Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.8% → 79.8%

Time: 15.0s

Alternatives: 25

Speedup: 0.7×

• 57.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
  (* j (- (* c t) (* i y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

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, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
  (* j (- (* c t) (* i y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

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, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5.8 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(-j, y, \mathsf{fma}\left(\mathsf{fma}\left(a, -t, y \cdot z\right), \frac{x}{i}, \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(b, -z, j \cdot t\right)}{i}, b \cdot a\right)\right)\right) \cdot i\\ \mathbf{elif}\;i \leq 6.5 \cdot 10^{-83}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)}{i} + b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -5.8e+66)
   (*
    (fma
     (- j)
     y
     (fma
      (fma a (- t) (* y z))
      (/ x i)
      (fma c (/ (fma b (- z) (* j t)) i) (* b a))))
    i)
   (if (<= i 6.5e-83)
     (+
      (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
      (* j (- (* c t) (* i y))))
     (*
      (fma
       (- j)
       y
       (+
        (/ (fma (fma (- t) a (* z y)) x (* (fma (- z) b (* j t)) c)) i)
        (* b a)))
      i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -5.8e+66) {
		tmp = fma(-j, y, fma(fma(a, -t, (y * z)), (x / i), fma(c, (fma(b, -z, (j * t)) / i), (b * a)))) * i;
	} else if (i <= 6.5e-83) {
		tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
	} else {
		tmp = fma(-j, y, ((fma(fma(-t, a, (z * y)), x, (fma(-z, b, (j * t)) * c)) / i) + (b * a))) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -5.8e+66)
		tmp = Float64(fma(Float64(-j), y, fma(fma(a, Float64(-t), Float64(y * z)), Float64(x / i), fma(c, Float64(fma(b, Float64(-z), Float64(j * t)) / i), Float64(b * a)))) * i);
	elseif (i <= 6.5e-83)
		tmp = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))));
	else
		tmp = Float64(fma(Float64(-j), y, Float64(Float64(fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-z), b, Float64(j * t)) * c)) / i) + Float64(b * a))) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -5.8e+66], N[(N[((-j) * y + N[(N[(a * (-t) + N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x / i), $MachinePrecision] + N[(c * N[(N[(b * (-z) + N[(j * t), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[i, 6.5e-83], N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-j) * y + N[(N[(N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -5.79999999999999972e66

   1. Initial program 65.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 86.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

         \[\leadsto \mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)}{i} + b \cdot a\right) \cdot i \]

      5. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)}{i} + b \cdot a\right) \cdot i \]

      6. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x + \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c}{i} + b \cdot a\right) \cdot i \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x + \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c}{i} + b \cdot a\right) \cdot i \]

      8. div-add N/A

         \[\leadsto \mathsf{fma}\left(-j, y, \left(\frac{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x}{i} + \frac{\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c}{i}\right) + b \cdot a\right) \cdot i \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-j, y, \left(\frac{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x}{i} + \frac{\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c}{i}\right) + b \cdot a\right) \cdot i \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x}{i} + \left(\frac{\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c}{i} + b \cdot a\right)\right) \cdot i \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x}{i} + \left(\frac{\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c}{i} + b \cdot a\right)\right) \cdot i \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(-j, y, \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot \frac{x}{i} + \left(\frac{\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c}{i} + b \cdot a\right)\right) \cdot i \]

   7. Applied rewrites 94.0%

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

   if -5.79999999999999972e66 < i < 6.5e-83

   1. Initial program 89.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   if 6.5e-83 < i

   1. Initial program 67.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 87.3%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.8 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(-j, y, \mathsf{fma}\left(\mathsf{fma}\left(a, -t, y \cdot z\right), \frac{x}{i}, \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(b, -z, j \cdot t\right)}{i}, b \cdot a\right)\right)\right) \cdot i\\ \mathbf{elif}\;i \leq 6.5 \cdot 10^{-83}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)}{i} + b \cdot a\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)}{i} + b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot a\right), \frac{i}{x}, z \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (* j (- (* c t) (* i y))))))
   (if (<= t_1 2e+290)
     t_1
     (if (<= t_1 INFINITY)
       (*
        (fma
         (- j)
         y
         (+
          (/ (fma (fma (- t) a (* z y)) x (* (fma (- z) b (* j t)) c)) i)
          (* b a)))
        i)
       (* (fma (- a) t (fma (fma (- j) y (* b a)) (/ i x) (* z y))) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
	double tmp;
	if (t_1 <= 2e+290) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(-j, y, ((fma(fma(-t, a, (z * y)), x, (fma(-z, b, (j * t)) * c)) / i) + (b * a))) * i;
	} else {
		tmp = fma(-a, t, fma(fma(-j, y, (b * a)), (i / x), (z * y))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
	tmp = 0.0
	if (t_1 <= 2e+290)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(-j), y, Float64(Float64(fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-z), b, Float64(j * t)) * c)) / i) + Float64(b * a))) * i);
	else
		tmp = Float64(fma(Float64(-a), t, fma(fma(Float64(-j), y, Float64(b * a)), Float64(i / x), Float64(z * y))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+290], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[((-j) * y + N[(N[(N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], N[(N[((-a) * t + N[(N[((-j) * y + N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(i / x), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < 2.00000000000000012e290

   1. Initial program 94.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   if 2.00000000000000012e290 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 84.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 91.0%

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

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 0.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      4. lower-*.f64 29.0

         \[\leadsto \left(i \cdot b\right) \cdot a \]

   8. Applied rewrites 29.0%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-j, y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)}{i} + b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot a\right), \frac{i}{x}, z \cdot y\right)\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot a\right), \frac{i}{x}, z \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (* j (- (* c t) (* i y))))))
   (if (<= t_1 INFINITY)
     t_1
     (* (fma (- a) t (fma (fma (- j) y (* b a)) (/ i x) (* z y))) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(-a, t, fma(fma(-j, y, (b * a)), (i / x), (z * y))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(-a), t, fma(fma(Float64(-j), y, Float64(b * a)), Float64(i / x), Float64(z * y))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[((-a) * t + N[(N[((-j) * y + N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(i / x), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 91.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 0.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      4. lower-*.f64 29.0

         \[\leadsto \left(i \cdot b\right) \cdot a \]

   8. Applied rewrites 29.0%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.0%

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

Alternative 4: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot a\right), \frac{i}{x}, z \cdot y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-144} \lor \neg \left(x \leq 2.35 \cdot 10^{+18}\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, \left(i \cdot b\right) \cdot a\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- t) a (* z y))))
   (if (<= x -3.7e+149)
     (fma t_1 x (* (fma (- z) b (* j t)) c))
     (if (<= x -7.4e-12)
       (* (fma (- a) t (fma (fma (- j) y (* b a)) (/ i x) (* z y))) x)
       (if (or (<= x -8.8e-144) (not (<= x 2.35e+18)))
         (+ (fma t_1 x (* (* i b) a)) (* j (- (* c t) (* i y))))
         (fma (fma (- i) y (* c t)) j (* (fma (- z) c (* i a)) b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y));
	double tmp;
	if (x <= -3.7e+149) {
		tmp = fma(t_1, x, (fma(-z, b, (j * t)) * c));
	} else if (x <= -7.4e-12) {
		tmp = fma(-a, t, fma(fma(-j, y, (b * a)), (i / x), (z * y))) * x;
	} else if ((x <= -8.8e-144) || !(x <= 2.35e+18)) {
		tmp = fma(t_1, x, ((i * b) * a)) + (j * ((c * t) - (i * y)));
	} else {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-z, c, (i * a)) * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-t), a, Float64(z * y))
	tmp = 0.0
	if (x <= -3.7e+149)
		tmp = fma(t_1, x, Float64(fma(Float64(-z), b, Float64(j * t)) * c));
	elseif (x <= -7.4e-12)
		tmp = Float64(fma(Float64(-a), t, fma(fma(Float64(-j), y, Float64(b * a)), Float64(i / x), Float64(z * y))) * x);
	elseif ((x <= -8.8e-144) || !(x <= 2.35e+18))
		tmp = Float64(fma(t_1, x, Float64(Float64(i * b) * a)) + Float64(j * Float64(Float64(c * t) - Float64(i * y))));
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-z), c, Float64(i * a)) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e+149], N[(t$95$1 * x + N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.4e-12], N[(N[((-a) * t + N[(N[((-j) * y + N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(i / x), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[Or[LessEqual[x, -8.8e-144], N[Not[LessEqual[x, 2.35e+18]], $MachinePrecision]], N[(N[(t$95$1 * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.69999999999999978e149

