Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.5% → 98.2%

Time: 9.4s

Alternatives: 10

Speedup: 1.4×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{t\_m}{\frac{\frac{-1}{y\_m}}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot \left(x - z\right)\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (<= t_m 6.5e-35)
     (/ t_m (/ (/ -1.0 y_m) (- z x)))
     (* (* y_m t_m) (- x z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 6.5e-35) {
		tmp = t_m / ((-1.0 / y_m) / (z - x));
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 6.5d-35) then
        tmp = t_m / (((-1.0d0) / y_m) / (z - x))
    else
        tmp = (y_m * t_m) * (x - z)
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 6.5e-35) {
		tmp = t_m / ((-1.0 / y_m) / (z - x));
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 6.5e-35:
		tmp = t_m / ((-1.0 / y_m) / (z - x))
	else:
		tmp = (y_m * t_m) * (x - z)
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 6.5e-35)
		tmp = Float64(t_m / Float64(Float64(-1.0 / y_m) / Float64(z - x)));
	else
		tmp = Float64(Float64(y_m * t_m) * Float64(x - z));
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 6.5e-35)
		tmp = t_m / ((-1.0 / y_m) / (z - x));
	else
		tmp = (y_m * t_m) * (x - z);
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 6.5e-35], N[(t$95$m / N[(N[(-1.0 / y$95$m), $MachinePrecision] / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.4999999999999999e-35

   1. Initial program 86.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 89.0

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. flip-- N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. clear-num N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. distribute-rgt-in N/A

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

      21. +-commutative N/A

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

   6. Applied rewrites 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. flip-- N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-frac2 N/A

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

      9. difference-of-squares N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-neg.f64 N/A

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

      15. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{\mathsf{neg}\left(\frac{1}{y}\right)}{\frac{\left(z + x\right) \cdot \left(\color{blue}{\left(0 - z\right)} + x\right)}{\mathsf{neg}\left(\left(z + x\right)\right)}}} \]

      16. associate-+l- N/A

          \[\leadsto \frac{t}{\frac{\mathsf{neg}\left(\frac{1}{y}\right)}{\frac{\left(z + x\right) \cdot \color{blue}{\left(0 - \left(z - x\right)\right)}}{\mathsf{neg}\left(\left(z + x\right)\right)}}} \]

      17. neg-sub0 N/A

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

      18. distribute-rgt-neg-in N/A

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

      19. difference-of-squares N/A

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

      20. frac-2neg N/A

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

      21. flip-- N/A

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

   8. Applied rewrites 89.1%

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

   if 6.4999999999999999e-35 < t

   1. Initial program 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{t}{\frac{\frac{-1}{y}}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+168}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+111}:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* (- z) y_m) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= z -3.7e+168)
       t_2
       (if (<= z 6e+111) (* (* (- x z) t_m) y_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-z * y_m) * t_m;
	double tmp;
	if (z <= -3.7e+168) {
		tmp = t_2;
	} else if (z <= 6e+111) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (-z * y_m) * t_m
    if (z <= (-3.7d+168)) then
        tmp = t_2
    else if (z <= 6d+111) then
        tmp = ((x - z) * t_m) * y_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (-z * y_m) * t_m;
	double tmp;
	if (z <= -3.7e+168) {
		tmp = t_2;
	} else if (z <= 6e+111) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (-z * y_m) * t_m
	tmp = 0
	if z <= -3.7e+168:
		tmp = t_2
	elif z <= 6e+111:
		tmp = ((x - z) * t_m) * y_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(Float64(-z) * y_m) * t_m)
	tmp = 0.0
	if (z <= -3.7e+168)
		tmp = t_2;
	elseif (z <= 6e+111)
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (-z * y_m) * t_m;
	tmp = 0.0;
	if (z <= -3.7e+168)
		tmp = t_2;
	elseif (z <= 6e+111)
		tmp = ((x - z) * t_m) * y_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[z, -3.7e+168], t$95$2, If[LessEqual[z, 6e+111], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.70000000000000009e168 or 6e111 < z

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -3.70000000000000009e168 < z < 6e111

