Data.Colour.Matrix:inverse from colour-2.3.3, B

Percentage Accurate: 91.4% → 95.9%

Time: 7.6s

Alternatives: 14

Speedup: 0.7×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - z \cdot t}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;{\left(\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}\right)}^{3} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (- (pow (/ (cbrt (* x y)) (cbrt a)) 3.0) (/ t (/ a z)))
     (if (<= t_1 5e+276) (/ t_1 a) (fma y (/ x a) (/ (- z) (/ a t)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = pow((cbrt((x * y)) / cbrt(a)), 3.0) - (t / (a / z));
	} else if (t_1 <= 5e+276) {
		tmp = t_1 / a;
	} else {
		tmp = fma(y, (x / a), (-z / (a / t)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64((Float64(cbrt(Float64(x * y)) / cbrt(a)) ^ 3.0) - Float64(t / Float64(a / z)));
	elseif (t_1 <= 5e+276)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(y, Float64(x / a), Float64(Float64(-z) / Float64(a / t)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Power[N[(N[Power[N[(x * y), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+276], N[(t$95$1 / a), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0

   1. Initial program 79.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 79.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. add-cube-cbrt 79.0%

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

      3. add-cube-cbrt 79.0%

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

      4. times-frac 79.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      5. fma-neg 79.0%

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

      6. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      7. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      8. associate-/l* 92.8%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z}{\frac{a}{t}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 92.8%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{z}{\frac{a}{t}}} \]

      2. associate-/r/ 96.5%

         \[\leadsto \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \color{blue}{\frac{z}{a} \cdot t} \]

      3. *-commutative 96.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}} - \frac{z}{a} \cdot t \]

      4. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\color{blue}{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}}{{\left(\sqrt[3]{a}\right)}^{2}} - \frac{z}{a} \cdot t \]

      5. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\color{blue}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} - \frac{z}{a} \cdot t \]

      6. times-frac 96.5%

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

      7. cube-unmult 96.5%

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

      8. associate-*l/ 79.0%

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

      9. *-commutative 79.0%

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

      10. associate-/l* 96.5%

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

   5. Simplified 96.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.00000000000000001e276

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 5.00000000000000001e276 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 78.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 78.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a} - \frac{z \cdot t}{a} \]

      3. *-un-lft-identity 78.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{1 \cdot a}} - \frac{z \cdot t}{a} \]

      4. times-frac 86.5%

         \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{x}{a}} - \frac{z \cdot t}{a} \]

      5. fma-neg 86.5%

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

      6. associate-/l* 97.2%

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

   3. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{1}, \frac{x}{a}, -\frac{z}{\frac{a}{t}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;{\left(\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}\right)}^{3} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (- (* x (/ y a)) (/ t (/ a z)))
     (if (<= t_1 5e+276) (/ t_1 a) (fma y (/ x a) (/ (- z) (/ a t)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x * (y / a)) - (t / (a / z));
	} else if (t_1 <= 5e+276) {
		tmp = t_1 / a;
	} else {
		tmp = fma(y, (x / a), (-z / (a / t)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(t / Float64(a / z)));
	elseif (t_1 <= 5e+276)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(y, Float64(x / a), Float64(Float64(-z) / Float64(a / t)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+276], N[(t$95$1 / a), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision] + N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0

   1. Initial program 79.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 79.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. add-cube-cbrt 79.0%

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

      3. add-cube-cbrt 79.0%

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

      4. times-frac 79.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      5. fma-neg 79.0%

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

      6. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      7. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      8. associate-/l* 92.8%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z}{\frac{a}{t}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 92.8%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{z}{\frac{a}{t}}} \]

      2. associate-/r/ 96.5%

         \[\leadsto \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \color{blue}{\frac{z}{a} \cdot t} \]

      3. *-commutative 96.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}} - \frac{z}{a} \cdot t \]

      4. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\color{blue}{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}}{{\left(\sqrt[3]{a}\right)}^{2}} - \frac{z}{a} \cdot t \]

      5. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\color{blue}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} - \frac{z}{a} \cdot t \]

      6. times-frac 96.5%

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

      7. cube-unmult 96.5%

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

      8. associate-*l/ 79.0%

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

      9. *-commutative 79.0%

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

      10. associate-/l* 96.5%

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

   5. Simplified 96.5%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}\right)}^{3} - \frac{t}{\frac{a}{z}}} \]
   6. Step-by-step derivation

      1. unpow3 96.5%

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

      2. frac-times 96.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{t}{\frac{a}{z}} \]

      3. times-frac 96.5%

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

      4. add-cube-cbrt 96.5%

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

      5. add-cube-cbrt 96.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{a}} - \frac{t}{\frac{a}{z}} \]

      6. associate-*r/ 96.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{a}} - \frac{t}{\frac{a}{z}} \]

      7. *-commutative 96.2%

         \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   7. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 5.00000000000000001e276

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 5.00000000000000001e276 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 78.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 78.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a} - \frac{z \cdot t}{a} \]

      3. *-un-lft-identity 78.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{1 \cdot a}} - \frac{z \cdot t}{a} \]

      4. times-frac 86.5%

         \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{x}{a}} - \frac{z \cdot t}{a} \]

      5. fma-neg 86.5%

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

      6. associate-/l* 97.2%

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

   3. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{1}, \frac{x}{a}, -\frac{z}{\frac{a}{t}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{a}, \frac{-z}{\frac{a}{t}}\right)\\ \end{array} \]