   1. Initial program 81.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 90.8%

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

   if -3.69999999999999978e149 < x < -7.39999999999999997e-12

   1. Initial program 59.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      4. lower-*.f64 28.9

         \[\leadsto \left(i \cdot b\right) \cdot a \]

   8. Applied rewrites 28.9%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 85.6%

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

   if -7.39999999999999997e-12 < x < -8.80000000000000025e-144 or 2.35e18 < x

   1. Initial program 80.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(\left(a \cdot \left(b \cdot i\right)\right) \cdot -1\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot \left(b \cdot i\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

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

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y \cdot z - t \cdot a, \color{blue}{x}, a \cdot \left(b \cdot i\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   if -8.80000000000000025e-144 < x < 2.35e18

   1. Initial program 82.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot t - i \cdot y\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(c \cdot z - a \cdot i\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, \color{blue}{j}, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(\left(c \cdot z - a \cdot i\right) \cdot b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

   5. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot a\right), \frac{i}{x}, z \cdot y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-144} \lor \neg \left(x \leq 2.35 \cdot 10^{+18}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, \mathsf{fma}\left(\mathsf{fma}\left(-j, y, b \cdot a\right), \frac{i}{x}, z \cdot y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-142}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot b\right) \cdot z\right) + \left(j \cdot t\right) \cdot c\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (fma (- t) a (* z y)) x (* (fma (- z) b (* j t)) c))))
   (if (<= x -3.7e+149)
     t_1
     (if (<= x -4.4e-52)
       (* (fma (- a) t (fma (fma (- j) y (* b a)) (/ i x) (* z y))) x)
       (if (<= x -5.7e-142)
         (+ (- (* x (- (* y z) (* t a))) (* (* c b) z)) (* (* j t) c))
         (if (<= x 4.8e+45)
           (fma (fma (- i) y (* c t)) j (* (fma (- z) c (* i a)) b))
           t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(fma(-t, a, (z * y)), x, (fma(-z, b, (j * t)) * c));
	double tmp;
	if (x <= -3.7e+149) {
		tmp = t_1;
	} else if (x <= -4.4e-52) {
		tmp = fma(-a, t, fma(fma(-j, y, (b * a)), (i / x), (z * y))) * x;
	} else if (x <= -5.7e-142) {
		tmp = ((x * ((y * z) - (t * a))) - ((c * b) * z)) + ((j * t) * c);
	} else if (x <= 4.8e+45) {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-z, c, (i * a)) * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-z), b, Float64(j * t)) * c))
	tmp = 0.0
	if (x <= -3.7e+149)
		tmp = t_1;
	elseif (x <= -4.4e-52)
		tmp = Float64(fma(Float64(-a), t, fma(fma(Float64(-j), y, Float64(b * a)), Float64(i / x), Float64(z * y))) * x);
	elseif (x <= -5.7e-142)
		tmp = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(c * b) * z)) + Float64(Float64(j * t) * c));
	elseif (x <= 4.8e+45)
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-z), c, Float64(i * a)) * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e+149], t$95$1, If[LessEqual[x, -4.4e-52], N[(N[((-a) * t + N[(N[((-j) * y + N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(i / x), $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -5.7e-142], N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * b), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e+45], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.69999999999999978e149 or 4.79999999999999979e45 < x

   1. Initial program 81.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 88.4%

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

   if -3.69999999999999978e149 < x < -4.40000000000000018e-52

   1. Initial program 61.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      4. lower-*.f64 29.2

         \[\leadsto \left(i \cdot b\right) \cdot a \]

   8. Applied rewrites 29.2%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 75.7%

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

   if -4.40000000000000018e-52 < x < -5.69999999999999995e-142

   1. Initial program 84.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{c \cdot \left(j \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 79.1

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot t\right) \cdot c \]

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot b\right) \cdot z\right) + \left(j \cdot t\right) \cdot c \]

      4. lower-*.f64 84.2

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot b\right) \cdot z\right) + \left(j \cdot t\right) \cdot c \]

   8. Applied rewrites 84.2%

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

   if -5.69999999999999995e-142 < x < 4.79999999999999979e45

   1. Initial program 83.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot t - i \cdot y\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(c \cdot z - a \cdot i\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, \color{blue}{j}, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(\left(c \cdot z - a \cdot i\right) \cdot b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

   5. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right)\\ t_2 := \mathsf{fma}\left(t\_1, x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right)\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-142}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot b\right) \cdot z\right) + \left(j \cdot t\right) \cdot c\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- t) a (* z y)))
        (t_2 (fma t_1 x (* (fma (- z) b (* j t)) c))))
   (if (<= x -1.75e+149)
     t_2
     (if (<= x -4.4e-52)
       (fma t_1 x (* (fma (- y) j (* b a)) i))
       (if (<= x -5.7e-142)
         (+ (- (* x (- (* y z) (* t a))) (* (* c b) z)) (* (* j t) c))
         (if (<= x 4.8e+45)
           (fma (fma (- i) y (* c t)) j (* (fma (- z) c (* i a)) b))
           t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y));
	double t_2 = fma(t_1, x, (fma(-z, b, (j * t)) * c));
	double tmp;
	if (x <= -1.75e+149) {
		tmp = t_2;
	} else if (x <= -4.4e-52) {
		tmp = fma(t_1, x, (fma(-y, j, (b * a)) * i));
	} else if (x <= -5.7e-142) {
		tmp = ((x * ((y * z) - (t * a))) - ((c * b) * z)) + ((j * t) * c);
	} else if (x <= 4.8e+45) {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-z, c, (i * a)) * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-t), a, Float64(z * y))
	t_2 = fma(t_1, x, Float64(fma(Float64(-z), b, Float64(j * t)) * c))
	tmp = 0.0
	if (x <= -1.75e+149)
		tmp = t_2;
	elseif (x <= -4.4e-52)
		tmp = fma(t_1, x, Float64(fma(Float64(-y), j, Float64(b * a)) * i));
	elseif (x <= -5.7e-142)
		tmp = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(c * b) * z)) + Float64(Float64(j * t) * c));
	elseif (x <= 4.8e+45)
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-z), c, Float64(i * a)) * b));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * x + N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e+149], t$95$2, If[LessEqual[x, -4.4e-52], N[(t$95$1 * x + N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.7e-142], N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * b), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e+45], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.75000000000000006e149 or 4.79999999999999979e45 < x

