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

Percentage Accurate: 91.6% → 97.5%

Time: 9.6s

Alternatives: 10

Speedup: 0.7×

• 28.3% 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: 91.6% 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: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-z}{a} \cdot t\\ t_2 := y \cdot x - t \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+280}:\\ \;\;\;\;\frac{t\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\frac{a}{x}}, -y, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z) a) t)) (t_2 (- (* y x) (* t z))))
   (if (<= t_2 (- INFINITY))
     (fma (/ y a) x t_1)
     (if (<= t_2 1e+280) (/ t_2 a) (fma (/ -1.0 (/ a x)) (- y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-z / a) * t;
	double t_2 = (y * x) - (t * z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((y / a), x, t_1);
	} else if (t_2 <= 1e+280) {
		tmp = t_2 / a;
	} else {
		tmp = fma((-1.0 / (a / x)), -y, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-z) / a) * t)
	t_2 = Float64(Float64(y * x) - Float64(t * z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(y / a), x, t_1);
	elseif (t_2 <= 1e+280)
		tmp = Float64(t_2 / a);
	else
		tmp = fma(Float64(-1.0 / Float64(a / x)), Float64(-y), t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(y / a), $MachinePrecision] * x + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e+280], N[(t$95$2 / a), $MachinePrecision], N[(N[(-1.0 / N[(a / x), $MachinePrecision]), $MachinePrecision] * (-y) + 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 58.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 1e280

   1. Initial program 99.0%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 1e280 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 80.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.4

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

   6. Applied rewrites 80.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lower-fma.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

      16. distribute-frac-neg N/A

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

      17. frac-2neg N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-/.f64 N/A

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

      21. un-div-inv N/A

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

      22. lift-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot z \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{-z}{a} \cdot t\right)\\ \mathbf{elif}\;y \cdot x - t \cdot z \leq 10^{+280}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\frac{a}{x}}, -y, \frac{-z}{a} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} \cdot x\\ t_2 := \frac{y \cdot x - t \cdot z}{a}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+261}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y a) x)) (t_2 (/ (- (* y x) (* t z)) a)))
   (if (<= t_2 -5e+261) t_1 (if (<= t_2 4e+301) (/ (* y x) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) * x;
	double t_2 = ((y * x) - (t * z)) / a;
	double tmp;
	if (t_2 <= -5e+261) {
		tmp = t_1;
	} else if (t_2 <= 4e+301) {
		tmp = (y * x) / 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) :: t_2
    real(8) :: tmp
    t_1 = (y / a) * x
    t_2 = ((y * x) - (t * z)) / a
    if (t_2 <= (-5d+261)) then
        tmp = t_1
    else if (t_2 <= 4d+301) then
        tmp = (y * x) / 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;
	double t_2 = ((y * x) - (t * z)) / a;
	double tmp;
	if (t_2 <= -5e+261) {
		tmp = t_1;
	} else if (t_2 <= 4e+301) {
		tmp = (y * x) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / a) * x
	t_2 = ((y * x) - (t * z)) / a
	tmp = 0
	if t_2 <= -5e+261:
		tmp = t_1
	elif t_2 <= 4e+301:
		tmp = (y * x) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / a) * x)
	t_2 = Float64(Float64(Float64(y * x) - Float64(t * z)) / a)
	tmp = 0.0
	if (t_2 <= -5e+261)
		tmp = t_1;
	elseif (t_2 <= 4e+301)
		tmp = Float64(Float64(y * x) / 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_2 = ((y * x) - (t * z)) / a;
	tmp = 0.0;
	if (t_2 <= -5e+261)
		tmp = t_1;
	elseif (t_2 <= 4e+301)
		tmp = (y * x) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+261], t$95$1, If[LessEqual[t$95$2, 4e+301], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < -5.0000000000000001e261 or 4.00000000000000021e301 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a)

   1. Initial program 79.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 51.5

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 57.8

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

   11. Applied rewrites 57.8%

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

   if -5.0000000000000001e261 < (/.f64 (-.f64 (*.f64 x y) (*.f64 z t)) a) < 4.00000000000000021e301

   1. Initial program 98.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.8

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

   5. Applied rewrites 52.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x - t \cdot z}{a} \leq -5 \cdot 10^{+261}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \mathbf{elif}\;\frac{y \cdot x - t \cdot z}{a} \leq 4 \cdot 10^{+301}:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, x, \frac{-z}{a} \cdot t\right)\\ t_2 := y \cdot x - t \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+249}:\\ \;\;\;\;\frac{t\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) x (* (/ (- z) a) t))) (t_2 (- (* y x) (* t z))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 1e+249) (/ t_2 a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), x, ((-z / a) * t));
	double t_2 = (y * x) - (t * z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+249) {
		tmp = t_2 / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), x, Float64(Float64(Float64(-z) / a) * t))
	t_2 = Float64(Float64(y * x) - Float64(t * z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+249)
		tmp = Float64(t_2 / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * x + N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+249], N[(t$95$2 / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -inf.0 or 9.9999999999999992e248 < (-.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999992e248

