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

Percentage Accurate: 91.5% → 95.7%

Time: 7.2s

Alternatives: 8

Speedup: 0.7×

• 28.4% 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: 91.5% 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: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := x \cdot y - z \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(\frac{x}{t} - \frac{z}{y}\right)\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, y \cdot \frac{x}{a}\right)\\ \end{array} \end{array} \]

FPCore

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))
     (* (* y (- (/ x t) (/ z y))) (/ t a))
     (if (<= t_1 4e+288) (/ t_1 a) (fma -1.0 (/ t (/ a z)) (* y (/ x a)))))))

C

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 = (y * ((x / t) - (z / y))) * (t / a);
	} else if (t_1 <= 4e+288) {
		tmp = t_1 / a;
	} else {
		tmp = fma(-1.0, (t / (a / z)), (y * (x / a)));
	}
	return tmp;
}

Julia

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(y * Float64(Float64(x / t) - Float64(z / y))) * Float64(t / a));
	elseif (t_1 <= 4e+288)
		tmp = Float64(t_1 / a);
	else
		tmp = fma(-1.0, Float64(t / Float64(a / z)), Float64(y * Float64(x / a)));
	end
	return tmp
end

Wolfram

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[(y * N[(N[(x / t), $MachinePrecision] - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+288], N[(t$95$1 / a), $MachinePrecision], N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 71.8%

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

      1. div-sub 56.4%

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

      2. associate-/l* 60.0%

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

      3. associate-/l* 84.4%

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

   3. Applied egg-rr 84.4%

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

      1. sub-neg 84.4%

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

      2. frac-2neg 84.4%

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

      3. associate-/r/ 84.4%

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

      4. distribute-rgt-neg-out 84.4%

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

      5. add-sqr-sqrt 30.6%

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

      6. sqrt-unprod 58.5%

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

      7. sqr-neg 58.5%

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

      8. sqrt-unprod 38.5%

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

      9. add-sqr-sqrt 42.4%

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

      10. associate-/r/ 46.2%

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

      11. frac-2neg 46.2%

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

      12. frac-add 38.5%

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

   5. Applied egg-rr 77.8%

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

      1. distribute-rgt-neg-out 77.8%

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

      2. sub-neg 77.8%

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

      3. sub-neg 77.8%

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

      4. distribute-lft-neg-out 77.8%

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

      5. distribute-neg-out 77.8%

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

      6. associate-*r/ 81.6%

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

      7. distribute-frac-neg 81.6%

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

      8. cancel-sign-sub-inv 81.6%

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

      9. *-commutative 81.6%

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

   7. Simplified 81.4%

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

   8. Taylor expanded in a around -inf 71.8%

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

      1. *-commutative 71.8%

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

      2. *-lft-identity 71.8%

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

      3. times-frac 89.0%

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

      4. associate-*l/ 89.0%

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

      5. /-rgt-identity 89.0%

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

   10. Simplified 89.0%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4e288

   1. Initial program 99.7%

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

   if 4e288 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 79.5%

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

      1. associate-/l* 90.9%

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

      2. fma-def 90.9%

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

      3. associate-/l* 96.7%

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

      4. associate-/r/ 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\left(y \cdot \left(\frac{x}{t} - \frac{z}{y}\right)\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 2: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{t}{a} \cdot \left(-z\right)\\ t_2 := \frac{x \cdot y}{a}\\ \mathbf{if}\;x \cdot y \leq -400000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-84}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

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) a)))
   (if (<= (* x y) -400000000.0)
     t_2
     (if (<= (* x y) -1e-33)
       t_1
       (if (<= (* x y) -4e-84)
         t_2
         (if (<= (* x y) 2e-60) t_1 (* x (/ y a))))))))

C

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) / a;
	double tmp;
	if ((x * y) <= -400000000.0) {
		tmp = t_2;
	} else if ((x * y) <= -1e-33) {
		tmp = t_1;
	} else if ((x * y) <= -4e-84) {
		tmp = t_2;
	} else if ((x * y) <= 2e-60) {
		tmp = t_1;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