   1. Initial program 81.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 88.4%

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

   if -1.75000000000000006e149 < x < -4.40000000000000018e-52

   1. Initial program 61.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 71.4%

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

   if -4.40000000000000018e-52 < x < -5.69999999999999995e-142

   1. Initial program 84.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{c \cdot \left(j \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 79.1

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot t\right) \cdot c \]

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot b\right) \cdot z\right) + \left(j \cdot t\right) \cdot c \]

      4. lower-*.f64 84.2

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot b\right) \cdot z\right) + \left(j \cdot t\right) \cdot c \]

   8. Applied rewrites 84.2%

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

   if -5.69999999999999995e-142 < x < 4.79999999999999979e45

   1. Initial program 83.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot t - i \cdot y\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(c \cdot z - a \cdot i\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, \color{blue}{j}, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(\left(c \cdot z - a \cdot i\right) \cdot b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

   5. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right)\\ t_2 := \mathsf{fma}\left(t\_1, x, \mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- t) a (* z y)))
        (t_2 (fma t_1 x (* (fma (- z) b (* j t)) c))))
   (if (<= x -1.75e+149)
     t_2
     (if (<= x -7e-27)
       (fma t_1 x (* (fma (- y) j (* b a)) i))
       (if (<= x 4.8e+45)
         (fma (fma (- i) y (* c t)) j (* (fma (- z) c (* i a)) b))
         t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y));
	double t_2 = fma(t_1, x, (fma(-z, b, (j * t)) * c));
	double tmp;
	if (x <= -1.75e+149) {
		tmp = t_2;
	} else if (x <= -7e-27) {
		tmp = fma(t_1, x, (fma(-y, j, (b * a)) * i));
	} else if (x <= 4.8e+45) {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-z, c, (i * a)) * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-t), a, Float64(z * y))
	t_2 = fma(t_1, x, Float64(fma(Float64(-z), b, Float64(j * t)) * c))
	tmp = 0.0
	if (x <= -1.75e+149)
		tmp = t_2;
	elseif (x <= -7e-27)
		tmp = fma(t_1, x, Float64(fma(Float64(-y), j, Float64(b * a)) * i));
	elseif (x <= 4.8e+45)
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-z), c, Float64(i * a)) * b));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * x + N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e+149], t$95$2, If[LessEqual[x, -7e-27], N[(t$95$1 * x + N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e+45], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75000000000000006e149 or 4.79999999999999979e45 < x

   1. Initial program 81.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

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

   5. Applied rewrites 88.4%

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

   if -1.75000000000000006e149 < x < -7.0000000000000003e-27

   1. Initial program 59.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 77.4%

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

   if -7.0000000000000003e-27 < x < 4.79999999999999979e45

   1. Initial program 82.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot t - i \cdot y\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(c \cdot z - a \cdot i\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, \color{blue}{j}, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(\left(c \cdot z - a \cdot i\right) \cdot b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{fma}\left(-c, \frac{z}{a}, i\right) \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* z y)) x)))
   (if (<= x -2.85e+84)
     t_1
     (if (<= x -1.9e-36)
       (* (fma (- y) j (* b a)) i)
       (if (<= x 1.75e-25)
         (* (fma (- i) y (* c t)) j)
         (if (<= x 2.7e+64) (* (* (fma (- c) (/ z a) i) a) b) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y)) * x;
	double tmp;
	if (x <= -2.85e+84) {
		tmp = t_1;
	} else if (x <= -1.9e-36) {
		tmp = fma(-y, j, (b * a)) * i;
	} else if (x <= 1.75e-25) {
		tmp = fma(-i, y, (c * t)) * j;
	} else if (x <= 2.7e+64) {
		tmp = (fma(-c, (z / a), i) * a) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -2.85e+84)
		tmp = t_1;
	elseif (x <= -1.9e-36)
		tmp = Float64(fma(Float64(-y), j, Float64(b * a)) * i);
	elseif (x <= 1.75e-25)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	elseif (x <= 2.7e+64)
		tmp = Float64(Float64(fma(Float64(-c), Float64(z / a), i) * a) * b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.85e+84], t$95$1, If[LessEqual[x, -1.9e-36], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[x, 1.75e-25], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[x, 2.7e+64], N[(N[(N[((-c) * N[(z / a), $MachinePrecision] + i), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.84999999999999985e84 or 2.7e64 < x

   1. Initial program 78.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -2.84999999999999985e84 < x < -1.89999999999999985e-36

   1. Initial program 54.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{i} \]

      2. distribute-lft-out-- N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y - a \cdot b\right)\right) \cdot i \]

      3. associate-*r* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(j \cdot y - a \cdot b\right) \cdot i\right)} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \left(i \cdot \color{blue}{\left(j \cdot y - a \cdot b\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \left(\left(j \cdot y - a \cdot b\right) \cdot \color{blue}{i}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y - a \cdot b\right)\right) \cdot \color{blue}{i} \]

      7. distribute-lft-out-- N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{i} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   if -1.89999999999999985e-36 < x < 1.7500000000000001e-25

   1. Initial program 81.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 57.0

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   if 1.7500000000000001e-25 < x < 2.7e64

   1. Initial program 99.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

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

         \[\leadsto \left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c\right)\right) \cdot z + a \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right) \cdot b \]

      5. *-lft-identity N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(1 \cdot a\right) \cdot i\right) \cdot b \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot a\right) \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot a\right)\right) \cdot i\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)\right) \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right) \cdot i\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot a\right)\right)\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot a\right)\right)\right) \cdot b \]

      13. distribute-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i\right)\right) \cdot a\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z - i \cdot a\right)\right)\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b \]

      16. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b \]

      17. lower-*.f64 N/A

          \[\leadsto \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot \color{blue}{b} \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b} \]

   6. Taylor expanded in a around inf

      \[\leadsto \left(a \cdot \left(i + -1 \cdot \frac{c \cdot z}{a}\right)\right) \cdot b \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(i + -1 \cdot \frac{c \cdot z}{a}\right) \cdot a\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(i + -1 \cdot \frac{c \cdot z}{a}\right) \cdot a\right) \cdot b \]

      3. +-commutative N/A

         \[\leadsto \left(\left(-1 \cdot \frac{c \cdot z}{a} + i\right) \cdot a\right) \cdot b \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\left(\mathsf{neg}\left(\frac{c \cdot z}{a}\right)\right) + i\right) \cdot a\right) \cdot b \]

      5. associate-/l* N/A

         \[\leadsto \left(\left(\left(\mathsf{neg}\left(c \cdot \frac{z}{a}\right)\right) + i\right) \cdot a\right) \cdot b \]

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

         \[\leadsto \left(\left(\left(\mathsf{neg}\left(c\right)\right) \cdot \frac{z}{a} + i\right) \cdot a\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \left(\left(\left(-1 \cdot c\right) \cdot \frac{z}{a} + i\right) \cdot a\right) \cdot b \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(-1 \cdot c, \frac{z}{a}, i\right) \cdot a\right) \cdot b \]

      9. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{neg}\left(c\right), \frac{z}{a}, i\right) \cdot a\right) \cdot b \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(-c, \frac{z}{a}, i\right) \cdot a\right) \cdot b \]

      11. lower-/.f64 75.6

          \[\leadsto \left(\mathsf{fma}\left(-c, \frac{z}{a}, i\right) \cdot a\right) \cdot b \]

