Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 74.2% → 82.5%

Time: 14.5s

Alternatives: 24

Speedup: 0.5×

• 57.6% 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: 74.2% 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: 82.5% accurate, 0.3× speedup

Math

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

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)
      (fma
       a
       x
       (-
        (fma
         j
         c
         (/ (fma (fma (- z) c (* i a)) b (* (fma (- i) j (* z x)) y)) t)))))
     (if (<= t_1 INFINITY) t_1 (* (fma (- x) t (* i b)) a)))))

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 * fma(a, x, -fma(j, c, (fma(fma(-z, c, (i * a)), b, (fma(-i, j, (z * x)) * y)) / t)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(-x, t, (i * b)) * a;
	}
	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 <= Float64(-Inf))
		tmp = Float64(Float64(-t) * fma(a, x, Float64(-fma(j, c, Float64(fma(fma(Float64(-z), c, Float64(i * a)), b, Float64(fma(Float64(-i), j, Float64(z * x)) * y)) / t)))));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	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)], N[((-t) * N[(a * x + (-N[(j * c + N[(N[(N[((-z) * c + N[(i * a), $MachinePrecision]), $MachinePrecision] * b + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $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)))) < -inf.0

   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

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

   5. Applied rewrites 92.7%

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

   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)))) < +inf.0

   1. Initial program 95.9%

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

   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 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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \cdot i\right) \cdot a \]

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \color{blue}{b \cdot i}\right) \cdot a \]

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

Alternative 2: 82.5% accurate, 0.5× speedup

Math

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

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 (- x) t (* i b)) a))))

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(-x, t, (i * b)) * a;
	}
	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(-x), t, Float64(i * b)) * a);
	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[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $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 92.6%

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

   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 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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \cdot i\right) \cdot a \]

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \color{blue}{b \cdot i}\right) \cdot a \]

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

Alternative 3: 82.5% accurate, 0.5× speedup

Math

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

FPCore

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

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[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], Infinity], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[((-i) * a + N[(c * z), $MachinePrecision]), $MachinePrecision] + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $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 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 92.6%

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

   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 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

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \cdot i\right) \cdot a \]

      9. mul-1-neg N/A

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

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

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \color{blue}{b \cdot i}\right) \cdot a \]

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 52.7

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

   5. Applied rewrites 52.7%

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

Alternative 4: 71.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{-15} \lor \neg \left(c \leq 4.5 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot t\right), c, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) t (* z y)) x)))
   (if (or (<= c -2.7e-15) (not (<= c 4.5e-75)))
     (fma (fma (- z) b (* j t)) c t_1)
     (fma (fma (- y) j (* b a)) i 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(-a, t, (z * y)) * x;
	double tmp;
	if ((c <= -2.7e-15) || !(c <= 4.5e-75)) {
		tmp = fma(fma(-z, b, (j * t)), c, t_1);
	} else {
		tmp = fma(fma(-y, j, (b * a)), i, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), t, Float64(z * y)) * x)
	tmp = 0.0
	if ((c <= -2.7e-15) || !(c <= 4.5e-75))
		tmp = fma(fma(Float64(-z), b, Float64(j * t)), c, t_1);
	else
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[Or[LessEqual[c, -2.7e-15], N[Not[LessEqual[c, 4.5e-75]], $MachinePrecision]], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c + t$95$1), $MachinePrecision], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.70000000000000009e-15 or 4.5000000000000003e-75 < c

   1. Initial program 66.1%

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

   3. Taylor expanded in i around 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 76.3%

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

   if -2.70000000000000009e-15 < c < 4.5000000000000003e-75

   1. Initial program 79.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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 75.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-15} \lor \neg \left(c \leq 4.5 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-z, b, j \cdot t\right), c, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+247}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-y, j, b \cdot a\right), i, \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(i \cdot b\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -5.2e+247)
   (* (fma i a (* (- z) c)) b)
   (if (<= b 1.55e+56)
     (fma (fma (- y) j (* b a)) i (* (fma (- a) t (* z y)) x))
     (fma (fma (- i) y (* c t)) j (* (* i b) a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -5.2e+247) {
		tmp = fma(i, a, (-z * c)) * b;
	} else if (b <= 1.55e+56) {
		tmp = fma(fma(-y, j, (b * a)), i, (fma(-a, t, (z * y)) * x));
	} else {
		tmp = fma(fma(-i, y, (c * t)), j, ((i * b) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -5.2e+247)
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	elseif (b <= 1.55e+56)
		tmp = fma(fma(Float64(-y), j, Float64(b * a)), i, Float64(fma(Float64(-a), t, Float64(z * y)) * x));
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(Float64(i * b) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -5.2e+247], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, 1.55e+56], N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.19999999999999981e247

