Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 96.5% → 98.2%

Time: 7.5s

Alternatives: 6

Speedup: N/A×

• 24.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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, N/A× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-10}:\\ \;\;\;\;\left(t\_m \cdot \left(x - z\right)\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \left(t\_m \cdot y\_m\right)\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (<= t_m 2.25e-10) (* (* t_m (- x z)) y_m) (* (- x z) (* t_m y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2.25e-10) {
		tmp = (t_m * (x - z)) * y_m;
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Fortran

y\_m =     private
y\_s =     private
t\_m =     private
t\_s =     private
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, y_s, x, y_m, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: y_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 <= 2.25d-10) then
        tmp = (t_m * (x - z)) * y_m
    else
        tmp = (x - z) * (t_m * y_m)
    end if
    code = t_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
assert x < y_m && y_m < z && z < t_m;
public static double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if (t_m <= 2.25e-10) {
		tmp = (t_m * (x - z)) * y_m;
	} else {
		tmp = (x - z) * (t_m * y_m);
	}
	return t_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
[x, y_m, z, t_m] = sort([x, y_m, z, t_m])
def code(t_s, y_s, x, y_m, z, t_m):
	tmp = 0
	if t_m <= 2.25e-10:
		tmp = (t_m * (x - z)) * y_m
	else:
		tmp = (x - z) * (t_m * y_m)
	return t_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if (t_m <= 2.25e-10)
		tmp = Float64(Float64(t_m * Float64(x - z)) * y_m);
	else
		tmp = Float64(Float64(x - z) * Float64(t_m * y_m));
	end
	return Float64(t_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
x, y_m, z, t_m = num2cell(sort([x, y_m, z, t_m])){:}
function tmp_2 = code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0;
	if (t_m <= 2.25e-10)
		tmp = (t_m * (x - z)) * y_m;
	else
		tmp = (x - z) * (t_m * y_m);
	end
	tmp_2 = t_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[LessEqual[t$95$m, 2.25e-10], N[(N[(t$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.25e-10

   1. Initial program 86.1%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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

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

      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. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 2.25e-10 < t

   1. Initial program 98.4%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 2: 95.9% accurate, N/A× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-42} \lor \neg \left(z \leq 2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x}{z}, -1, 1\right) \cdot y\_m\right) \cdot \left(-z\right)\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\left(t\_m \cdot \left(x - z\right)\right) \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (or (<= z -2.6e-42) (not (<= z 2e-115)))
     (* (* (* (fma (/ x z) -1.0 1.0) y_m) (- z)) t_m)
     (* (* t_m (- x z)) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -2.6e-42) || !(z <= 2e-115)) {
		tmp = ((fma((x / z), -1.0, 1.0) * y_m) * -z) * t_m;
	} else {
		tmp = (t_m * (x - z)) * y_m;
	}
	return t_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((z <= -2.6e-42) || !(z <= 2e-115))
		tmp = Float64(Float64(Float64(fma(Float64(x / z), -1.0, 1.0) * y_m) * Float64(-z)) * t_m);
	else
		tmp = Float64(Float64(t_m * Float64(x - z)) * y_m);
	end
	return Float64(t_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[Or[LessEqual[z, -2.6e-42], N[Not[LessEqual[z, 2e-115]], $MachinePrecision]], N[(N[(N[(N[(N[(x / z), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * (-z)), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * N[(x - z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e-42 or 2.0000000000000001e-115 < z

   1. Initial program 87.7%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

   if -2.6e-42 < z < 2.0000000000000001e-115

   1. Initial program 93.5%

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]
   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. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 93.5

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

   4. Applied rewrites 93.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-42} \lor \neg \left(z \leq 2 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x}{z}, -1, 1\right) \cdot y\right) \cdot \left(-z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(x - z\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, N/A× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ [x, y_m, z, t_m] = \mathsf{sort}([x, y_m, z, t_m])\\ \\ t\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-12} \lor \neg \left(z \leq 2.4 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x}{z}, -1, 1\right) \cdot y\_m\right) \cdot \left(-z\right)\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t\_m, z \cdot \frac{y\_m}{x}, t\_m \cdot y\_m\right) \cdot x\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
NOTE: x, y_m, z, and t_m should be sorted in increasing order before calling this function.
(FPCore (t_s y_s x y_m z t_m)
 :precision binary64
 (*
  t_s
  (*
   y_s
   (if (or (<= z -4.8e-12) (not (<= z 2.4e-115)))
     (* (* (* (fma (/ x z) -1.0 1.0) y_m) (- z)) t_m)
     (* (fma (- t_m) (* z (/ y_m x)) (* t_m y_m)) x)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
assert(x < y_m && y_m < z && z < t_m);
double code(double t_s, double y_s, double x, double y_m, double z, double t_m) {
	double tmp;
	if ((z <= -4.8e-12) || !(z <= 2.4e-115)) {
		tmp = ((fma((x / z), -1.0, 1.0) * y_m) * -z) * t_m;
	} else {
		tmp = fma(-t_m, (z * (y_m / x)), (t_m * y_m)) * x;
	}
	return t_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
t\_m = abs(t)
t\_s = copysign(1.0, t)
x, y_m, z, t_m = sort([x, y_m, z, t_m])
function code(t_s, y_s, x, y_m, z, t_m)
	tmp = 0.0
	if ((z <= -4.8e-12) || !(z <= 2.4e-115))
		tmp = Float64(Float64(Float64(fma(Float64(x / z), -1.0, 1.0) * y_m) * Float64(-z)) * t_m);
	else
		tmp = Float64(fma(Float64(-t_m), Float64(z * Float64(y_m / x)), Float64(t_m * y_m)) * x);
	end
	return Float64(t_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * If[Or[LessEqual[z, -4.8e-12], N[Not[LessEqual[z, 2.4e-115]], $MachinePrecision]], N[(N[(N[(N[(N[(x / z), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision] * (-z)), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[((-t$95$m) * N[(z * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.79999999999999974e-12 or 2.40000000000000021e-115 < z

   1. Initial program 87.3%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.4

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

   8. Applied rewrites 90.4%

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

   if -4.79999999999999974e-12 < z < 2.40000000000000021e-115

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 86.2

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

   7. Applied rewrites 86.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-12} \lor \neg \left(z \leq 2.4 \cdot 10^{-115}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{x}{z}, -1, 1\right) \cdot y\right) \cdot \left(-z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, z \cdot \frac{y}{x}, t \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * N[(N[(N[((-z) * N[(y$95$m / x), $MachinePrecision] + y$95$m), $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in z around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 82.2

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 83.4

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

8. Applied rewrites 83.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 78.5%

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

Alternative 5: 84.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * N[(N[((-t$95$m) * N[(N[(z * y$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 76.9%

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

Alternative 6: 82.9% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, 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[t$95$s_, y$95$s_, x_, y$95$m_, z_, t$95$m_] := N[(t$95$s * N[(y$95$s * N[(N[((-t$95$m) * N[(z * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 76.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 75.1

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

7. Applied rewrites 75.1%

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

Reproduce

herbie shell --seed 2025057 
(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))