   8. Applied rewrites 75.6%

      \[\leadsto \left(\mathsf{fma}\left(-c, \frac{z}{a}, i\right) \cdot a\right) \cdot b \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.02 \cdot 10^{-17}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \leq 4.9 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;j \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(\left(-y\right) \cdot j, i, \left(b \cdot a\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1.02e-17)
   (+ (* (* z x) y) (* j (- (* c t) (* i y))))
   (if (<= j 4.9e+51)
     (fma (fma (- t) a (* z y)) x (* (* i b) a))
     (if (<= j 1.4e+148)
       (fma (* y z) x (fma (* (- y) j) i (* (* b a) i)))
       (* (* (fma c (/ t i) (- y)) i) j)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1.02e-17) {
		tmp = ((z * x) * y) + (j * ((c * t) - (i * y)));
	} else if (j <= 4.9e+51) {
		tmp = fma(fma(-t, a, (z * y)), x, ((i * b) * a));
	} else if (j <= 1.4e+148) {
		tmp = fma((y * z), x, fma((-y * j), i, ((b * a) * i)));
	} else {
		tmp = (fma(c, (t / i), -y) * i) * j;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1.02e-17)
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(j * Float64(Float64(c * t) - Float64(i * y))));
	elseif (j <= 4.9e+51)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(Float64(i * b) * a));
	elseif (j <= 1.4e+148)
		tmp = fma(Float64(y * z), x, fma(Float64(Float64(-y) * j), i, Float64(Float64(b * a) * i)));
	else
		tmp = Float64(Float64(fma(c, Float64(t / i), Float64(-y)) * i) * j);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1.02e-17], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 4.9e+51], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.4e+148], N[(N[(y * z), $MachinePrecision] * x + N[(N[((-y) * j), $MachinePrecision] * i + N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(t / i), $MachinePrecision] + (-y)), $MachinePrecision] * i), $MachinePrecision] * j), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.01999999999999997e-17

   1. Initial program 78.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

         \[\leadsto \left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. lower-*.f64 76.5

         \[\leadsto \left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Applied rewrites 76.5%

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

   if -1.01999999999999997e-17 < j < 4.89999999999999983e51

   1. Initial program 75.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if 4.89999999999999983e51 < j < 1.3999999999999999e148

   1. Initial program 89.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 78.5

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

   8. Applied rewrites 78.5%

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, i \cdot \mathsf{fma}\left(-y, j, b \cdot a\right)\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, i \cdot \left(\left(-y\right) \cdot j + b \cdot a\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \left(\left(-y\right) \cdot j\right) \cdot i + \left(b \cdot a\right) \cdot i\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(\left(-y\right) \cdot j, i, \left(b \cdot a\right) \cdot i\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(\left(-y\right) \cdot j, i, \left(b \cdot a\right) \cdot i\right)\right) \]

      7. lower-*.f64 78.6

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(\left(-y\right) \cdot j, i, \left(b \cdot a\right) \cdot i\right)\right) \]

   10. Applied rewrites 78.6%

       \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(\left(-y\right) \cdot j, i, \left(b \cdot a\right) \cdot i\right)\right) \]

   if 1.3999999999999999e148 < j

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 78.6

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in i around inf

      \[\leadsto \left(i \cdot \left(-1 \cdot y + \frac{c \cdot t}{i}\right)\right) \cdot j \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{c \cdot t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      4. associate-/l* N/A

         \[\leadsto \left(\left(c \cdot \frac{t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, \mathsf{neg}\left(y\right)\right) \cdot i\right) \cdot j \]

      8. lower-neg.f64 81.7

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]

   8. Applied rewrites 81.7%

      \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-27} \lor \neg \left(x \leq 6 \cdot 10^{-32}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= x -7e-27) (not (<= x 6e-32)))
   (fma (fma (- t) a (* z y)) x (* (fma (- y) j (* b a)) i))
   (fma (fma (- i) y (* c t)) j (* (fma (- z) c (* i a)) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((x <= -7e-27) || !(x <= 6e-32)) {
		tmp = fma(fma(-t, a, (z * y)), x, (fma(-y, j, (b * a)) * i));
	} else {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-z, c, (i * a)) * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((x <= -7e-27) || !(x <= 6e-32))
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(fma(Float64(-y), j, Float64(b * a)) * i));
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-z), c, Float64(i * a)) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[x, -7e-27], N[Not[LessEqual[x, 6e-32]], $MachinePrecision]], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.0000000000000003e-27 or 6.0000000000000001e-32 < x

   1. Initial program 75.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 76.4%

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

   if -7.0000000000000003e-27 < x < 6.0000000000000001e-32

   1. Initial program 81.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot t - i \cdot y\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(c \cdot z - a \cdot i\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, \color{blue}{j}, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(\left(c \cdot z - a \cdot i\right) \cdot b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

   5. Applied rewrites 81.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-27} \lor \neg \left(x \leq 6 \cdot 10^{-32}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+84} \lor \neg \left(x \leq 8.5 \cdot 10^{+18}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= x -7.2e+84) (not (<= x 8.5e+18)))
   (fma (fma (- t) a (* z y)) x (* (* i b) a))
   (fma (fma (- i) y (* c t)) j (* (fma (- z) c (* i a)) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((x <= -7.2e+84) || !(x <= 8.5e+18)) {
		tmp = fma(fma(-t, a, (z * y)), x, ((i * b) * a));
	} else {
		tmp = fma(fma(-i, y, (c * t)), j, (fma(-z, c, (i * a)) * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((x <= -7.2e+84) || !(x <= 8.5e+18))
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(Float64(i * b) * a));
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(fma(Float64(-z), c, Float64(i * a)) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[x, -7.2e+84], N[Not[LessEqual[x, 8.5e+18]], $MachinePrecision]], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999999e84 or 8.5e18 < x

   1. Initial program 80.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.0

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

   8. Applied rewrites 78.0%

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

   if -7.1999999999999999e84 < x < 8.5e18

   1. Initial program 77.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) - b \cdot \left(c \cdot z - a \cdot i\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto j \cdot \left(c \cdot t - i \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot t - i \cdot y\right) \cdot j + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(c \cdot z - a \cdot i\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, \color{blue}{j}, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t, j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - a \cdot i\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(b \cdot \left(c \cdot z - a \cdot i\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{neg}\left(\left(c \cdot z - a \cdot i\right) \cdot b\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b\right) \]

   5. Applied rewrites 77.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+84} \lor \neg \left(x \leq 8.5 \cdot 10^{+18}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.02 \cdot 10^{-17}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;j \leq 4.9 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;j \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1.02e-17)
   (+ (* (* z x) y) (* j (- (* c t) (* i y))))
   (if (<= j 4.9e+51)
     (fma (fma (- t) a (* z y)) x (* (* i b) a))
     (if (<= j 1.4e+148)
       (fma (* y z) x (* (fma (- y) j (* b a)) i))
       (* (* (fma c (/ t i) (- y)) i) j)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1.02e-17) {
		tmp = ((z * x) * y) + (j * ((c * t) - (i * y)));
	} else if (j <= 4.9e+51) {
		tmp = fma(fma(-t, a, (z * y)), x, ((i * b) * a));
	} else if (j <= 1.4e+148) {
		tmp = fma((y * z), x, (fma(-y, j, (b * a)) * i));
	} else {
		tmp = (fma(c, (t / i), -y) * i) * j;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1.02e-17)
		tmp = Float64(Float64(Float64(z * x) * y) + Float64(j * Float64(Float64(c * t) - Float64(i * y))));
	elseif (j <= 4.9e+51)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(Float64(i * b) * a));
	elseif (j <= 1.4e+148)
		tmp = fma(Float64(y * z), x, Float64(fma(Float64(-y), j, Float64(b * a)) * i));
	else
		tmp = Float64(Float64(fma(c, Float64(t / i), Float64(-y)) * i) * j);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1.02e-17], N[(N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 4.9e+51], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.4e+148], N[(N[(y * z), $MachinePrecision] * x + N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(t / i), $MachinePrecision] + (-y)), $MachinePrecision] * i), $MachinePrecision] * j), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.01999999999999997e-17