Alternative 3: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+299}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+299)))
     (- (* x (/ y a)) (/ t (/ a z)))
     (/ t_1 a))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+299)) {
		tmp = (x * (y / a)) - (t / (a / z));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+299)) {
		tmp = (x * (y / a)) - (t / (a / z));
	} else {
		tmp = t_1 / a;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+299):
		tmp = (x * (y / a)) - (t / (a / z))
	else:
		tmp = t_1 / a
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+299))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(t / Float64(a / z)));
	else
		tmp = Float64(t_1 / a);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+299)))
		tmp = (x * (y / a)) - (t / (a / z));
	else
		tmp = t_1 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+299]], $MachinePrecision]], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 1.0000000000000001e299 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 76.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 76.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. add-cube-cbrt 76.5%

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

      3. add-cube-cbrt 76.5%

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

      4. times-frac 76.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      5. fma-neg 76.5%

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

      6. pow2 76.5%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      7. pow2 76.5%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      8. associate-/l* 88.2%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   3. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z}{\frac{a}{t}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 88.2%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{z}{\frac{a}{t}}} \]

      2. associate-/r/ 89.8%

         \[\leadsto \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \color{blue}{\frac{z}{a} \cdot t} \]

      3. *-commutative 89.8%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}} - \frac{z}{a} \cdot t \]

      4. unpow2 89.8%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\color{blue}{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}}{{\left(\sqrt[3]{a}\right)}^{2}} - \frac{z}{a} \cdot t \]

      5. unpow2 89.8%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\color{blue}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} - \frac{z}{a} \cdot t \]

      6. times-frac 89.8%

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

      7. cube-unmult 89.8%

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

      8. associate-*l/ 76.5%

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

      9. *-commutative 76.5%

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

      10. associate-/l* 89.9%

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

   5. Simplified 89.9%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}\right)}^{3} - \frac{t}{\frac{a}{z}}} \]
   6. Step-by-step derivation

      1. unpow3 89.9%

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

      2. frac-times 89.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{t}{\frac{a}{z}} \]

      3. times-frac 89.9%

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

      4. add-cube-cbrt 89.9%

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

      5. add-cube-cbrt 89.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{a}} - \frac{t}{\frac{a}{z}} \]

      6. associate-*r/ 98.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{a}} - \frac{t}{\frac{a}{z}} \]

      7. *-commutative 98.1%

         \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   7. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e299

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty \lor \neg \left(x \cdot y - z \cdot t \leq 10^{+299}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \end{array} \]

Alternative 4: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{t}{\frac{a}{z}}\\ t_2 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - t_1\\ \mathbf{elif}\;t_2 \leq 10^{+299}:\\ \;\;\;\;\frac{t_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - t_1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ t (/ a z))) (t_2 (- (* x y) (* z t))))
   (if (<= t_2 (- INFINITY))
     (- (* x (/ y a)) t_1)
     (if (<= t_2 1e+299) (/ t_2 a) (- (/ x (/ a y)) t_1)))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t / (a / z);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (x * (y / a)) - t_1;
	} else if (t_2 <= 1e+299) {
		tmp = t_2 / a;
	} else {
		tmp = (x / (a / y)) - t_1;
	}
	return tmp;
}

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t / (a / z);
	double t_2 = (x * y) - (z * t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * (y / a)) - t_1;
	} else if (t_2 <= 1e+299) {
		tmp = t_2 / a;
	} else {
		tmp = (x / (a / y)) - t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = t / (a / z)
	t_2 = (x * y) - (z * t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (x * (y / a)) - t_1
	elif t_2 <= 1e+299:
		tmp = t_2 / a
	else:
		tmp = (x / (a / y)) - t_1
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(t / Float64(a / z))
	t_2 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(y / a)) - t_1);
	elseif (t_2 <= 1e+299)
		tmp = Float64(t_2 / a);
	else
		tmp = Float64(Float64(x / Float64(a / y)) - t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t / (a / z);
	t_2 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (x * (y / a)) - t_1;
	elseif (t_2 <= 1e+299)
		tmp = t_2 / a;
	else
		tmp = (x / (a / y)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e+299], N[(t$95$2 / a), $MachinePrecision], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0

   1. Initial program 79.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 79.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. add-cube-cbrt 79.0%

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

      3. add-cube-cbrt 79.0%

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

      4. times-frac 79.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      5. fma-neg 79.0%

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

      6. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      7. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      8. associate-/l* 92.8%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z}{\frac{a}{t}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 92.8%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{z}{\frac{a}{t}}} \]

      2. associate-/r/ 96.5%

         \[\leadsto \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \color{blue}{\frac{z}{a} \cdot t} \]

      3. *-commutative 96.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}} - \frac{z}{a} \cdot t \]

      4. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\color{blue}{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}}{{\left(\sqrt[3]{a}\right)}^{2}} - \frac{z}{a} \cdot t \]

      5. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\color{blue}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} - \frac{z}{a} \cdot t \]

      6. times-frac 96.5%

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

      7. cube-unmult 96.5%

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

      8. associate-*l/ 79.0%

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

      9. *-commutative 79.0%

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

      10. associate-/l* 96.5%

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

   5. Simplified 96.5%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}\right)}^{3} - \frac{t}{\frac{a}{z}}} \]
   6. Step-by-step derivation

      1. unpow3 96.5%

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

      2. frac-times 96.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{t}{\frac{a}{z}} \]