   1. Initial program 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot z \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{-z}{a} \cdot t\right)\\ \mathbf{elif}\;y \cdot x - t \cdot z \leq 10^{+249}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, x, \frac{-z}{a} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* y x) (* t z)) (- INFINITY))
   (* (/ (- (* (/ x z) y) t) a) z)
   (/ (fma y x (* (- z) t)) a)))

C

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

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(y * x) - Float64(t * z)) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(x / z) * y) - t) / a) * z);
	else
		tmp = Float64(fma(y, x, Float64(Float64(-z) * t)) / a);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 58.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 89.2

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

   8. Applied rewrites 89.2%

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

   10. Applied rewrites 82.5%

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

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

   1. Initial program 95.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 96.3

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

   4. Applied rewrites 96.3%

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

4. Final simplification 94.8%

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

Alternative 5: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* t z) -2e+19)
   (* (/ -1.0 a) (* t z))
   (if (<= (* t z) 400000.0) (/ (* y x) a) (/ (- z) (/ a t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = (-1.0 / a) * (t * z);
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / a;
	} else {
		tmp = -z / (a / t);
	}
	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) :: tmp
    if ((t * z) <= (-2d+19)) then
        tmp = ((-1.0d0) / a) * (t * z)
    else if ((t * z) <= 400000.0d0) then
        tmp = (y * x) / a
    else
        tmp = -z / (a / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = (-1.0 / a) * (t * z);
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / a;
	} else {
		tmp = -z / (a / t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t * z) <= -2e+19:
		tmp = (-1.0 / a) * (t * z)
	elif (t * z) <= 400000.0:
		tmp = (y * x) / a
	else:
		tmp = -z / (a / t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(t * z) <= -2e+19)
		tmp = Float64(Float64(-1.0 / a) * Float64(t * z));
	elseif (Float64(t * z) <= 400000.0)
		tmp = Float64(Float64(y * x) / a);
	else
		tmp = Float64(Float64(-z) / Float64(a / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t * z) <= -2e+19)
		tmp = (-1.0 / a) * (t * z);
	elseif ((t * z) <= 400000.0)
		tmp = (y * x) / a;
	else
		tmp = -z / (a / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(t * z), $MachinePrecision], -2e+19], N[(N[(-1.0 / a), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 400000.0], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], N[((-z) / N[(a / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e19

   1. Initial program 95.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. associate-/r* N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(t \cdot z\right)} \cdot \frac{-1}{a} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 73.0

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

   7. Applied rewrites 73.0%

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

   if -2e19 < (*.f64 z t) < 4e5

   1. Initial program 92.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 4e5 < (*.f64 z t)

   1. Initial program 87.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 78.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* t z) -2e+19)
   (* (/ -1.0 a) (* t z))
   (if (<= (* t z) 400000.0) (/ (* y x) a) (* (/ (- t) a) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = (-1.0 / a) * (t * z);
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / a;
	} else {
		tmp = (-t / a) * z;
	}
	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) :: tmp
    if ((t * z) <= (-2d+19)) then
        tmp = ((-1.0d0) / a) * (t * z)
    else if ((t * z) <= 400000.0d0) then
        tmp = (y * x) / a
    else
        tmp = (-t / a) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = (-1.0 / a) * (t * z);
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / a;
	} else {
		tmp = (-t / a) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t * z) <= -2e+19:
		tmp = (-1.0 / a) * (t * z)
	elif (t * z) <= 400000.0:
		tmp = (y * x) / a
	else:
		tmp = (-t / a) * z
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(t * z) <= -2e+19)
		tmp = Float64(Float64(-1.0 / a) * Float64(t * z));
	elseif (Float64(t * z) <= 400000.0)
		tmp = Float64(Float64(y * x) / a);
	else
		tmp = Float64(Float64(Float64(-t) / a) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t * z) <= -2e+19)
		tmp = (-1.0 / a) * (t * z);
	elseif ((t * z) <= 400000.0)
		tmp = (y * x) / a;
	else
		tmp = (-t / a) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(t * z), $MachinePrecision], -2e+19], N[(N[(-1.0 / a), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 400000.0], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], N[(N[((-t) / a), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e19

   1. Initial program 95.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. remove-double-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. associate-/r* N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(t \cdot z\right)} \cdot \frac{-1}{a} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 73.0