   1. Initial program 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 3: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(x \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\left(-z\right) \cdot \left(y\_m \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* x y_m) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= x -5.9e-33) t_2 (if (<= x 4e-21) (* (- z) (* y_m t_m)) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -5.9e-33) {
		tmp = t_2;
	} else if (x <= 4e-21) {
		tmp = -z * (y_m * t_m);
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x * y_m) * t_m
    if (x <= (-5.9d-33)) then
        tmp = t_2
    else if (x <= 4d-21) then
        tmp = -z * (y_m * t_m)
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -5.9e-33) {
		tmp = t_2;
	} else if (x <= 4e-21) {
		tmp = -z * (y_m * t_m);
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (x * y_m) * t_m
	tmp = 0
	if x <= -5.9e-33:
		tmp = t_2
	elif x <= 4e-21:
		tmp = -z * (y_m * t_m)
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(x * y_m) * t_m)
	tmp = 0.0
	if (x <= -5.9e-33)
		tmp = t_2;
	elseif (x <= 4e-21)
		tmp = Float64(Float64(-z) * Float64(y_m * t_m));
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (x * y_m) * t_m;
	tmp = 0.0;
	if (x <= -5.9e-33)
		tmp = t_2;
	elseif (x <= 4e-21)
		tmp = -z * (y_m * t_m);
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[x, -5.9e-33], t$95$2, If[LessEqual[x, 4e-21], N[((-z) * N[(y$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.89999999999999985e-33 or 3.99999999999999963e-21 < x

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.89999999999999985e-33 < x < 3.99999999999999963e-21

   1. Initial program 91.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.8

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

   7. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 83.7

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

   9. Applied rewrites 83.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-33}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\left(-z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(x \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* x y_m) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= x -5.9e-33)
       t_2
       (if (<= x 1.05e-18) (* (* (- z) y_m) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -5.9e-33) {
		tmp = t_2;
	} else if (x <= 1.05e-18) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x * y_m) * t_m
    if (x <= (-5.9d-33)) then
        tmp = t_2
    else if (x <= 1.05d-18) then
        tmp = (-z * y_m) * t_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -5.9e-33) {
		tmp = t_2;
	} else if (x <= 1.05e-18) {
		tmp = (-z * y_m) * t_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (x * y_m) * t_m
	tmp = 0
	if x <= -5.9e-33:
		tmp = t_2
	elif x <= 1.05e-18:
		tmp = (-z * y_m) * t_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(x * y_m) * t_m)
	tmp = 0.0
	if (x <= -5.9e-33)
		tmp = t_2;
	elseif (x <= 1.05e-18)
		tmp = Float64(Float64(Float64(-z) * y_m) * t_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (x * y_m) * t_m;
	tmp = 0.0;
	if (x <= -5.9e-33)
		tmp = t_2;
	elseif (x <= 1.05e-18)
		tmp = (-z * y_m) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[x, -5.9e-33], t$95$2, If[LessEqual[x, 1.05e-18], N[(N[((-z) * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.89999999999999985e-33 or 1.05e-18 < x

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.89999999999999985e-33 < x < 1.05e-18

   1. Initial program 91.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-33}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(-z\right) \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ \begin{array}{l} t_2 := \left(x \cdot y\_m\right) \cdot t\_m\\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(-t\_m\right) \cdot z\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (let* ((t_2 (* (* x y_m) t_m)))
   (*
    y_s
    (*
     t_s
     (if (<= x -5.9e-33) t_2 (if (<= x 4e-21) (* (* (- t_m) z) y_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -5.9e-33) {
		tmp = t_2;
	} else if (x <= 4e-21) {
		tmp = (-t_m * z) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x * y_m) * t_m
    if (x <= (-5.9d-33)) then
        tmp = t_2
    else if (x <= 4d-21) then
        tmp = (-t_m * z) * y_m
    else
        tmp = t_2
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double t_2 = (x * y_m) * t_m;
	double tmp;
	if (x <= -5.9e-33) {
		tmp = t_2;
	} else if (x <= 4e-21) {
		tmp = (-t_m * z) * y_m;
	} else {
		tmp = t_2;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	t_2 = (x * y_m) * t_m
	tmp = 0
	if x <= -5.9e-33:
		tmp = t_2
	elif x <= 4e-21:
		tmp = (-t_m * z) * y_m
	else:
		tmp = t_2
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	t_2 = Float64(Float64(x * y_m) * t_m)
	tmp = 0.0
	if (x <= -5.9e-33)
		tmp = t_2;
	elseif (x <= 4e-21)
		tmp = Float64(Float64(Float64(-t_m) * z) * y_m);
	else
		tmp = t_2;
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	t_2 = (x * y_m) * t_m;
	tmp = 0.0;
	if (x <= -5.9e-33)
		tmp = t_2;
	elseif (x <= 4e-21)
		tmp = (-t_m * z) * y_m;
	else
		tmp = t_2;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(y$95$s * N[(t$95$s * If[LessEqual[x, -5.9e-33], t$95$2, If[LessEqual[x, 4e-21], N[(N[((-t$95$m) * z), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.89999999999999985e-33 or 3.99999999999999963e-21 < x