   1. Initial program 78.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

         \[\leadsto \left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. lower-*.f64 76.5

         \[\leadsto \left(z \cdot x\right) \cdot y + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Applied rewrites 76.5%

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

   if -1.01999999999999997e-17 < j < 4.89999999999999983e51

   1. Initial program 75.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if 4.89999999999999983e51 < j < 1.3999999999999999e148

   1. Initial program 89.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 78.5

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

   8. Applied rewrites 78.5%

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

   if 1.3999999999999999e148 < j

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 78.6

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in i around inf

      \[\leadsto \left(i \cdot \left(-1 \cdot y + \frac{c \cdot t}{i}\right)\right) \cdot j \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{c \cdot t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      4. associate-/l* N/A

         \[\leadsto \left(\left(c \cdot \frac{t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, \mathsf{neg}\left(y\right)\right) \cdot i\right) \cdot j \]

      8. lower-neg.f64 81.7

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]

   8. Applied rewrites 81.7%

      \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{elif}\;j \leq 4.9 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{elif}\;j \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1e-14)
   (* (fma (- i) y (* c t)) j)
   (if (<= j 4.9e+51)
     (fma (fma (- t) a (* z y)) x (* (* i b) a))
     (if (<= j 1.4e+148)
       (fma (* y z) x (* (fma (- y) j (* b a)) i))
       (* (* (fma c (/ t i) (- y)) i) j)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1e-14) {
		tmp = fma(-i, y, (c * t)) * j;
	} else if (j <= 4.9e+51) {
		tmp = fma(fma(-t, a, (z * y)), x, ((i * b) * a));
	} else if (j <= 1.4e+148) {
		tmp = fma((y * z), x, (fma(-y, j, (b * a)) * i));
	} else {
		tmp = (fma(c, (t / i), -y) * i) * j;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1e-14)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	elseif (j <= 4.9e+51)
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(Float64(i * b) * a));
	elseif (j <= 1.4e+148)
		tmp = fma(Float64(y * z), x, Float64(fma(Float64(-y), j, Float64(b * a)) * i));
	else
		tmp = Float64(Float64(fma(c, Float64(t / i), Float64(-y)) * i) * j);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1e-14], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[j, 4.9e+51], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.4e+148], N[(N[(y * z), $MachinePrecision] * x + N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(t / i), $MachinePrecision] + (-y)), $MachinePrecision] * i), $MachinePrecision] * j), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -9.99999999999999999e-15

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 67.7

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   if -9.99999999999999999e-15 < j < 4.89999999999999983e51

   1. Initial program 75.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.8

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

   8. Applied rewrites 65.8%

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

   if 4.89999999999999983e51 < j < 1.3999999999999999e148

   1. Initial program 89.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 78.5

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

   8. Applied rewrites 78.5%

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

   if 1.3999999999999999e148 < j

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 78.6

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in i around inf

      \[\leadsto \left(i \cdot \left(-1 \cdot y + \frac{c \cdot t}{i}\right)\right) \cdot j \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{c \cdot t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      4. associate-/l* N/A

         \[\leadsto \left(\left(c \cdot \frac{t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, \mathsf{neg}\left(y\right)\right) \cdot i\right) \cdot j \]

      8. lower-neg.f64 81.7

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]

   8. Applied rewrites 81.7%

      \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* z y)) x)))
   (if (<= x -2.85e+84)
     t_1
     (if (<= x -1.9e-36)
       (* (fma (- y) j (* b a)) i)
       (if (<= x 8.5e-27)
         (* (fma (- i) y (* c t)) j)
         (if (<= x 2.7e+64) (* (fma (- z) c (* i a)) b) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y)) * x;
	double tmp;
	if (x <= -2.85e+84) {
		tmp = t_1;
	} else if (x <= -1.9e-36) {
		tmp = fma(-y, j, (b * a)) * i;
	} else if (x <= 8.5e-27) {
		tmp = fma(-i, y, (c * t)) * j;
	} else if (x <= 2.7e+64) {
		tmp = fma(-z, c, (i * a)) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -2.85e+84)
		tmp = t_1;
	elseif (x <= -1.9e-36)
		tmp = Float64(fma(Float64(-y), j, Float64(b * a)) * i);
	elseif (x <= 8.5e-27)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	elseif (x <= 2.7e+64)
		tmp = Float64(fma(Float64(-z), c, Float64(i * a)) * b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.85e+84], t$95$1, If[LessEqual[x, -1.9e-36], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[x, 8.5e-27], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[x, 2.7e+64], N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.84999999999999985e84 or 2.7e64 < x

   1. Initial program 78.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -2.84999999999999985e84 < x < -1.89999999999999985e-36

   1. Initial program 54.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{i} \]

      2. distribute-lft-out-- N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y - a \cdot b\right)\right) \cdot i \]

      3. associate-*r* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(j \cdot y - a \cdot b\right) \cdot i\right)} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \left(i \cdot \color{blue}{\left(j \cdot y - a \cdot b\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \left(\left(j \cdot y - a \cdot b\right) \cdot \color{blue}{i}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y - a \cdot b\right)\right) \cdot \color{blue}{i} \]

      7. distribute-lft-out-- N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{i} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   if -1.89999999999999985e-36 < x < 8.50000000000000033e-27

   1. Initial program 81.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 57.0

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   if 8.50000000000000033e-27 < x < 2.7e64

   1. Initial program 99.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

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

         \[\leadsto \left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c\right)\right) \cdot z + a \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right) \cdot b \]

      5. *-lft-identity N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(1 \cdot a\right) \cdot i\right) \cdot b \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot a\right) \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot a\right)\right) \cdot i\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)\right) \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right) \cdot i\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot a\right)\right)\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot a\right)\right)\right) \cdot b \]

      13. distribute-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i\right)\right) \cdot a\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z - i \cdot a\right)\right)\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b \]

      16. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b \]

      17. lower-*.f64 N/A

          \[\leadsto \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot \color{blue}{b} \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-142}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* z y)) x)))
   (if (<= x -4.8e+106)
     t_1
     (if (<= x -3.2e+14)
       (* (fma (- b) c (* y x)) z)
       (if (<= x -5.7e-142)
         (* (fma (- a) x (* j c)) t)
         (if (<= x 6.5e+15) (* (fma (- i) y (* c t)) j) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y)) * x;
	double tmp;
	if (x <= -4.8e+106) {
		tmp = t_1;
	} else if (x <= -3.2e+14) {
		tmp = fma(-b, c, (y * x)) * z;
	} else if (x <= -5.7e-142) {
		tmp = fma(-a, x, (j * c)) * t;
	} else if (x <= 6.5e+15) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -4.8e+106)
		tmp = t_1;
	elseif (x <= -3.2e+14)
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	elseif (x <= -5.7e-142)
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	elseif (x <= 6.5e+15)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.8e+106], t$95$1, If[LessEqual[x, -3.2e+14], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[x, -5.7e-142], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[x, 6.5e+15], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.8000000000000001e106 or 6.5e15 < x

   1. Initial program 82.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -4.8000000000000001e106 < x < -3.2e14

   1. Initial program 53.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot y + \left(\mathsf{neg}\left(b\right)\right) \cdot c\right) \cdot z \]

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

         \[\leadsto \left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right) \cdot z \]