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) :: t_2
    real(8) :: tmp
    t_1 = (t / a) * -z
    t_2 = (x * y) / a
    if ((x * y) <= (-400000000.0d0)) then
        tmp = t_2
    else if ((x * y) <= (-1d-33)) then
        tmp = t_1
    else if ((x * y) <= (-4d-84)) then
        tmp = t_2
    else if ((x * y) <= 2d-60) then
        tmp = t_1
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

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) / a;
	double tmp;
	if ((x * y) <= -400000000.0) {
		tmp = t_2;
	} else if ((x * y) <= -1e-33) {
		tmp = t_1;
	} else if ((x * y) <= -4e-84) {
		tmp = t_2;
	} else if ((x * y) <= 2e-60) {
		tmp = t_1;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (t / a) * -z
	t_2 = (x * y) / a
	tmp = 0
	if (x * y) <= -400000000.0:
		tmp = t_2
	elif (x * y) <= -1e-33:
		tmp = t_1
	elif (x * y) <= -4e-84:
		tmp = t_2
	elif (x * y) <= 2e-60:
		tmp = t_1
	else:
		tmp = x * (y / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(t / a) * Float64(-z))
	t_2 = Float64(Float64(x * y) / a)
	tmp = 0.0
	if (Float64(x * y) <= -400000000.0)
		tmp = t_2;
	elseif (Float64(x * y) <= -1e-33)
		tmp = t_1;
	elseif (Float64(x * y) <= -4e-84)
		tmp = t_2;
	elseif (Float64(x * y) <= 2e-60)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (t / a) * -z;
	t_2 = (x * y) / a;
	tmp = 0.0;
	if ((x * y) <= -400000000.0)
		tmp = t_2;
	elseif ((x * y) <= -1e-33)
		tmp = t_1;
	elseif ((x * y) <= -4e-84)
		tmp = t_2;
	elseif ((x * y) <= 2e-60)
		tmp = t_1;
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

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[(t / a), $MachinePrecision] * (-z)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -400000000.0], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], -1e-33], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -4e-84], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], 2e-60], t$95$1, N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4e8 or -1.0000000000000001e-33 < (*.f64 x y) < -4.0000000000000001e-84

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 73.9%

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

   if -4e8 < (*.f64 x y) < -1.0000000000000001e-33 or -4.0000000000000001e-84 < (*.f64 x y) < 1.9999999999999999e-60

   1. Initial program 97.4%

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

      1. div-inv 97.2%

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

      2. fma-neg 97.2%

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

      3. *-commutative 97.2%

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

      4. distribute-rgt-neg-in 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. associate-*l/ 77.6%

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

      3. distribute-rgt-neg-in 77.6%

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

   6. Simplified 77.6%

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

   if 1.9999999999999999e-60 < (*.f64 x y)

   1. Initial program 90.3%

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

   2. Taylor expanded in x around inf 66.9%

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

      1. associate-*r/ 71.0%

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

   4. Simplified 71.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -400000000:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 3: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ \mathbf{if}\;x \cdot y \leq -400000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

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) a)))
   (if (<= (* x y) -400000000.0)
     t_1
     (if (<= (* x y) -1e-33)
       (/ (- z) (/ a t))
       (if (<= (* x y) -4e-84)
         t_1
         (if (<= (* x y) 2e-60) (* (/ t a) (- z)) (* x (/ y a))))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if ((x * y) <= -400000000.0) {
		tmp = t_1;
	} else if ((x * y) <= -1e-33) {
		tmp = -z / (a / t);
	} else if ((x * y) <= -4e-84) {
		tmp = t_1;
	} else if ((x * y) <= 2e-60) {
		tmp = (t / a) * -z;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