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   if -5.89999999999999985e-33 < x < 3.99999999999999963e-21

   1. Initial program 91.2%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-33}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(-t\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5200000000:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot \left(x - z\right)\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (*
  y_s
  (*
   t_s
   (if (<= t_m 5200000000.0)
     (* (* (- x z) t_m) y_m)
     (* (* y_m t_m) (- x z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 5200000000.0) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 5200000000.0d0) then
        tmp = ((x - z) * t_m) * y_m
    else
        tmp = (y_m * t_m) * (x - z)
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 5200000000.0) {
		tmp = ((x - z) * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * (x - z);
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 5200000000.0:
		tmp = ((x - z) * t_m) * y_m
	else:
		tmp = (y_m * t_m) * (x - z)
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 5200000000.0)
		tmp = Float64(Float64(Float64(x - z) * t_m) * y_m);
	else
		tmp = Float64(Float64(y_m * t_m) * Float64(x - z));
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 5200000000.0)
		tmp = ((x - z) * t_m) * y_m;
	else
		tmp = (y_m * t_m) * (x - z);
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 5200000000.0], N[(N[(N[(x - z), $MachinePrecision] * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.2e9

   1. Initial program 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 5.2e9 < t

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5200000000:\\ \;\;\;\;\left(\left(x - z\right) \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left(x \cdot y\_m\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (if (<= t_m 2e+20) (* (* x y_m) t_m) (* (* y_m t_m) x)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2e+20) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = (y_m * t_m) * x;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2d+20) then
        tmp = (x * y_m) * t_m
    else
        tmp = (y_m * t_m) * x
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2e+20) {
		tmp = (x * y_m) * t_m;
	} else {
		tmp = (y_m * t_m) * x;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 2e+20:
		tmp = (x * y_m) * t_m
	else:
		tmp = (y_m * t_m) * x
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 2e+20)
		tmp = Float64(Float64(x * y_m) * t_m);
	else
		tmp = Float64(Float64(y_m * t_m) * x);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 2e+20)
		tmp = (x * y_m) * t_m;
	else
		tmp = (y_m * t_m) * x;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 2e+20], N[(N[(x * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2e20

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.5

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

   5. Applied rewrites 52.5%

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

   if 2e20 < t

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-*r* N/A

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

      8. flip-- N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.2

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

   7. Applied rewrites 67.2%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\left(x \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-89}:\\ \;\;\;\;\left(x \cdot t\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot t\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (if (<= t_m 4.5e-89) (* (* x t_m) y_m) (* (* y_m t_m) x)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 4.5e-89) {
		tmp = (x * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * x;
	}
	return y_s * (t_s * tmp);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 4.5d-89) then
        tmp = (x * t_m) * y_m
    else
        tmp = (y_m * t_m) * x
    end if
    code = y_s * (t_s * tmp)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 4.5e-89) {
		tmp = (x * t_m) * y_m;
	} else {
		tmp = (y_m * t_m) * x;
	}
	return y_s * (t_s * tmp);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 4.5e-89:
		tmp = (x * t_m) * y_m
	else:
		tmp = (y_m * t_m) * x
	return y_s * (t_s * tmp)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 4.5e-89)
		tmp = Float64(Float64(x * t_m) * y_m);
	else
		tmp = Float64(Float64(y_m * t_m) * x);
	end
	return Float64(y_s * Float64(t_s * tmp))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(y_s, t_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 4.5e-89)
		tmp = (x * t_m) * y_m;
	else
		tmp = (y_m * t_m) * x;
	end
	tmp_2 = y_s * (t_s * tmp);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * If[LessEqual[t$95$m, 4.5e-89], N[(N[(x * t$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(y$95$m * t$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.4999999999999999e-89