      5. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right) \cdot z \]

      6. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b \cdot c\right)\right) + x \cdot y\right) \cdot z \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + x \cdot y\right) \cdot z \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(b\right), c, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if -3.2e14 < x < -5.69999999999999995e-142

   1. Initial program 70.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

   5. Applied rewrites 54.1%

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

   if -5.69999999999999995e-142 < x < 6.5e15

   1. Initial program 81.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 56.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -5.2 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{elif}\;j \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -5.2e+73)
   (* (fma (- i) y (* c t)) j)
   (if (<= j 1.4e+148)
     (fma (* y z) x (* (fma (- y) j (* b a)) i))
     (* (* (fma c (/ t i) (- y)) i) j))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -5.2e+73) {
		tmp = fma(-i, y, (c * t)) * j;
	} else if (j <= 1.4e+148) {
		tmp = fma((y * z), x, (fma(-y, j, (b * a)) * i));
	} else {
		tmp = (fma(c, (t / i), -y) * i) * j;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -5.2e+73)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	elseif (j <= 1.4e+148)
		tmp = fma(Float64(y * z), x, Float64(fma(Float64(-y), j, Float64(b * a)) * i));
	else
		tmp = Float64(Float64(fma(c, Float64(t / i), Float64(-y)) * i) * j);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -5.2e+73], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[j, 1.4e+148], N[(N[(y * z), $MachinePrecision] * x + N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(t / i), $MachinePrecision] + (-y)), $MachinePrecision] * i), $MachinePrecision] * j), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -5.2000000000000001e73

   1. Initial program 76.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 72.2

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   if -5.2000000000000001e73 < j < 1.3999999999999999e148

   1. Initial program 78.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 59.5

         \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

   8. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(y \cdot z, x, \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\right) \]

   if 1.3999999999999999e148 < j

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 78.6

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]

   6. Taylor expanded in i around inf

      \[\leadsto \left(i \cdot \left(-1 \cdot y + \frac{c \cdot t}{i}\right)\right) \cdot j \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot y + \frac{c \cdot t}{i}\right) \cdot i\right) \cdot j \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{c \cdot t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      4. associate-/l* N/A

         \[\leadsto \left(\left(c \cdot \frac{t}{i} + -1 \cdot y\right) \cdot i\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -1 \cdot y\right) \cdot i\right) \cdot j \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, \mathsf{neg}\left(y\right)\right) \cdot i\right) \cdot j \]

      8. lower-neg.f64 81.7

         \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]

   8. Applied rewrites 81.7%

      \[\leadsto \left(\mathsf{fma}\left(c, \frac{t}{i}, -y\right) \cdot i\right) \cdot j \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 53.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* z y)) x)))
   (if (<= x -2.85e+84)
     t_1
     (if (<= x -1.9e-36)
       (* (fma (- y) j (* b a)) i)
       (if (<= x 6.5e+15) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y)) * x;
	double tmp;
	if (x <= -2.85e+84) {
		tmp = t_1;
	} else if (x <= -1.9e-36) {
		tmp = fma(-y, j, (b * a)) * i;
	} else if (x <= 6.5e+15) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -2.85e+84)
		tmp = t_1;
	elseif (x <= -1.9e-36)
		tmp = Float64(fma(Float64(-y), j, Float64(b * a)) * i);
	elseif (x <= 6.5e+15)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.85e+84], t$95$1, If[LessEqual[x, -1.9e-36], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[x, 6.5e+15], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.84999999999999985e84 or 6.5e15 < x

   1. Initial program 80.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -2.84999999999999985e84 < x < -1.89999999999999985e-36

   1. Initial program 54.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{i} \]

      2. distribute-lft-out-- N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y - a \cdot b\right)\right) \cdot i \]

      3. associate-*r* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(j \cdot y - a \cdot b\right) \cdot i\right)} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \left(i \cdot \color{blue}{\left(j \cdot y - a \cdot b\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \left(\left(j \cdot y - a \cdot b\right) \cdot \color{blue}{i}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y - a \cdot b\right)\right) \cdot \color{blue}{i} \]

      7. distribute-lft-out-- N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{i} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   if -1.89999999999999985e-36 < x < 6.5e15

   1. Initial program 81.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 56.5

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 53.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, z \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* z y)) x)))
   (if (<= x -3.4e+84)
     t_1
     (if (<= x -3e-36)
       (* (fma (- t) x (* i b)) a)
       (if (<= x 6.5e+15) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-t, a, (z * y)) * x;
	double tmp;
	if (x <= -3.4e+84) {
		tmp = t_1;
	} else if (x <= -3e-36) {
		tmp = fma(-t, x, (i * b)) * a;
	} else if (x <= 6.5e+15) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(z * y)) * x)
	tmp = 0.0
	if (x <= -3.4e+84)
		tmp = t_1;
	elseif (x <= -3e-36)
		tmp = Float64(fma(Float64(-t), x, Float64(i * b)) * a);
	elseif (x <= 6.5e+15)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.4e+84], t$95$1, If[LessEqual[x, -3e-36], N[(N[((-t) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 6.5e+15], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3999999999999998e84 or 6.5e15 < x

   1. Initial program 80.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -3.3999999999999998e84 < x < -3.0000000000000002e-36

   1. Initial program 54.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot \color{blue}{a} \]

      3. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) - \left(b \cdot i\right) \cdot -1\right) \cdot a \]

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

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(b \cdot i\right)\right) \cdot -1\right) \cdot a \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot t\right) \cdot x + \left(\mathsf{neg}\left(b \cdot i\right)\right) \cdot -1\right) \cdot a \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + \left(\mathsf{neg}\left(b \cdot i\right)\right) \cdot -1\right) \cdot a \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + \left(\mathsf{neg}\left(\left(b \cdot i\right) \cdot -1\right)\right)\right) \cdot a \]

      8. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \cdot a \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)\right) \cdot a \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + 1 \cdot \left(b \cdot i\right)\right) \cdot a \]

      11. *-lft-identity N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + b \cdot i\right) \cdot a \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(t\right), x, b \cdot i\right) \cdot a \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-t, x, b \cdot i\right) \cdot a \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a \]

      15. lower-*.f64 55.4

          \[\leadsto \mathsf{fma}\left(-t, x, i \cdot b\right) \cdot a \]

   5. Applied rewrites 55.4%

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

   if -3.0000000000000002e-36 < x < 6.5e15

   1. Initial program 81.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 56.5

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 51.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -9 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -9e+94)
     t_1
     (if (<= z -1.3e-56)
       (* (fma (- a) x (* j c)) t)
       (if (<= z 7.6e+56) (* (fma (- i) y (* c t)) j) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-b, c, (y * x)) * z;
	double tmp;
	if (z <= -9e+94) {
		tmp = t_1;
	} else if (z <= -1.3e-56) {
		tmp = fma(-a, x, (j * c)) * t;
	} else if (z <= 7.6e+56) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-b), c, Float64(y * x)) * z)
	tmp = 0.0
	if (z <= -9e+94)
		tmp = t_1;
	elseif (z <= -1.3e-56)
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	elseif (z <= 7.6e+56)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -9e+94], t$95$1, If[LessEqual[z, -1.3e-56], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 7.6e+56], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.99999999999999944e94 or 7.59999999999999991e56 < z

   1. Initial program 70.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot y + \left(\mathsf{neg}\left(b\right)\right) \cdot c\right) \cdot z \]

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

         \[\leadsto \left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right) \cdot z \]

      5. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right) \cdot z \]

