      1. +-commutative 67.7%

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

      2. *-commutative 67.7%

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

      3. *-commutative 67.7%

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

      4. metadata-eval 67.7%

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

      5. distribute-rgt-neg-in 67.7%

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

      6. distribute-lft-neg-in 67.7%

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

      7. distribute-rgt-out 67.7%

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

      8. unsub-neg 67.7%

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

      9. associate-/l* 67.7%

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

   5. Simplified 67.7%

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

      1. unpow2 67.7%

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

      2. *-un-lft-identity 67.7%

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

      3. times-frac 75.0%

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

   7. Applied egg-rr 75.0%

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

      1. /-rgt-identity 75.0%

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

      2. associate-*r* 80.9%

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

      3. clear-num 80.9%

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

      4. un-div-inv 80.9%

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

   9. Applied egg-rr 80.9%

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

       1. add-cube-cbrt 80.8%

          \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot x - t \cdot \left(4 \cdot \left(\frac{y \cdot z}{\frac{t}{z}} - y\right)\right)} \cdot \sqrt[3]{x \cdot x - t \cdot \left(4 \cdot \left(\frac{y \cdot z}{\frac{t}{z}} - y\right)\right)}\right) \cdot \sqrt[3]{x \cdot x - t \cdot \left(4 \cdot \left(\frac{y \cdot z}{\frac{t}{z}} - y\right)\right)}} \]

   11. Applied egg-rr 81.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, x, t \cdot \left(\mathsf{fma}\left(y, z \cdot \frac{z}{t}, -y\right) \cdot -4\right)\right)} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(x, x, t \cdot \left(\mathsf{fma}\left(y, z \cdot \frac{z}{t}, -y\right) \cdot -4\right)\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+308)
   (fma (* y 4.0) (- t (* z z)) (* x x))
   (+ (* 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+308) {
		tmp = fma((y * 4.0), (t - (z * z)), (x * x));
	} 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+308)
		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(z * y) / Float64(t / z))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+308], 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[(z * y), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e308

   1. Initial program 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. distribute-lft-neg-out 97.9%

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

      3. +-commutative 97.9%

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

      4. distribute-lft-neg-out 97.9%

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

      5. distribute-lft-neg-in 97.9%

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

      6. distribute-rgt-neg-in 97.9%

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

      7. fma-define 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. remove-double-neg 99.4%

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

      12. 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 1e308 < (*.f64 z z)

   1. Initial program 67.7%

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

   3. Taylor expanded in t around inf 67.7%

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

      1. +-commutative 67.7%

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

      2. *-commutative 67.7%

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

      3. *-commutative 67.7%

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

      4. metadata-eval 67.7%

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

      5. distribute-rgt-neg-in 67.7%

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

      6. distribute-lft-neg-in 67.7%

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

      7. distribute-rgt-out 67.7%

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

      8. unsub-neg 67.7%

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

      9. associate-/l* 67.7%

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

   5. Simplified 67.7%

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

      1. unpow2 67.7%

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

      2. *-un-lft-identity 67.7%

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

      3. times-frac 75.0%

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

   7. Applied egg-rr 75.0%

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

      1. /-rgt-identity 75.0%

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

      2. associate-*r* 80.9%

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

      3. clear-num 80.9%

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

      4. un-div-inv 80.9%

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

   9. Applied egg-rr 80.9%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 4, t - z \cdot z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.35e+154)
   (fma x x (* (- (* z z) t) (* y -4.0)))
   (+ (* 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 <= 1.35e+154) {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	} 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 (z <= 1.35e+154)
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	else
		tmp = Float64(Float64(x * x) + Float64(t * Float64(4.0 * Float64(y - Float64(Float64(z * y) / Float64(t / z))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.35e+154], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $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 z < 1.35000000000000003e154