      3. times-frac 96.5%

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

      4. add-cube-cbrt 96.5%

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

      5. add-cube-cbrt 96.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{a}} - \frac{t}{\frac{a}{z}} \]

      6. associate-*r/ 96.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{a}} - \frac{t}{\frac{a}{z}} \]

      7. *-commutative 96.2%

         \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   7. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e299

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 1.0000000000000001e299 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 74.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 74.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. associate-/l* 87.1%

         \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]

      3. associate-/l* 99.9%

         \[\leadsto \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]

   4. Taylor expanded in z around 0 87.1%

      \[\leadsto \frac{x}{\frac{a}{y}} - \color{blue}{\frac{t \cdot z}{a}} \]
   5. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \frac{x}{\frac{a}{y}} - \color{blue}{\frac{t}{\frac{a}{z}}} \]

   6. Simplified 99.9%

      \[\leadsto \frac{x}{\frac{a}{y}} - \color{blue}{\frac{t}{\frac{a}{z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+299}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 5: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 10^{+299}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))))
   (if (<= t_1 (- INFINITY))
     (- (* x (/ y a)) (/ t (/ a z)))
     (if (<= t_1 1e+299) (/ t_1 a) (- (/ x (/ a y)) (/ z (/ a t)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x * (y / a)) - (t / (a / z));
	} else if (t_1 <= 1e+299) {
		tmp = t_1 / a;
	} else {
		tmp = (x / (a / y)) - (z / (a / t));
	}
	return tmp;
}

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * (y / a)) - (t / (a / z));
	} else if (t_1 <= 1e+299) {
		tmp = t_1 / a;
	} else {
		tmp = (x / (a / y)) - (z / (a / t));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) - (z * t)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x * (y / a)) - (t / (a / z))
	elif t_1 <= 1e+299:
		tmp = t_1 / a
	else:
		tmp = (x / (a / y)) - (z / (a / t))
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(y / a)) - Float64(t / Float64(a / z)));
	elseif (t_1 <= 1e+299)
		tmp = Float64(t_1 / a);
	else
		tmp = Float64(Float64(x / Float64(a / y)) - Float64(z / Float64(a / t)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) - (z * t);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x * (y / a)) - (t / (a / z));
	elseif (t_1 <= 1e+299)
		tmp = t_1 / a;
	else
		tmp = (x / (a / y)) - (z / (a / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], N[(t$95$1 / a), $MachinePrecision], N[(N[(x / N[(a / y), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0

   1. Initial program 79.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 79.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. add-cube-cbrt 79.0%

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

      3. add-cube-cbrt 79.0%

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

      4. times-frac 79.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}} - \frac{z \cdot t}{a} \]

      5. fma-neg 79.0%

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

      6. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      7. pow2 79.0%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z \cdot t}{a}\right) \]

      8. associate-/l* 92.8%

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\color{blue}{\frac{z}{\frac{a}{t}}}\right) \]

   3. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}, \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}, -\frac{z}{\frac{a}{t}}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 92.8%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{z}{\frac{a}{t}}} \]

      2. associate-/r/ 96.5%

         \[\leadsto \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \color{blue}{\frac{z}{a} \cdot t} \]

      3. *-commutative 96.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{{\left(\sqrt[3]{x \cdot y}\right)}^{2}}{{\left(\sqrt[3]{a}\right)}^{2}}} - \frac{z}{a} \cdot t \]

      4. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\color{blue}{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}}{{\left(\sqrt[3]{a}\right)}^{2}} - \frac{z}{a} \cdot t \]

      5. unpow2 96.5%

         \[\leadsto \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\color{blue}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} - \frac{z}{a} \cdot t \]

      6. times-frac 96.5%

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

      7. cube-unmult 96.5%

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

      8. associate-*l/ 79.0%

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

      9. *-commutative 79.0%

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

      10. associate-/l* 96.5%

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

   5. Simplified 96.5%

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}}\right)}^{3} - \frac{t}{\frac{a}{z}}} \]
   6. Step-by-step derivation

      1. unpow3 96.5%

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

      2. frac-times 96.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\sqrt[3]{x \cdot y}}{\sqrt[3]{a}} - \frac{t}{\frac{a}{z}} \]

      3. times-frac 96.5%

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

      4. add-cube-cbrt 96.5%

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

      5. add-cube-cbrt 96.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{a}} - \frac{t}{\frac{a}{z}} \]

      6. associate-*r/ 96.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{a}} - \frac{t}{\frac{a}{z}} \]

      7. *-commutative 96.2%

         \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   7. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{\frac{y}{a} \cdot x} - \frac{t}{\frac{a}{z}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1.0000000000000001e299

   1. Initial program 99.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 1.0000000000000001e299 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 74.4%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Step-by-step derivation

      1. div-sub 74.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}} \]

      2. associate-/l* 87.1%

         \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{z \cdot t}{a} \]