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 = (x * y) / a
    if ((x * y) <= (-400000000.0d0)) then
        tmp = t_1
    else if ((x * y) <= (-1d-33)) then
        tmp = -z / (a / t)
    else if ((x * y) <= (-4d-84)) then
        tmp = t_1
    else if ((x * y) <= 2d-60) then
        tmp = (t / a) * -z
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if ((x * y) <= -400000000.0) {
		tmp = t_1;
	} else if ((x * y) <= -1e-33) {
		tmp = -z / (a / t);
	} else if ((x * y) <= -4e-84) {
		tmp = t_1;
	} else if ((x * y) <= 2e-60) {
		tmp = (t / a) * -z;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) / a
	tmp = 0
	if (x * y) <= -400000000.0:
		tmp = t_1
	elif (x * y) <= -1e-33:
		tmp = -z / (a / t)
	elif (x * y) <= -4e-84:
		tmp = t_1
	elif (x * y) <= 2e-60:
		tmp = (t / a) * -z
	else:
		tmp = x * (y / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) / a)
	tmp = 0.0
	if (Float64(x * y) <= -400000000.0)
		tmp = t_1;
	elseif (Float64(x * y) <= -1e-33)
		tmp = Float64(Float64(-z) / Float64(a / t));
	elseif (Float64(x * y) <= -4e-84)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-60)
		tmp = Float64(Float64(t / a) * Float64(-z));
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) / a;
	tmp = 0.0;
	if ((x * y) <= -400000000.0)
		tmp = t_1;
	elseif ((x * y) <= -1e-33)
		tmp = -z / (a / t);
	elseif ((x * y) <= -4e-84)
		tmp = t_1;
	elseif ((x * y) <= 2e-60)
		tmp = (t / a) * -z;
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

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] / a), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -400000000.0], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -1e-33], N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4e-84], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-60], N[(N[(t / a), $MachinePrecision] * (-z)), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4e8 or -1.0000000000000001e-33 < (*.f64 x y) < -4.0000000000000001e-84

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 73.9%

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

   if -4e8 < (*.f64 x y) < -1.0000000000000001e-33

   1. Initial program 86.9%

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

   2. Taylor expanded in x around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. associate-*r* 73.9%

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

      3. neg-mul-1 73.9%

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

   4. Simplified 73.9%

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

      1. associate-/l* 73.2%

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

      2. add-sqr-sqrt 29.7%

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

      3. sqrt-unprod 30.0%

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

      4. sqr-neg 30.0%

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

      5. sqrt-unprod 0.6%

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

      6. add-sqr-sqrt 2.0%

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

      7. associate-/r/ 1.8%

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

   6. Applied egg-rr 1.8%

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

      1. associate-*l/ 1.8%

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

      2. associate-*r/ 1.8%

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

      3. add-sqr-sqrt 0.4%

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

      4. sqrt-unprod 30.1%

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

      5. sqr-neg 30.1%

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

      6. sqrt-unprod 29.7%

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

      7. add-sqr-sqrt 82.5%

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

      8. distribute-lft-neg-in 82.5%

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

      9. *-commutative 82.5%

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

      10. associate-/r/ 86.8%

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

      11. distribute-neg-frac 86.8%

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

   8. Applied egg-rr 86.8%

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

   if -4.0000000000000001e-84 < (*.f64 x y) < 1.9999999999999999e-60

   1. Initial program 98.0%

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

      1. div-inv 97.9%

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

      2. fma-neg 97.9%

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

      3. *-commutative 97.9%

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

      4. distribute-rgt-neg-in 97.9%

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

   3. Applied egg-rr 97.9%

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

   4. Taylor expanded in x around 0 87.7%

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

      1. mul-1-neg 87.7%

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

      2. associate-*l/ 77.0%

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

      3. distribute-rgt-neg-in 77.0%

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

   6. Simplified 77.0%

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

   if 1.9999999999999999e-60 < (*.f64 x y)

   1. Initial program 90.3%

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

   2. Taylor expanded in x around inf 66.9%

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

      1. associate-*r/ 71.0%

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

   4. Simplified 71.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -400000000:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\frac{t}{a} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 4: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{a}\\ \mathbf{if}\;x \cdot y \leq -400000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+26}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

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) a)))
   (if (<= (* x y) -400000000.0)
     t_1
     (if (<= (* x y) -1e-33)
       (/ (- z) (/ a t))
       (if (<= (* x y) -4e-84)
         t_1
         (if (<= (* x y) 1e+26) (/ (* t (- z)) a) (* x (/ y a))))))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if ((x * y) <= -400000000.0) {
		tmp = t_1;
	} else if ((x * y) <= -1e-33) {
		tmp = -z / (a / t);
	} else if ((x * y) <= -4e-84) {
		tmp = t_1;
	} else if ((x * y) <= 1e+26) {
		tmp = (t * -z) / a;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Fortran