   1. Initial program 95.0%

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

      1. fmm-def 96.4%

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

      2. distribute-lft-neg-in 96.4%

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

      3. *-commutative 96.4%

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

      4. distribute-rgt-neg-in 96.4%

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

      5. metadata-eval 96.4%

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

   3. Simplified 96.4%

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

   if 1.35000000000000003e154 < z

   1. Initial program 64.4%

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

   3. Taylor expanded in t around inf 64.4%

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

      1. +-commutative 64.4%

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

      2. *-commutative 64.4%

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

      3. *-commutative 64.4%

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

      4. metadata-eval 64.4%

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

      5. distribute-rgt-neg-in 64.4%

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

      6. distribute-lft-neg-in 64.4%

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

      7. distribute-rgt-out 64.4%

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

      8. unsub-neg 64.4%

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

      9. associate-/l* 64.4%

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

   5. Simplified 64.4%

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

      1. unpow2 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. times-frac 71.5%

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

   7. Applied egg-rr 71.5%

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

      1. /-rgt-identity 71.5%

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

      2. associate-*r* 78.6%

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

      3. clear-num 78.6%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+308)
   (+ (* 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+308) {
		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+308) 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+308) {
		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+308:
		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+308)
		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+308)
		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+308], 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) < 1e308

   1. Initial program 97.9%

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

   if 1e308 < (*.f64 z z)

   1. Initial program 67.7%

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

   3. Taylor expanded in t around inf 67.7%

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

      1. +-commutative 67.7%

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

      2. *-commutative 67.7%

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

      3. *-commutative 67.7%

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

      4. metadata-eval 67.7%

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

      5. distribute-rgt-neg-in 67.7%

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

      6. distribute-lft-neg-in 67.7%

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

      7. distribute-rgt-out 67.7%

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

      8. unsub-neg 67.7%

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

      9. associate-/l* 67.7%

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

   5. Simplified 67.7%

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

      1. unpow2 67.7%

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

      2. *-un-lft-identity 67.7%

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

      3. times-frac 75.0%

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

   7. Applied egg-rr 75.0%

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

      1. /-rgt-identity 75.0%

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

      2. associate-*r* 80.9%

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

      3. clear-num 80.9%

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

      4. un-div-inv 80.9%

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

   9. Applied egg-rr 80.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+308}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + t \cdot \left(4 \cdot \left(y - \frac{z \cdot y}{\frac{t}{z}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.45 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2.45e+297) (+ (* 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) <= 2.45e+297) {
		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) <= 2.45d+297) 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) <= 2.45e+297) {
		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) <= 2.45e+297:
		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) <= 2.45e+297)
		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) <= 2.45e+297)
		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], 2.45e+297], 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) < 2.45000000000000014e297

   1. Initial program 93.3%

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

   if 2.45000000000000014e297 < (*.f64 x x)

   1. Initial program 80.3%

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

   3. Taylor expanded in y around 0 80.3%

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

   5. Simplified 91.8%

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

      1. --rgt-identity 91.8%

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.45 \cdot 10^{+297}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 155000:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.7e-171)
   (* 4.0 (* y t))
   (if (<= x 155000.0) (* -4.0 (* (* z z) y)) (* x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.7e-171) {
		tmp = 4.0 * (y * t);
	} else if (x <= 155000.0) {
		tmp = -4.0 * ((z * z) * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 2.7d-171) then
        tmp = 4.0d0 * (y * t)
    else if (x <= 155000.0d0) then
        tmp = (-4.0d0) * ((z * z) * y)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.7e-171) {
		tmp = 4.0 * (y * t);
	} else if (x <= 155000.0) {
		tmp = -4.0 * ((z * z) * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.7e-171:
		tmp = 4.0 * (y * t)
	elif x <= 155000.0:
		tmp = -4.0 * ((z * z) * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.7e-171)
		tmp = Float64(4.0 * Float64(y * t));
	elseif (x <= 155000.0)
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.7e-171)
		tmp = 4.0 * (y * t);
	elseif (x <= 155000.0)
		tmp = -4.0 * ((z * z) * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.7e-171], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 155000.0], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.70000000000000014e-171