      3. associate-/l* 99.9%

         \[\leadsto \frac{x}{\frac{a}{y}} - \color{blue}{\frac{z}{\frac{a}{t}}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 10^{+299}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 6: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \left(-t\right) \cdot \frac{z}{a}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-285}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- t) (/ z a))))
   (if (<= (* x y) -2e-40)
     (/ (* x y) a)
     (if (<= (* x y) -2e-261)
       t_1
       (if (<= (* x y) 5e-285)
         (* z (/ (- t) a))
         (if (<= (* x y) 2e+47) t_1 (/ y (/ a x))))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = -t * (z / a);
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = (x * y) / a;
	} else if ((x * y) <= -2e-261) {
		tmp = t_1;
	} else if ((x * y) <= 5e-285) {
		tmp = z * (-t / a);
	} else if ((x * y) <= 2e+47) {
		tmp = t_1;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t * (z / a)
    if ((x * y) <= (-2d-40)) then
        tmp = (x * y) / a
    else if ((x * y) <= (-2d-261)) then
        tmp = t_1
    else if ((x * y) <= 5d-285) then
        tmp = z * (-t / a)
    else if ((x * y) <= 2d+47) then
        tmp = t_1
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -t * (z / a);
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = (x * y) / a;
	} else if ((x * y) <= -2e-261) {
		tmp = t_1;
	} else if ((x * y) <= 5e-285) {
		tmp = z * (-t / a);
	} else if ((x * y) <= 2e+47) {
		tmp = t_1;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = -t * (z / a)
	tmp = 0
	if (x * y) <= -2e-40:
		tmp = (x * y) / a
	elif (x * y) <= -2e-261:
		tmp = t_1
	elif (x * y) <= 5e-285:
		tmp = z * (-t / a)
	elif (x * y) <= 2e+47:
		tmp = t_1
	else:
		tmp = y / (a / x)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(-t) * Float64(z / a))
	tmp = 0.0
	if (Float64(x * y) <= -2e-40)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= -2e-261)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e-285)
		tmp = Float64(z * Float64(Float64(-t) / a));
	elseif (Float64(x * y) <= 2e+47)
		tmp = t_1;
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = -t * (z / a);
	tmp = 0.0;
	if ((x * y) <= -2e-40)
		tmp = (x * y) / a;
	elseif ((x * y) <= -2e-261)
		tmp = t_1;
	elseif ((x * y) <= 5e-285)
		tmp = z * (-t / a);
	elseif ((x * y) <= 2e+47)
		tmp = t_1;
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-40], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-261], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e-285], N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+47], t$95$1, N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -1.9999999999999999e-40

   1. Initial program 93.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 77.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]

   if -1.9999999999999999e-40 < (*.f64 x y) < -1.99999999999999997e-261 or 5.00000000000000018e-285 < (*.f64 x y) < 2.0000000000000001e47

   1. Initial program 97.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 73.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 73.4%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 73.4%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 66.9%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 66.9%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 66.9%

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

      6. distribute-frac-neg 66.9%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 66.9%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]

   if -1.99999999999999997e-261 < (*.f64 x y) < 5.00000000000000018e-285

   1. Initial program 93.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 90.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 90.5%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 90.5%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 90.4%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 90.4%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 90.4%

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

      6. distribute-frac-neg 90.4%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 90.4%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg 90.4%

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

      2. distribute-rgt-neg-out 90.4%

         \[\leadsto \color{blue}{-t \cdot \frac{z}{a}} \]

      3. clear-num 90.4%

         \[\leadsto -t \cdot \color{blue}{\frac{1}{\frac{a}{z}}} \]

      4. div-inv 91.9%

         \[\leadsto -\color{blue}{\frac{t}{\frac{a}{z}}} \]

      5. associate-/r/ 81.8%

         \[\leadsto -\color{blue}{\frac{t}{a} \cdot z} \]

   6. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{-\frac{t}{a} \cdot z} \]

   if 2.0000000000000001e47 < (*.f64 x y)

   1. Initial program 91.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 75.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ 79.4%

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
   5. Step-by-step derivation

      1. clear-num 79.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{x}}} \cdot y \]

      2. associate-*l/ 79.4%

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{a}{x}}} \]

      3. *-un-lft-identity 79.4%

         \[\leadsto \frac{\color{blue}{y}}{\frac{a}{x}} \]

   6. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-285}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 7: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-285}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-40)
   (/ (* x y) a)
   (if (<= (* x y) -2e-261)
     (* (- t) (/ z a))
     (if (<= (* x y) 5e-285)
       (* z (/ (- t) a))
       (if (<= (* x y) 2e+47) (/ (- t) (/ a z)) (/ y (/ a x)))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = (x * y) / a;
	} else if ((x * y) <= -2e-261) {
		tmp = -t * (z / a);
	} else if ((x * y) <= 5e-285) {
		tmp = z * (-t / a);
	} else if ((x * y) <= 2e+47) {
		tmp = -t / (a / z);
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d-40)) then
        tmp = (x * y) / a
    else if ((x * y) <= (-2d-261)) then
        tmp = -t * (z / a)
    else if ((x * y) <= 5d-285) then
        tmp = z * (-t / a)
    else if ((x * y) <= 2d+47) then
        tmp = -t / (a / z)
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = (x * y) / a;
	} else if ((x * y) <= -2e-261) {
		tmp = -t * (z / a);
	} else if ((x * y) <= 5e-285) {
		tmp = z * (-t / a);
	} else if ((x * y) <= 2e+47) {
		tmp = -t / (a / z);
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-40:
		tmp = (x * y) / a
	elif (x * y) <= -2e-261:
		tmp = -t * (z / a)
	elif (x * y) <= 5e-285:
		tmp = z * (-t / a)
	elif (x * y) <= 2e+47:
		tmp = -t / (a / z)
	else:
		tmp = y / (a / x)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-40)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= -2e-261)
		tmp = Float64(Float64(-t) * Float64(z / a));
	elseif (Float64(x * y) <= 5e-285)
		tmp = Float64(z * Float64(Float64(-t) / a));
	elseif (Float64(x * y) <= 2e+47)
		tmp = Float64(Float64(-t) / Float64(a / z));
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-40)
		tmp = (x * y) / a;
	elseif ((x * y) <= -2e-261)
		tmp = -t * (z / a);
	elseif ((x * y) <= 5e-285)
		tmp = z * (-t / a);
	elseif ((x * y) <= 2e+47)
		tmp = -t / (a / z);
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-40], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e-261], N[((-t) * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-285], N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+47], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 x y) < -1.9999999999999999e-40