   1. Initial program 66.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 93.8

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

   11. Applied rewrites 93.8%

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

   if -5.19999999999999981e247 < b < 1.55000000000000002e56

   1. Initial program 72.9%

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

   3. Taylor expanded in 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%

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

   if 1.55000000000000002e56 < b

   1. Initial program 71.7%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.4

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 74.4

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

   7. Applied rewrites 74.4%

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

Alternative 6: 52.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, a, y \cdot z\right) \cdot x\\ t_2 := \mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i\\ \mathbf{if}\;i \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq -4.2 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq -1.1 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(-z, b, j \cdot t\right) \cdot c\\ \mathbf{elif}\;i \leq 9 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- t) a (* y z)) x)) (t_2 (* (fma (- y) j (* b a)) i)))
   (if (<= i -5.4e-5)
     t_2
     (if (<= i -4.2e-73)
       t_1
       (if (<= i -1.1e-260)
         (* (fma (- z) b (* j t)) c)
         (if (<= i 9e-82)
           t_1
           (if (<= i 1.6e+77) (* (fma (- a) x (* j c)) t) 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, (y * z)) * x;
	double t_2 = fma(-y, j, (b * a)) * i;
	double tmp;
	if (i <= -5.4e-5) {
		tmp = t_2;
	} else if (i <= -4.2e-73) {
		tmp = t_1;
	} else if (i <= -1.1e-260) {
		tmp = fma(-z, b, (j * t)) * c;
	} else if (i <= 9e-82) {
		tmp = t_1;
	} else if (i <= 1.6e+77) {
		tmp = fma(-a, x, (j * c)) * t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-t), a, Float64(y * z)) * x)
	t_2 = Float64(fma(Float64(-y), j, Float64(b * a)) * i)
	tmp = 0.0
	if (i <= -5.4e-5)
		tmp = t_2;
	elseif (i <= -4.2e-73)
		tmp = t_1;
	elseif (i <= -1.1e-260)
		tmp = Float64(fma(Float64(-z), b, Float64(j * t)) * c);
	elseif (i <= 9e-82)
		tmp = t_1;
	elseif (i <= 1.6e+77)
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-t) * a + N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-y) * j + N[(b * a), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -5.4e-5], t$95$2, If[LessEqual[i, -4.2e-73], t$95$1, If[LessEqual[i, -1.1e-260], N[(N[((-z) * b + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[i, 9e-82], t$95$1, If[LessEqual[i, 1.6e+77], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -5.3999999999999998e-5 or 1.6000000000000001e77 < i

   1. Initial program 65.0%

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

   3. Taylor expanded in i around 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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if -5.3999999999999998e-5 < i < -4.1999999999999997e-73 or -1.10000000000000008e-260 < i < 8.9999999999999997e-82

   1. Initial program 83.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 46.5

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

   5. Applied rewrites 46.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   if -4.1999999999999997e-73 < i < -1.10000000000000008e-260

   1. Initial program 74.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   if 8.9999999999999997e-82 < i < 1.6000000000000001e77