   1. Initial program 90.1%

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

      1. fmm-def 91.6%

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

      2. distribute-lft-neg-in 91.6%

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

      3. *-commutative 91.6%

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

      4. distribute-rgt-neg-in 91.6%

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

      5. metadata-eval 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in t around inf 41.1%

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

      1. *-commutative 41.1%

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

   7. Simplified 41.1%

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

   if 2.70000000000000014e-171 < x < 155000

   1. Initial program 94.8%

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

   3. Taylor expanded in t around inf 86.7%

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

      1. +-commutative 86.7%

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

      2. *-commutative 86.7%

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

      3. *-commutative 86.7%

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

      4. metadata-eval 86.7%

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

      5. distribute-rgt-neg-in 86.7%

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

      6. distribute-lft-neg-in 86.7%

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

      7. distribute-rgt-out 86.7%

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

      8. unsub-neg 86.7%

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

      9. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in z around inf 56.5%

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

      1. *-commutative 56.5%

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

   8. Simplified 56.5%

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

      1. unpow2 56.5%

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

   10. Applied egg-rr 56.5%

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

   if 155000 < x

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0 88.4%

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

   5. Simplified 72.5%

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

      1. --rgt-identity 72.5%

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

   7. Applied egg-rr 72.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 155000:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+116}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.2e+116) (- (* x x) (* y (* t -4.0))) (* -4.0 (* (* z z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+116) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.2d+116) then
        tmp = (x * x) - (y * (t * (-4.0d0)))
    else
        tmp = (-4.0d0) * ((z * z) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.2e+116) {
		tmp = (x * x) - (y * (t * -4.0));
	} else {
		tmp = -4.0 * ((z * z) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.2e+116:
		tmp = (x * x) - (y * (t * -4.0))
	else:
		tmp = -4.0 * ((z * z) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.2e+116)
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	else
		tmp = Float64(-4.0 * Float64(Float64(z * z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.2e+116)
		tmp = (x * x) - (y * (t * -4.0));
	else
		tmp = -4.0 * ((z * z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.2e+116], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e116

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0 77.7%

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

      1. *-commutative 77.7%

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

      2. *-commutative 77.7%

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

      3. associate-*l* 77.7%

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

   5. Simplified 77.7%

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

   if 1.2e116 < z

   1. Initial program 70.3%

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

   3. Taylor expanded in t around inf 68.2%

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

      1. +-commutative 68.2%

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

      2. *-commutative 68.2%

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

      3. *-commutative 68.2%

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

      4. metadata-eval 68.2%

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

      5. distribute-rgt-neg-in 68.2%

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

      6. distribute-lft-neg-in 68.2%

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

      7. distribute-rgt-out 68.2%

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

      8. unsub-neg 68.2%

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

      9. associate-/l* 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in z around inf 70.4%

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

      1. *-commutative 70.4%

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

   8. Simplified 70.4%

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

      1. unpow2 70.4%

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

   10. Applied egg-rr 70.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+116}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(z \cdot z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.062:\\ \;\;\;\;4 \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 0.062) (* 4.0 (* y t)) (* x x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 0.062) {
		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 <= 0.062d0) 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 <= 0.062) {
		tmp = 4.0 * (y * t);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 0.062:
		tmp = 4.0 * (y * t)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 0.062)
		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 <= 0.062)
		tmp = 4.0 * (y * t);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 0.062], N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.062

   1. Initial program 91.0%

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

      1. fmm-def 92.2%

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

      2. distribute-lft-neg-in 92.2%

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

      3. *-commutative 92.2%

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

      4. distribute-rgt-neg-in 92.2%

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

      5. metadata-eval 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in t around inf 39.6%

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

      1. *-commutative 39.6%

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

   7. Simplified 39.6%

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

   if 0.062 < x

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0 88.7%

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

   5. Simplified 71.0%

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

      1. --rgt-identity 71.0%

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

   7. Applied egg-rr 71.0%

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

Alternative 9: 41.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0 90.2%

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

5. Simplified 42.1%

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

   1. --rgt-identity 42.1%

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

7. Applied egg-rr 42.1%

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

Developer Target 1: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(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))))