   1. Initial program 93.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 77.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]

   if -1.9999999999999999e-40 < (*.f64 x y) < -1.99999999999999997e-261

   1. Initial program 99.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 83.8%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 83.8%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 73.5%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 73.5%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 73.5%

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

      6. distribute-frac-neg 73.5%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 73.5%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]

   if -1.99999999999999997e-261 < (*.f64 x y) < 5.00000000000000018e-285

   1. Initial program 93.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 90.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 90.5%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 90.5%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 90.4%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 90.4%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 90.4%

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

      6. distribute-frac-neg 90.4%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 90.4%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg 90.4%

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

      2. distribute-rgt-neg-out 90.4%

         \[\leadsto \color{blue}{-t \cdot \frac{z}{a}} \]

      3. clear-num 90.4%

         \[\leadsto -t \cdot \color{blue}{\frac{1}{\frac{a}{z}}} \]

      4. div-inv 91.9%

         \[\leadsto -\color{blue}{\frac{t}{\frac{a}{z}}} \]

      5. associate-/r/ 81.8%

         \[\leadsto -\color{blue}{\frac{t}{a} \cdot z} \]

   6. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{-\frac{t}{a} \cdot z} \]

   if 5.00000000000000018e-285 < (*.f64 x y) < 2.0000000000000001e47

   1. Initial program 96.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 65.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 65.9%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 65.9%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 62.1%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 62.1%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 62.1%

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

      6. distribute-frac-neg 62.1%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 62.1%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 30.7%

         \[\leadsto \color{blue}{\sqrt{t \cdot \frac{-z}{a}} \cdot \sqrt{t \cdot \frac{-z}{a}}} \]

      2. sqrt-unprod 24.5%

         \[\leadsto \color{blue}{\sqrt{\left(t \cdot \frac{-z}{a}\right) \cdot \left(t \cdot \frac{-z}{a}\right)}} \]

      3. swap-sqr 18.7%

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot t\right) \cdot \left(\frac{-z}{a} \cdot \frac{-z}{a}\right)}} \]

      4. distribute-frac-neg 18.7%

         \[\leadsto \sqrt{\left(t \cdot t\right) \cdot \left(\color{blue}{\left(-\frac{z}{a}\right)} \cdot \frac{-z}{a}\right)} \]

      5. distribute-frac-neg 18.7%

         \[\leadsto \sqrt{\left(t \cdot t\right) \cdot \left(\left(-\frac{z}{a}\right) \cdot \color{blue}{\left(-\frac{z}{a}\right)}\right)} \]

      6. sqr-neg 18.7%

         \[\leadsto \sqrt{\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{z}{a} \cdot \frac{z}{a}\right)}} \]

      7. swap-sqr 24.5%

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot \frac{z}{a}\right) \cdot \left(t \cdot \frac{z}{a}\right)}} \]

      8. clear-num 24.5%

         \[\leadsto \sqrt{\left(t \cdot \color{blue}{\frac{1}{\frac{a}{z}}}\right) \cdot \left(t \cdot \frac{z}{a}\right)} \]

      9. div-inv 24.6%

         \[\leadsto \sqrt{\color{blue}{\frac{t}{\frac{a}{z}}} \cdot \left(t \cdot \frac{z}{a}\right)} \]

      10. clear-num 24.6%

          \[\leadsto \sqrt{\frac{t}{\frac{a}{z}} \cdot \left(t \cdot \color{blue}{\frac{1}{\frac{a}{z}}}\right)} \]

      11. div-inv 24.6%

          \[\leadsto \sqrt{\frac{t}{\frac{a}{z}} \cdot \color{blue}{\frac{t}{\frac{a}{z}}}} \]

      12. sqrt-unprod 8.8%

          \[\leadsto \color{blue}{\sqrt{\frac{t}{\frac{a}{z}}} \cdot \sqrt{\frac{t}{\frac{a}{z}}}} \]

      13. add-sqr-sqrt 9.2%

          \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

      14. frac-2neg 9.2%

          \[\leadsto \color{blue}{\frac{-t}{-\frac{a}{z}}} \]

      15. distribute-neg-frac 9.2%

          \[\leadsto \frac{-t}{\color{blue}{\frac{-a}{z}}} \]

      16. add-sqr-sqrt 4.3%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}} \]

      17. sqrt-unprod 20.9%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{\sqrt{z \cdot z}}}} \]

      18. sqr-neg 20.9%

          \[\leadsto \frac{-t}{\frac{-a}{\sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}} \]

      19. sqrt-unprod 24.3%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}} \]

      20. add-sqr-sqrt 62.3%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{-z}}} \]

      21. frac-2neg 62.3%

          \[\leadsto \frac{-t}{\color{blue}{\frac{a}{z}}} \]

   6. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{\frac{-t}{\frac{a}{z}}} \]

   if 2.0000000000000001e47 < (*.f64 x y)