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 = (x * y) / a
    if ((x * y) <= (-400000000.0d0)) then
        tmp = t_1
    else if ((x * y) <= (-1d-33)) then
        tmp = -z / (a / t)
    else if ((x * y) <= (-4d-84)) then
        tmp = t_1
    else if ((x * y) <= 1d+26) then
        tmp = (t * -z) / a
    else
        tmp = x * (y / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / a;
	double tmp;
	if ((x * y) <= -400000000.0) {
		tmp = t_1;
	} else if ((x * y) <= -1e-33) {
		tmp = -z / (a / t);
	} else if ((x * y) <= -4e-84) {
		tmp = t_1;
	} else if ((x * y) <= 1e+26) {
		tmp = (t * -z) / a;
	} else {
		tmp = x * (y / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	t_1 = (x * y) / a
	tmp = 0
	if (x * y) <= -400000000.0:
		tmp = t_1
	elif (x * y) <= -1e-33:
		tmp = -z / (a / t)
	elif (x * y) <= -4e-84:
		tmp = t_1
	elif (x * y) <= 1e+26:
		tmp = (t * -z) / a
	else:
		tmp = x * (y / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) / a)
	tmp = 0.0
	if (Float64(x * y) <= -400000000.0)
		tmp = t_1;
	elseif (Float64(x * y) <= -1e-33)
		tmp = Float64(Float64(-z) / Float64(a / t));
	elseif (Float64(x * y) <= -4e-84)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+26)
		tmp = Float64(Float64(t * Float64(-z)) / a);
	else
		tmp = Float64(x * Float64(y / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * y) / a;
	tmp = 0.0;
	if ((x * y) <= -400000000.0)
		tmp = t_1;
	elseif ((x * y) <= -1e-33)
		tmp = -z / (a / t);
	elseif ((x * y) <= -4e-84)
		tmp = t_1;
	elseif ((x * y) <= 1e+26)
		tmp = (t * -z) / a;
	else
		tmp = x * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

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] / a), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -400000000.0], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -1e-33], N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4e-84], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+26], N[(N[(t * (-z)), $MachinePrecision] / a), $MachinePrecision], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4e8 or -1.0000000000000001e-33 < (*.f64 x y) < -4.0000000000000001e-84

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 73.9%

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

   if -4e8 < (*.f64 x y) < -1.0000000000000001e-33

   1. Initial program 86.9%

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

   2. Taylor expanded in x around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. associate-*r* 73.9%

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

      3. neg-mul-1 73.9%

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

   4. Simplified 73.9%

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

      1. associate-/l* 73.2%

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

      2. add-sqr-sqrt 29.7%

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

      3. sqrt-unprod 30.0%

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

      4. sqr-neg 30.0%

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

      5. sqrt-unprod 0.6%

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

      6. add-sqr-sqrt 2.0%

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

      7. associate-/r/ 1.8%

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

   6. Applied egg-rr 1.8%

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

      1. associate-*l/ 1.8%

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

      2. associate-*r/ 1.8%

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

      3. add-sqr-sqrt 0.4%

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

      4. sqrt-unprod 30.1%

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

      5. sqr-neg 30.1%

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

      6. sqrt-unprod 29.7%

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

      7. add-sqr-sqrt 82.5%

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

      8. distribute-lft-neg-in 82.5%

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

      9. *-commutative 82.5%

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

      10. associate-/r/ 86.8%

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

      11. distribute-neg-frac 86.8%

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

   8. Applied egg-rr 86.8%

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

   if -4.0000000000000001e-84 < (*.f64 x y) < 1.00000000000000005e26

   1. Initial program 98.2%

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

   2. Taylor expanded in x around 0 84.2%

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

      1. associate-*r/ 84.2%

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

      2. associate-*r* 84.2%

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

      3. neg-mul-1 84.2%

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

   4. Simplified 84.2%

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

   if 1.00000000000000005e26 < (*.f64 x y)

   1. Initial program 88.1%

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

   2. Taylor expanded in x around inf 71.3%

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

      1. associate-*r/ 78.0%

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

   4. Simplified 78.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -400000000:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-84}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y \leq 10^{+26}:\\ \;\;\;\;\frac{t \cdot \left(-z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \end{array} \]

Alternative 5: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+307}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

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+307) (/ (- (* x y) (* z t)) a) (/ (- z) (/ a t))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= 1e+307) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -z / (a / t);
	}
	return tmp;
}