      6. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b \cdot c\right)\right) + x \cdot y\right) \cdot z \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + x \cdot y\right) \cdot z \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(b\right), c, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if -8.99999999999999944e94 < z < -1.29999999999999998e-56

   1. Initial program 76.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

   5. Applied rewrites 57.7%

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

   if -1.29999999999999998e-56 < z < 7.59999999999999991e56

   1. Initial program 85.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

         \[\leadsto \left(c \cdot t + \left(\mathsf{neg}\left(i\right)\right) \cdot y\right) \cdot j \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot y + c \cdot t\right) \cdot j \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), y, c \cdot t\right) \cdot j \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

      7. lower-*.f64 54.4

         \[\leadsto \mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j \]

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+94} \lor \neg \left(z \leq 1.25 \cdot 10^{+44}\right):\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -9e+94) (not (<= z 1.25e+44)))
   (* (fma (- b) c (* y x)) z)
   (* (fma (- a) x (* j c)) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -9e+94) || !(z <= 1.25e+44)) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = fma(-a, x, (j * c)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -9e+94) || !(z <= 1.25e+44))
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -9e+94], N[Not[LessEqual[z, 1.25e+44]], $MachinePrecision]], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999944e94 or 1.2499999999999999e44 < z

   1. Initial program 68.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

         \[\leadsto \left(x \cdot y + \left(\mathsf{neg}\left(b\right)\right) \cdot c\right) \cdot z \]

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

         \[\leadsto \left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right) \cdot z \]

      5. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right) \cdot z \]

      6. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b \cdot c\right)\right) + x \cdot y\right) \cdot z \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + x \cdot y\right) \cdot z \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(b\right), c, x \cdot y\right) \cdot z \]

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if -8.99999999999999944e94 < z < 1.2499999999999999e44

   1. Initial program 84.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

   5. Applied rewrites 47.6%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+94} \lor \neg \left(z \leq 1.25 \cdot 10^{+44}\right):\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.15 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\left(y \cdot z\right) \cdot x\\ \mathbf{elif}\;c \leq 7 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-t\right) \cdot a\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -2.15e+107)
   (* (* (- b) c) z)
   (if (<= c 5e-17)
     (* (* y z) x)
     (if (<= c 7e+100) (* (* (- t) a) x) (* (* j t) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -2.15e+107) {
		tmp = (-b * c) * z;
	} else if (c <= 5e-17) {
		tmp = (y * z) * x;
	} else if (c <= 7e+100) {
		tmp = (-t * a) * x;
	} else {
		tmp = (j * t) * c;
	}
	return tmp;
}

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, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (c <= (-2.15d+107)) then
        tmp = (-b * c) * z
    else if (c <= 5d-17) then
        tmp = (y * z) * x
    else if (c <= 7d+100) then
        tmp = (-t * a) * x
    else
        tmp = (j * t) * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -2.15e+107) {
		tmp = (-b * c) * z;
	} else if (c <= 5e-17) {
		tmp = (y * z) * x;
	} else if (c <= 7e+100) {
		tmp = (-t * a) * x;
	} else {
		tmp = (j * t) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if c <= -2.15e+107:
		tmp = (-b * c) * z
	elif c <= 5e-17:
		tmp = (y * z) * x
	elif c <= 7e+100:
		tmp = (-t * a) * x
	else:
		tmp = (j * t) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -2.15e+107)
		tmp = Float64(Float64(Float64(-b) * c) * z);
	elseif (c <= 5e-17)
		tmp = Float64(Float64(y * z) * x);
	elseif (c <= 7e+100)
		tmp = Float64(Float64(Float64(-t) * a) * x);
	else
		tmp = Float64(Float64(j * t) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (c <= -2.15e+107)
		tmp = (-b * c) * z;
	elseif (c <= 5e-17)
		tmp = (y * z) * x;
	elseif (c <= 7e+100)
		tmp = (-t * a) * x;
	else
		tmp = (j * t) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -2.15e+107], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[c, 5e-17], N[(N[(y * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[c, 7e+100], N[(N[((-t) * a), $MachinePrecision] * x), $MachinePrecision], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.15e107

   1. Initial program 64.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

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

         \[\leadsto \left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c\right)\right) \cdot z + a \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + a \cdot i\right) \cdot b \]

      5. *-lft-identity N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(1 \cdot a\right) \cdot i\right) \cdot b \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot a\right) \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(-1 \cdot a\right)\right) \cdot i\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right)\right)\right) \cdot i\right) \cdot b \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a\right)\right) \cdot i\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(a \cdot i\right)\right)\right)\right)\right) \cdot b \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot a\right)\right)\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right) \cdot a\right)\right)\right) \cdot b \]

      13. distribute-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i\right)\right) \cdot a\right)\right)\right) \cdot b \]

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

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z - i \cdot a\right)\right)\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(c \cdot z - a \cdot i\right)\right)\right) \cdot b \]

      16. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot b \]

      17. lower-*.f64 N/A

          \[\leadsto \left(-1 \cdot \left(c \cdot z - a \cdot i\right)\right) \cdot \color{blue}{b} \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, c, i \cdot a\right) \cdot b} \]

   6. Taylor expanded in z around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(b \cdot \left(c \cdot z\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\left(b \cdot c\right) \cdot z\right) \]

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

         \[\leadsto \left(\mathsf{neg}\left(b \cdot c\right)\right) \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot c\right)\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot c\right)\right) \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot b\right) \cdot c\right) \cdot z \]

      7. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c\right) \cdot z \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c\right) \cdot z \]

      9. lower-neg.f64 51.8

         \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]

   8. Applied rewrites 51.8%

      \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot \color{blue}{z} \]

   if -2.15e107 < c < 4.9999999999999999e-17

   1. Initial program 85.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

      1. lower-*.f64 31.8

         \[\leadsto \left(y \cdot z\right) \cdot x \]

   8. Applied rewrites 31.8%

      \[\leadsto \left(y \cdot z\right) \cdot x \]

   if 4.9999999999999999e-17 < c < 6.99999999999999953e100

   1. Initial program 73.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(-1 \cdot \left(a \cdot t\right)\right) \cdot x \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a \cdot t\right)\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(t \cdot a\right)\right) \cdot x \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot a\right) \cdot x \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot t\right) \cdot a\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot t\right) \cdot a\right) \cdot x \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot a\right) \cdot x \]

      7. lower-neg.f64 37.0

         \[\leadsto \left(\left(-t\right) \cdot a\right) \cdot x \]

   8. Applied rewrites 37.0%

      \[\leadsto \left(\left(-t\right) \cdot a\right) \cdot x \]

   if 6.99999999999999953e100 < c

   1. Initial program 73.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(j \cdot t\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(j \cdot t\right) \cdot c \]

      3. lower-*.f64 61.5

         \[\leadsto \left(j \cdot t\right) \cdot c \]

   8. Applied rewrites 61.5%

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

Alternative 22: 42.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.05 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;i \leq 6.7 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.05e+69)
   (* (* (- i) j) y)
   (if (<= i 6.7e+156) (* (fma (- a) x (* j c)) t) (* (* b a) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.05e+69) {
		tmp = (-i * j) * y;
	} else if (i <= 6.7e+156) {
		tmp = fma(-a, x, (j * c)) * t;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.05e+69)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (i <= 6.7e+156)
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	else
		tmp = Float64(Float64(b * a) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.05e+69], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[i, 6.7e+156], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.05000000000000008e69

   1. Initial program 66.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in j around inf

      \[\leadsto -1 \cdot \color{blue}{\left(i \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(i\right)\right) \cdot \left(j \cdot y\right) \]