   1. Initial program 91.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 75.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ 79.4%

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
   5. Step-by-step derivation

      1. clear-num 79.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{x}}} \cdot y \]

      2. associate-*l/ 79.4%

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{a}{x}}} \]

      3. *-un-lft-identity 79.4%

         \[\leadsto \frac{\color{blue}{y}}{\frac{a}{x}} \]

   6. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-285}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 8: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-38}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+141}:\\ \;\;\;\;-\frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* z t) -1e-38)
   (/ (- t) (/ a z))
   (if (<= (* z t) 5e-22)
     (/ (* x y) a)
     (if (<= (* z t) 4e+141) (- (/ (* z t) a)) (* z (/ (- t) a))))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -1e-38) {
		tmp = -t / (a / z);
	} else if ((z * t) <= 5e-22) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 4e+141) {
		tmp = -((z * t) / a);
	} else {
		tmp = z * (-t / a);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z * t) <= (-1d-38)) then
        tmp = -t / (a / z)
    else if ((z * t) <= 5d-22) then
        tmp = (x * y) / a
    else if ((z * t) <= 4d+141) then
        tmp = -((z * t) / a)
    else
        tmp = z * (-t / a)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= -1e-38) {
		tmp = -t / (a / z);
	} else if ((z * t) <= 5e-22) {
		tmp = (x * y) / a;
	} else if ((z * t) <= 4e+141) {
		tmp = -((z * t) / a);
	} else {
		tmp = z * (-t / a);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= -1e-38:
		tmp = -t / (a / z)
	elif (z * t) <= 5e-22:
		tmp = (x * y) / a
	elif (z * t) <= 4e+141:
		tmp = -((z * t) / a)
	else:
		tmp = z * (-t / a)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= -1e-38)
		tmp = Float64(Float64(-t) / Float64(a / z));
	elseif (Float64(z * t) <= 5e-22)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(z * t) <= 4e+141)
		tmp = Float64(-Float64(Float64(z * t) / a));
	else
		tmp = Float64(z * Float64(Float64(-t) / a));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= -1e-38)
		tmp = -t / (a / z);
	elseif ((z * t) <= 5e-22)
		tmp = (x * y) / a;
	elseif ((z * t) <= 4e+141)
		tmp = -((z * t) / a);
	else
		tmp = z * (-t / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e-38], N[((-t) / N[(a / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-22], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+141], (-N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -9.9999999999999996e-39

   1. Initial program 92.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 76.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 76.2%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 76.2%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 73.0%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 73.0%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 73.0%

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

      6. distribute-frac-neg 73.0%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 73.0%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 41.2%

         \[\leadsto \color{blue}{\sqrt{t \cdot \frac{-z}{a}} \cdot \sqrt{t \cdot \frac{-z}{a}}} \]

      2. sqrt-unprod 36.3%

         \[\leadsto \color{blue}{\sqrt{\left(t \cdot \frac{-z}{a}\right) \cdot \left(t \cdot \frac{-z}{a}\right)}} \]

      3. swap-sqr 24.0%

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot t\right) \cdot \left(\frac{-z}{a} \cdot \frac{-z}{a}\right)}} \]

      4. distribute-frac-neg 24.0%

         \[\leadsto \sqrt{\left(t \cdot t\right) \cdot \left(\color{blue}{\left(-\frac{z}{a}\right)} \cdot \frac{-z}{a}\right)} \]

      5. distribute-frac-neg 24.0%

         \[\leadsto \sqrt{\left(t \cdot t\right) \cdot \left(\left(-\frac{z}{a}\right) \cdot \color{blue}{\left(-\frac{z}{a}\right)}\right)} \]

      6. sqr-neg 24.0%

         \[\leadsto \sqrt{\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{z}{a} \cdot \frac{z}{a}\right)}} \]

      7. swap-sqr 36.3%

         \[\leadsto \sqrt{\color{blue}{\left(t \cdot \frac{z}{a}\right) \cdot \left(t \cdot \frac{z}{a}\right)}} \]

      8. clear-num 36.2%

         \[\leadsto \sqrt{\left(t \cdot \color{blue}{\frac{1}{\frac{a}{z}}}\right) \cdot \left(t \cdot \frac{z}{a}\right)} \]

      9. div-inv 36.2%

         \[\leadsto \sqrt{\color{blue}{\frac{t}{\frac{a}{z}}} \cdot \left(t \cdot \frac{z}{a}\right)} \]

      10. clear-num 36.2%

          \[\leadsto \sqrt{\frac{t}{\frac{a}{z}} \cdot \left(t \cdot \color{blue}{\frac{1}{\frac{a}{z}}}\right)} \]

      11. div-inv 36.2%

          \[\leadsto \sqrt{\frac{t}{\frac{a}{z}} \cdot \color{blue}{\frac{t}{\frac{a}{z}}}} \]

      12. sqrt-unprod 0.9%

          \[\leadsto \color{blue}{\sqrt{\frac{t}{\frac{a}{z}}} \cdot \sqrt{\frac{t}{\frac{a}{z}}}} \]

      13. add-sqr-sqrt 1.3%

          \[\leadsto \color{blue}{\frac{t}{\frac{a}{z}}} \]

      14. frac-2neg 1.3%

          \[\leadsto \color{blue}{\frac{-t}{-\frac{a}{z}}} \]

      15. distribute-neg-frac 1.3%

          \[\leadsto \frac{-t}{\color{blue}{\frac{-a}{z}}} \]