Fortran

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+307) then
        tmp = ((x * y) - (z * t)) / a
    else
        tmp = -z / (a / t)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z * t) <= 1e+307) {
		tmp = ((x * y) - (z * t)) / a;
	} else {
		tmp = -z / (a / t);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (z * t) <= 1e+307:
		tmp = ((x * y) - (z * t)) / a
	else:
		tmp = -z / (a / t)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z * t) <= 1e+307)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * t)) / a);
	else
		tmp = Float64(Float64(-z) / Float64(a / t));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z * t) <= 1e+307)
		tmp = ((x * y) - (z * t)) / a;
	else
		tmp = -z / (a / t);
	end
	tmp_2 = tmp;
end

Wolfram

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+307], N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.99999999999999986e306

   1. Initial program 96.6%

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

   if 9.99999999999999986e306 < (*.f64 z t)

   1. Initial program 54.3%

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

   2. Taylor expanded in x around 0 54.3%

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

      1. associate-*r/ 54.3%

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

      2. associate-*r* 54.3%

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

      3. neg-mul-1 54.3%

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

   4. Simplified 54.3%

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

      1. associate-/l* 99.9%

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

      2. add-sqr-sqrt 49.7%

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

      3. sqrt-unprod 37.1%

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

      4. sqr-neg 37.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 0.1%

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

      7. associate-/r/ 0.1%

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

   6. Applied egg-rr 0.1%

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

      1. associate-*l/ 0.0%

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

      2. associate-*r/ 0.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 37.1%

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

      5. sqr-neg 37.1%

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

      6. sqrt-unprod 49.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. distribute-lft-neg-in 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-/r/ 99.7%

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

      11. distribute-neg-frac 99.7%

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

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{-z}{\frac{a}{t}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+307}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]

Alternative 6: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

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+89) (/ (* x y) a) (* y (/ x a))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 2e+89) {
		tmp = (x * y) / a;
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Fortran

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+89) then
        tmp = (x * y) / a
    else
        tmp = y * (x / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= 2e+89) {
		tmp = (x * y) / a;
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= 2e+89:
		tmp = (x * y) / a
	else:
		tmp = y * (x / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= 2e+89)
		tmp = Float64(Float64(x * y) / a);
	else
		tmp = Float64(y * Float64(x / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= 2e+89)
		tmp = (x * y) / a;
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

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+89], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.99999999999999999e89

   1. Initial program 95.8%

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

   2. Taylor expanded in x around inf 41.0%

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

   if 1.99999999999999999e89 < (*.f64 x y)

   1. Initial program 87.4%

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

   2. Taylor expanded in x around inf 72.5%

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

      1. associate-*l/ 84.9%

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

   4. Simplified 84.9%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 7: 51.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{-189}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a 6.2e-189) (* x (/ y a)) (* y (/ x a))))

C

assert(z < t);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 6.2e-189) {
		tmp = x * (y / a);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Fortran

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 (a <= 6.2d-189) then
        tmp = x * (y / a)
    else
        tmp = y * (x / a)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 6.2e-189) {
		tmp = x * (y / a);
	} else {
		tmp = y * (x / a);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t, a):
	tmp = 0
	if a <= 6.2e-189:
		tmp = x * (y / a)
	else:
		tmp = y * (x / a)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 6.2e-189)
		tmp = Float64(x * Float64(y / a));
	else
		tmp = Float64(y * Float64(x / a));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 6.2e-189)
		tmp = x * (y / a);
	else
		tmp = y * (x / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[a, 6.2e-189], N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.2000000000000001e-189

   1. Initial program 96.0%

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

   2. Taylor expanded in x around inf 52.1%

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

      1. associate-*r/ 50.7%

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

   4. Simplified 50.7%

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

   if 6.2000000000000001e-189 < a

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 38.5%

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

      1. associate-*l/ 39.9%

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

   4. Simplified 39.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{-189}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a}\\ \end{array} \]

Alternative 8: 51.2% accurate, 1.8× speedup

Math

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

FPCore

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 94.3%

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

2. Taylor expanded in x around inf 46.6%

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

   1. associate-*r/ 45.3%

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

4. Simplified 45.3%

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

5. Final simplification 45.3%

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

Developer target: 91.9% 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 2023308 
(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))