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

         \[\leadsto \left(-i\right) \cdot \left(j \cdot y\right) \]

      5. lower-*.f64 51.1

         \[\leadsto \left(-i\right) \cdot \left(j \cdot y\right) \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(-i\right) \cdot \color{blue}{\left(j \cdot y\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-i\right) \cdot \left(j \cdot \color{blue}{y}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-i\right) \cdot \left(j \cdot y\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(-i\right) \cdot j\right) \cdot y \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-i\right) \cdot j\right) \cdot y \]

      5. lower-*.f64 55.0

         \[\leadsto \left(\left(-i\right) \cdot j\right) \cdot y \]

   10. Applied rewrites 55.0%

       \[\leadsto \color{blue}{\left(\left(-i\right) \cdot j\right) \cdot y} \]

   if -1.05000000000000008e69 < i < 6.7e156

   1. Initial program 83.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

   5. Applied rewrites 46.9%

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

   if 6.7e156 < i

   1. Initial program 64.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      4. lower-*.f64 37.9

         \[\leadsto \left(i \cdot b\right) \cdot a \]

   8. Applied rewrites 37.9%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      2. lift-*.f64 N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      3. associate-*l* N/A

         \[\leadsto i \cdot \left(b \cdot \color{blue}{a}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot i \]

      5. lower-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot i \]

      6. lower-*.f64 49.1

         \[\leadsto \left(b \cdot a\right) \cdot i \]

   10. Applied rewrites 49.1%

       \[\leadsto \left(b \cdot a\right) \cdot i \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 31.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot x\right) \cdot y\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-36}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z x) y)))
   (if (<= x -3.5e+84)
     t_1
     (if (<= x -3.1e-36)
       (* (* i b) a)
       (if (<= x 3.6e+15) (* (* j t) c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (x <= -3.5e+84) {
		tmp = t_1;
	} else if (x <= -3.1e-36) {
		tmp = (i * b) * a;
	} else if (x <= 3.6e+15) {
		tmp = (j * t) * c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * x) * y
    if (x <= (-3.5d+84)) then
        tmp = t_1
    else if (x <= (-3.1d-36)) then
        tmp = (i * b) * a
    else if (x <= 3.6d+15) then
        tmp = (j * t) * c
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (z * x) * y;
	double tmp;
	if (x <= -3.5e+84) {
		tmp = t_1;
	} else if (x <= -3.1e-36) {
		tmp = (i * b) * a;
	} else if (x <= 3.6e+15) {
		tmp = (j * t) * c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * x) * y
	tmp = 0
	if x <= -3.5e+84:
		tmp = t_1
	elif x <= -3.1e-36:
		tmp = (i * b) * a
	elif x <= 3.6e+15:
		tmp = (j * t) * c
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * x) * y)
	tmp = 0.0
	if (x <= -3.5e+84)
		tmp = t_1;
	elseif (x <= -3.1e-36)
		tmp = Float64(Float64(i * b) * a);
	elseif (x <= 3.6e+15)
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * x) * y;
	tmp = 0.0;
	if (x <= -3.5e+84)
		tmp = t_1;
	elseif (x <= -3.1e-36)
		tmp = (i * b) * a;
	elseif (x <= 3.6e+15)
		tmp = (j * t) * c;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x, -3.5e+84], t$95$1, If[LessEqual[x, -3.1e-36], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 3.6e+15], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999999e84 or 3.6e15 < x

   1. Initial program 80.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i \cdot j\right)\right) + x \cdot z\right) \cdot y \]

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

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot j + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right) \cdot y \]

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z \cdot x\right) \cdot y \]

      2. lower-*.f64 46.6

         \[\leadsto \left(z \cdot x\right) \cdot y \]

   8. Applied rewrites 46.6%

      \[\leadsto \left(z \cdot x\right) \cdot y \]

   if -3.4999999999999999e84 < x < -3.0999999999999999e-36

   1. Initial program 54.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      4. lower-*.f64 38.3

         \[\leadsto \left(i \cdot b\right) \cdot a \]

   8. Applied rewrites 38.3%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   if -3.0999999999999999e-36 < x < 3.6e15

   1. Initial program 81.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(j \cdot t\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(j \cdot t\right) \cdot c \]

      3. lower-*.f64 36.9

         \[\leadsto \left(j \cdot t\right) \cdot c \]

   8. Applied rewrites 36.9%

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

Alternative 24: 30.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+70} \lor \neg \left(a \leq 5.7 \cdot 10^{-61}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= a -1e+70) (not (<= a 5.7e-61))) (* (* b a) i) (* (* j t) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((a <= -1e+70) || !(a <= 5.7e-61)) {
		tmp = (b * a) * i;
	} else {
		tmp = (j * t) * c;
	}
	return tmp;
}

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, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if ((a <= (-1d+70)) .or. (.not. (a <= 5.7d-61))) then
        tmp = (b * a) * i
    else
        tmp = (j * t) * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((a <= -1e+70) || !(a <= 5.7e-61)) {
		tmp = (b * a) * i;
	} else {
		tmp = (j * t) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (a <= -1e+70) or not (a <= 5.7e-61):
		tmp = (b * a) * i
	else:
		tmp = (j * t) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((a <= -1e+70) || !(a <= 5.7e-61))
		tmp = Float64(Float64(b * a) * i);
	else
		tmp = Float64(Float64(j * t) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((a <= -1e+70) || ~((a <= 5.7e-61)))
		tmp = (b * a) * i;
	else
		tmp = (j * t) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[a, -1e+70], N[Not[LessEqual[a, 5.7e-61]], $MachinePrecision]], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000007e70 or 5.70000000000000005e-61 < a

   1. Initial program 72.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in b around inf

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot i\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      4. lower-*.f64 38.1

         \[\leadsto \left(i \cdot b\right) \cdot a \]

   8. Applied rewrites 38.1%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      2. lift-*.f64 N/A

         \[\leadsto \left(i \cdot b\right) \cdot a \]

      3. associate-*l* N/A

         \[\leadsto i \cdot \left(b \cdot \color{blue}{a}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot i \]

      5. lower-*.f64 N/A

         \[\leadsto \left(b \cdot a\right) \cdot i \]

      6. lower-*.f64 39.7

         \[\leadsto \left(b \cdot a\right) \cdot i \]

   10. Applied rewrites 39.7%

       \[\leadsto \left(b \cdot a\right) \cdot i \]

   if -1.00000000000000007e70 < a < 5.70000000000000005e-61

   1. Initial program 84.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(j \cdot t\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(j \cdot t\right) \cdot c \]

      3. lower-*.f64 31.4

         \[\leadsto \left(j \cdot t\right) \cdot c \]

   8. Applied rewrites 31.4%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+70} \lor \neg \left(a \leq 5.7 \cdot 10^{-61}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 22.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(j \cdot t\right) \cdot c \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* j t) c))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (j * t) * c;
}

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, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (j * t) * c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (j * t) * c;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (j * t) * c

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(j * t) * c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (j * t) * c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 78.4%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

5. Applied rewrites 39.4%

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

6. Taylor expanded in x around 0

   \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(j \cdot t\right) \cdot c \]

   2. lower-*.f64 N/A

      \[\leadsto \left(j \cdot t\right) \cdot c \]

   3. lower-*.f64 26.6

      \[\leadsto \left(j \cdot t\right) \cdot c \]

8. Applied rewrites 26.6%

   \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]
9. Add Preprocessing

Developer Target 1: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

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, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2025026 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))