      16. add-sqr-sqrt 0.9%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}} \]

      17. sqrt-unprod 24.5%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{\sqrt{z \cdot z}}}} \]

      18. sqr-neg 24.5%

          \[\leadsto \frac{-t}{\frac{-a}{\sqrt{\color{blue}{\left(-z\right) \cdot \left(-z\right)}}}} \]

      19. sqrt-unprod 35.6%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}} \]

      20. add-sqr-sqrt 73.1%

          \[\leadsto \frac{-t}{\frac{-a}{\color{blue}{-z}}} \]

      21. frac-2neg 73.1%

          \[\leadsto \frac{-t}{\color{blue}{\frac{a}{z}}} \]

   6. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{\frac{-t}{\frac{a}{z}}} \]

   if -9.9999999999999996e-39 < (*.f64 z t) < 4.99999999999999954e-22

   1. Initial program 97.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 83.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]

   if 4.99999999999999954e-22 < (*.f64 z t) < 4.00000000000000007e141

   1. Initial program 99.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 65.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ 65.5%

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

      2. mul-1-neg 65.5%

         \[\leadsto \frac{\color{blue}{-t \cdot z}}{a} \]

      3. distribute-rgt-neg-in 65.5%

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

   4. Simplified 65.5%

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

   if 4.00000000000000007e141 < (*.f64 z t)

   1. Initial program 83.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 73.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 73.0%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 73.0%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 86.3%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 86.3%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 86.3%

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

      6. distribute-frac-neg 86.3%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 86.3%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg 86.3%

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

      2. distribute-rgt-neg-out 86.3%

         \[\leadsto \color{blue}{-t \cdot \frac{z}{a}} \]

      3. clear-num 86.4%

         \[\leadsto -t \cdot \color{blue}{\frac{1}{\frac{a}{z}}} \]

      4. div-inv 86.4%

         \[\leadsto -\color{blue}{\frac{t}{\frac{a}{z}}} \]

      5. associate-/r/ 83.7%

         \[\leadsto -\color{blue}{\frac{t}{a} \cdot z} \]

   6. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{-\frac{t}{a} \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-38}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+141}:\\ \;\;\;\;-\frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 9: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+47}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -2e-40)
   (/ (* x y) a)
   (if (<= (* x y) 2e+47) (* z (/ (- t) a)) (/ y (/ a x)))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 2e+47) {
		tmp = z * (-t / a);
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-2d-40)) then
        tmp = (x * y) / a
    else if ((x * y) <= 2d+47) then
        tmp = z * (-t / a)
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -2e-40) {
		tmp = (x * y) / a;
	} else if ((x * y) <= 2e+47) {
		tmp = z * (-t / a);
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -2e-40:
		tmp = (x * y) / a
	elif (x * y) <= 2e+47:
		tmp = z * (-t / a)
	else:
		tmp = y / (a / x)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e-40)
		tmp = Float64(Float64(x * y) / a);
	elseif (Float64(x * y) <= 2e+47)
		tmp = Float64(z * Float64(Float64(-t) / a));
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -2e-40)
		tmp = (x * y) / a;
	elseif ((x * y) <= 2e+47)
		tmp = z * (-t / a);
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e-40], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+47], N[(z * N[((-t) / a), $MachinePrecision]), $MachinePrecision], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.9999999999999999e-40

   1. Initial program 93.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 77.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]

   if -1.9999999999999999e-40 < (*.f64 x y) < 2.0000000000000001e47

   1. Initial program 96.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around 0 77.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg 77.7%

         \[\leadsto \color{blue}{-\frac{t \cdot z}{a}} \]

      2. *-commutative 77.7%

         \[\leadsto -\frac{\color{blue}{z \cdot t}}{a} \]

      3. associate-*l/ 72.9%

         \[\leadsto -\color{blue}{\frac{z}{a} \cdot t} \]

      4. *-commutative 72.9%

         \[\leadsto -\color{blue}{t \cdot \frac{z}{a}} \]

      5. distribute-rgt-neg-in 72.9%

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

      6. distribute-frac-neg 72.9%

         \[\leadsto t \cdot \color{blue}{\frac{-z}{a}} \]

   4. Simplified 72.9%

      \[\leadsto \color{blue}{t \cdot \frac{-z}{a}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg 72.9%

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

      2. distribute-rgt-neg-out 72.9%

         \[\leadsto \color{blue}{-t \cdot \frac{z}{a}} \]

      3. clear-num 72.2%

         \[\leadsto -t \cdot \color{blue}{\frac{1}{\frac{a}{z}}} \]

      4. div-inv 72.7%

         \[\leadsto -\color{blue}{\frac{t}{\frac{a}{z}}} \]

      5. associate-/r/ 70.3%

         \[\leadsto -\color{blue}{\frac{t}{a} \cdot z} \]

   6. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{-\frac{t}{a} \cdot z} \]

   if 2.0000000000000001e47 < (*.f64 x y)

   1. Initial program 91.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 75.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ 79.4%

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
   5. Step-by-step derivation

      1. clear-num 79.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{x}}} \cdot y \]

      2. associate-*l/ 79.4%

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{a}{x}}} \]

      3. *-un-lft-identity 79.4%

         \[\leadsto \frac{\color{blue}{y}}{\frac{a}{x}} \]