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

   7. Applied rewrites 73.0%

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

   if -2e19 < (*.f64 z t) < 4e5

   1. Initial program 92.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 4e5 < (*.f64 z t)

   1. Initial program 87.2%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 20.4

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

   5. Applied rewrites 20.4%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 78.0

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

   8. Applied rewrites 78.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{a} \cdot \left(t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t}{a} \cdot z\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- t) a) z)))
   (if (<= (* t z) -2e+19) t_1 (if (<= (* t z) 400000.0) (/ (* y x) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-t / a) * z;
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = t_1;
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / 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 = (-t / a) * z
    if ((t * z) <= (-2d+19)) then
        tmp = t_1
    else if ((t * z) <= 400000.0d0) then
        tmp = (y * x) / 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 = (-t / a) * z;
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = t_1;
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (-t / a) * z
	tmp = 0
	if (t * z) <= -2e+19:
		tmp = t_1
	elif (t * z) <= 400000.0:
		tmp = (y * x) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-t) / a) * z)
	tmp = 0.0
	if (Float64(t * z) <= -2e+19)
		tmp = t_1;
	elseif (Float64(t * z) <= 400000.0)
		tmp = Float64(Float64(y * x) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (-t / a) * z;
	tmp = 0.0;
	if ((t * z) <= -2e+19)
		tmp = t_1;
	elseif ((t * z) <= 400000.0)
		tmp = (y * x) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-t) / a), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+19], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 400000.0], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e19 or 4e5 < (*.f64 z t)

   1. Initial program 90.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 74.9

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

   8. Applied rewrites 74.9%

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

   if -2e19 < (*.f64 z t) < 4e5

   1. Initial program 92.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-z}{a} \cdot t\\ \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z) a) t)))
   (if (<= (* t z) -2e+19) t_1 (if (<= (* t z) 400000.0) (/ (* y x) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (-z / a) * t;
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = t_1;
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / 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 = (-z / a) * t
    if ((t * z) <= (-2d+19)) then
        tmp = t_1
    else if ((t * z) <= 400000.0d0) then
        tmp = (y * x) / 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 = (-z / a) * t;
	double tmp;
	if ((t * z) <= -2e+19) {
		tmp = t_1;
	} else if ((t * z) <= 400000.0) {
		tmp = (y * x) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (-z / a) * t
	tmp = 0
	if (t * z) <= -2e+19:
		tmp = t_1
	elif (t * z) <= 400000.0:
		tmp = (y * x) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(-z) / a) * t)
	tmp = 0.0
	if (Float64(t * z) <= -2e+19)
		tmp = t_1;
	elseif (Float64(t * z) <= 400000.0)
		tmp = Float64(Float64(y * x) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (-z / a) * t;
	tmp = 0.0;
	if ((t * z) <= -2e+19)
		tmp = t_1;
	elseif ((t * z) <= 400000.0)
		tmp = (y * x) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -2e+19], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 400000.0], N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e19 or 4e5 < (*.f64 z t)

   1. Initial program 90.9%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if -2e19 < (*.f64 z t) < 4e5

   1. Initial program 92.8%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \mathbf{elif}\;t \cdot z \leq 400000:\\ \;\;\;\;\frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* t z) 5e+302) (/ (- (* y x) (* t z)) a) (* (/ (- t) a) z)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t * z) <= 5e+302) {
		tmp = ((y * x) - (t * z)) / a;
	} else {
		tmp = (-t / a) * z;
	}
	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) :: tmp
    if ((t * z) <= 5d+302) then
        tmp = ((y * x) - (t * z)) / a
    else
        tmp = (-t / a) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t * z) <= 5e+302) {
		tmp = ((y * x) - (t * z)) / a;
	} else {
		tmp = (-t / a) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t * z) <= 5e+302:
		tmp = ((y * x) - (t * z)) / a
	else:
		tmp = (-t / a) * z
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(t * z) <= 5e+302)
		tmp = Float64(Float64(Float64(y * x) - Float64(t * z)) / a);
	else
		tmp = Float64(Float64(Float64(-t) / a) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t * z) <= 5e+302)
		tmp = ((y * x) - (t * z)) / a;
	else
		tmp = (-t / a) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(t * z), $MachinePrecision], 5e+302], N[(N[(N[(y * x), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[((-t) / a), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 5e302

   1. Initial program 94.5%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   if 5e302 < (*.f64 z t)

   1. Initial program 52.7%

      \[\frac{x \cdot y - z \cdot t}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{y \cdot x - t \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 = (y / a) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

   \[\frac{x \cdot y - z \cdot t}{a} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 50.6

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

5. Applied rewrites 50.6%

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

6. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 49.3

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

8. Applied rewrites 49.3%

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

9. Taylor expanded in t around 0

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

    1. *-commutative N/A

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

    2. associate-*l/ N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 48.2

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

11. Applied rewrites 48.2%

    \[\leadsto \color{blue}{\frac{y}{a} \cdot x} \]
12. Add Preprocessing

Developer Target 1: 91.3% accurate, 0.5× 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 2024255 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -246868496869954800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6309831121978371/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z)))))

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