   1. Initial program 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.0

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

   7. Applied rewrites 53.0%

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

   if 4.4999999999999999e-89 < t

   1. Initial program 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-*r* N/A

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

      8. flip-- N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 66.3

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 61.1

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

   7. Applied rewrites 61.1%

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

Alternative 9: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \left(\left(\left(x - z\right) \cdot y\_m\right) \cdot t\_m\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (* (* (- x z) y_m) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	return y_s * (t_s * (((x - z) * y_m) * t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = y_s * (t_s * (((x - z) * y_m) * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	return y_s * (t_s * (((x - z) * y_m) * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	return y_s * (t_s * (((x - z) * y_m) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	return Float64(y_s * Float64(t_s * Float64(Float64(Float64(x - z) * y_m) * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp = code(y_s, t_s, x, y_m, z, t_m)
	tmp = y_s * (t_s * (((x - z) * y_m) * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * N[(N[(N[(x - z), $MachinePrecision] * y$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt-out-- N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 91.1

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

4. Applied rewrites 91.1%

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

Alternative 10: 55.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ y\_s \cdot \left(t\_s \cdot \left(\left(y\_m \cdot t\_m\right) \cdot x\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (y_s t_s x y_m z t_m)
 :precision binary64
 (* y_s (* t_s (* (* y_m t_m) x))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x < y_m && y_m < z && z < t_m);
double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	return y_s * (t_s * ((y_m * t_m) * x));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, t_s, x, y_m, z, t_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = y_s * (t_s * ((y_m * t_m) * x))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x < y_m && y_m < z && z < t_m;
public static double code(double y_s, double t_s, double x, double y_m, double z, double t_m) {
	return y_s * (t_s * ((y_m * t_m) * x));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(y_s, t_s, x, y_m, z, t_m):
	return y_s * (t_s * ((y_m * t_m) * x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(y_s, t_s, x, y_m, z, t_m)
	return Float64(y_s * Float64(t_s * Float64(Float64(y_m * t_m) * x)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp = code(y_s, t_s, x, y_m, z, t_m)
	tmp = y_s * (t_s * ((y_m * t_m) * x));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
code[y$95$s_, t$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(y$95$s * N[(t$95$s * N[(N[(y$95$m * t$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-rgt-out-- N/A

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

   7. associate-*r* N/A

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

   8. flip-- N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. difference-of-squares N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower--.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 62.6

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

4. Applied rewrites 62.6%

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

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 56.6

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

7. Applied rewrites 56.6%

   \[\leadsto \color{blue}{\left(y \cdot t\right) \cdot x} \]
8. Add Preprocessing

Developer Target 1: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< t -9.231879582886777e-80)
   (* (* y t) (- x z))
   (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t < (-9.231879582886777d-80)) then
        tmp = (y * t) * (x - z)
    else if (t < 2.543067051564877d+83) then
        tmp = y * (t * (x - z))
    else
        tmp = (y * (x - z)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t < -9.231879582886777e-80:
		tmp = (y * t) * (x - z)
	elif t < 2.543067051564877e+83:
		tmp = y * (t * (x - z))
	else:
		tmp = (y * (x - z)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t < -9.231879582886777e-80)
		tmp = Float64(Float64(y * t) * Float64(x - z));
	elseif (t < 2.543067051564877e+83)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(y * Float64(x - z)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t < -9.231879582886777e-80)
		tmp = (y * t) * (x - z);
	elseif (t < 2.543067051564877e+83)
		tmp = y * (t * (x - z));
	else
		tmp = (y * (x - z)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[t, -9.231879582886777e-80], N[(N[(y * t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], If[Less[t, 2.543067051564877e+83], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -9231879582886777/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* y t) (- x z)) (if (< t 254306705156487700000000000000000000000000000000000000000000000000000000000000000000) (* y (* t (- x z))) (* (* y (- x z)) t))))

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