   6. Applied egg-rr 79.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+47}:\\ \;\;\;\;z \cdot \frac{-t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 10: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) 5e+276) (/ (- (* x y) (* z t)) a) (/ y (/ a x))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 5e+276) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= 5d+276) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 5e+276) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= 5e+276:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = y / (a / x)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= 5e+276)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= 5e+276)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], 5e+276], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 5.00000000000000001e276

   1. Initial program 96.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   if 5.00000000000000001e276 < (*.f64 x y)

   1. Initial program 74.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 74.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
   5. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{x}}} \cdot y \]

      2. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{a}{x}}} \]

      3. *-un-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{y}}{\frac{a}{x}} \]

   6. Applied egg-rr 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 11: 51.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) 5e+276) (/ (* x y) a) (/ y (/ a x))))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 5e+276) {
		tmp = (x * y) / a;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= 5d+276) then
        tmp = (x * y) / a
    else
        tmp = y / (a / x)
    end if
    code = tmp
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 5e+276) {
		tmp = (x * y) / a;
	} else {
		tmp = y / (a / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= 5e+276:
		tmp = (x * y) / a
	else:
		tmp = y / (a / x)
	return tmp

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= 5e+276)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(y / Float64(a / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= 5e+276)
		tmp = (x * y) / a;
	else
		tmp = y / (a / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], 5e+276], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 5.00000000000000001e276

   1. Initial program 96.1%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 52.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]

   if 5.00000000000000001e276 < (*.f64 x y)

   1. Initial program 74.6%

      \[\frac{x \cdot y - z \cdot t}{a} \]

   2. Taylor expanded in x around inf 74.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
   5. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{x}}} \cdot y \]

      2. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{a}{x}}} \]

      3. *-un-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{y}}{\frac{a}{x}} \]

   6. Applied egg-rr 99.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 12: 50.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ x \cdot \frac{y}{a} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* x (/ y a)))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	return x * (y / a);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x * (y / a)
end function

Java

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

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	return x * (y / a)

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	return Float64(x * Float64(y / a))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a)
	tmp = x * (y / a);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\frac{x \cdot y - z \cdot t}{a} \]

2. Taylor expanded in x around inf 53.7%

   \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
3. Step-by-step derivation

   1. associate-*r/ 50.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]

4. Simplified 50.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{a}} \]

5. Final simplification 50.7%

   \[\leadsto x \cdot \frac{y}{a} \]

Alternative 13: 50.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ y \cdot \frac{x}{a} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* y (/ x a)))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	return y * (x / a);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = y * (x / a)
end function

Java

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

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	return y * (x / a)

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	return Float64(y * Float64(x / a))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a)
	tmp = y * (x / a);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\frac{x \cdot y - z \cdot t}{a} \]

2. Taylor expanded in x around inf 53.7%

   \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
3. Step-by-step derivation

   1. associate-*l/ 52.3%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

4. Simplified 52.3%

   \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

5. Final simplification 52.3%

   \[\leadsto y \cdot \frac{x}{a} \]

Alternative 14: 50.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \\ \frac{y}{\frac{a}{x}} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (/ y (/ a x)))

C

assert(x < y);
assert(z < t);
double code(double x, double y, double z, double t, double a) {
	return y / (a / x);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = y / (a / x)
end function

Java

assert x < y;
assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	return y / (a / x);
}

Python

[x, y] = sort([x, y])
[z, t] = sort([z, t])
def code(x, y, z, t, a):
	return y / (a / x)

Julia

x, y = sort([x, y])
z, t = sort([z, t])
function code(x, y, z, t, a)
	return Float64(y / Float64(a / x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t, a)
	tmp = y / (a / x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\frac{x \cdot y - z \cdot t}{a} \]

2. Taylor expanded in x around inf 53.7%

   \[\leadsto \color{blue}{\frac{x \cdot y}{a}} \]
3. Step-by-step derivation

   1. associate-*l/ 52.3%

      \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]

4. Simplified 52.3%

   \[\leadsto \color{blue}{\frac{x}{a} \cdot y} \]
5. Step-by-step derivation

   1. clear-num 52.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{x}}} \cdot y \]

   2. associate-*l/ 52.4%

      \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{a}{x}}} \]

   3. *-un-lft-identity 52.4%

      \[\leadsto \frac{\color{blue}{y}}{\frac{a}{x}} \]

6. Applied egg-rr 52.4%

   \[\leadsto \color{blue}{\frac{y}{\frac{a}{x}}} \]

7. Final simplification 52.4%

   \[\leadsto \frac{y}{\frac{a}{x}} \]

Developer target: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (/ y a) x) (* (/ t a) z))))
   (if (< z -2.468684968699548e+170)
     t_1
     (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y / a) * x) - ((t / a) * z)
    if (z < (-2.468684968699548d+170)) then
        tmp = t_1
    else if (z < 6.309831121978371d-71) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y / a) * x) - ((t / a) * z);
	double tmp;
	if (z < -2.468684968699548e+170) {
		tmp = t_1;
	} else if (z < 6.309831121978371e-71) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y / a) * x) - ((t / a) * z)
	tmp = 0
	if z < -2.468684968699548e+170:
		tmp = t_1
	elif z < 6.309831121978371e-71:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y / a) * x) - Float64(Float64(t / a) * z))
	tmp = 0.0
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y / a) * x) - ((t / a) * z);
	tmp = 0.0;
	if (z < -2.468684968699548e+170)
		tmp = t_1;
	elseif (z < 6.309831121978371e-71)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.468684968699548e+170], t$95$1, If[Less[z, 6.309831121978371e-71], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))