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 56.3

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

   5. Applied rewrites 56.3%

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

Alternative 7: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -7.5 \cdot 10^{-7} \lor \neg \left(i \leq 8.6 \cdot 10^{-122}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, b \cdot \left(\left(-z\right) \cdot c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= i -7.5e-7) (not (<= i 8.6e-122)))
   (fma (fma (- i) y (* c t)) j (* (* i b) a))
   (fma (fma (- t) a (* z y)) x (* b (* (- z) 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 ((i <= -7.5e-7) || !(i <= 8.6e-122)) {
		tmp = fma(fma(-i, y, (c * t)), j, ((i * b) * a));
	} else {
		tmp = fma(fma(-t, a, (z * y)), x, (b * (-z * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((i <= -7.5e-7) || !(i <= 8.6e-122))
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(Float64(i * b) * a));
	else
		tmp = fma(fma(Float64(-t), a, Float64(z * y)), x, Float64(b * Float64(Float64(-z) * c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[i, -7.5e-7], N[Not[LessEqual[i, 8.6e-122]], $MachinePrecision]], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * a + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(b * N[((-z) * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -7.5000000000000002e-7 or 8.60000000000000037e-122 < i

   1. Initial program 67.8%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.3

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 66.9

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

   7. Applied rewrites 66.9%

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

   if -7.5000000000000002e-7 < i < 8.60000000000000037e-122

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.1

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

   7. Applied rewrites 68.1%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -7.5 \cdot 10^{-7} \lor \neg \left(i \leq 8.6 \cdot 10^{-122}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-t, a, z \cdot y\right), x, b \cdot \left(\left(-z\right) \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-265}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- b) c (* y x)) z)))
   (if (<= z -5.2e+30)
     t_1
     (if (<= z -5.8e-207)
       (* (fma (- y) i (* c t)) j)
       (if (<= z -3.5e-265)
         (* (* a b) i)
         (if (<= z 2.35e+86) (* (fma (- a) x (* j c)) t) 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 <= -5.2e+30) {
		tmp = t_1;
	} else if (z <= -5.8e-207) {
		tmp = fma(-y, i, (c * t)) * j;
	} else if (z <= -3.5e-265) {
		tmp = (a * b) * i;
	} else if (z <= 2.35e+86) {
		tmp = fma(-a, x, (j * c)) * t;
	} 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 <= -5.2e+30)
		tmp = t_1;
	elseif (z <= -5.8e-207)
		tmp = Float64(fma(Float64(-y), i, Float64(c * t)) * j);
	elseif (z <= -3.5e-265)
		tmp = Float64(Float64(a * b) * i);
	elseif (z <= 2.35e+86)
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	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, -5.2e+30], t$95$1, If[LessEqual[z, -5.8e-207], N[(N[((-y) * i + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[z, -3.5e-265], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[z, 2.35e+86], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.19999999999999977e30 or 2.3500000000000001e86 < z

   1. Initial program 60.6%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -5.19999999999999977e30 < z < -5.80000000000000022e-207

   1. Initial program 79.5%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 58.2

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 58.2

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 47.5

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

   10. Applied rewrites 47.5%

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

   if -5.80000000000000022e-207 < z < -3.50000000000000015e-265

   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 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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

   if -3.50000000000000015e-265 < z < 2.3500000000000001e86

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.8

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

   5. Applied rewrites 52.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-265}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ t_2 := \mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{+172}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) x (* j c)) t)) (t_2 (* (fma i a (* (- z) c)) b)))
   (if (<= b -6.6e+172)
     t_2
     (if (<= b -8e-30)
       t_1
       (if (<= b -1.6e-295)
         (* (fma (- y) i (* c t)) j)
         (if (<= b 3.4e-58) 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 = fma(-a, x, (j * c)) * t;
	double t_2 = fma(i, a, (-z * c)) * b;
	double tmp;
	if (b <= -6.6e+172) {
		tmp = t_2;
	} else if (b <= -8e-30) {
		tmp = t_1;
	} else if (b <= -1.6e-295) {
		tmp = fma(-y, i, (c * t)) * j;
	} else if (b <= 3.4e-58) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), x, Float64(j * c)) * t)
	t_2 = Float64(fma(i, a, Float64(Float64(-z) * c)) * b)
	tmp = 0.0
	if (b <= -6.6e+172)
		tmp = t_2;
	elseif (b <= -8e-30)
		tmp = t_1;
	elseif (b <= -1.6e-295)
		tmp = Float64(fma(Float64(-y), i, Float64(c * t)) * j);
	elseif (b <= 3.4e-58)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -6.6e+172], t$95$2, If[LessEqual[b, -8e-30], t$95$1, If[LessEqual[b, -1.6e-295], N[(N[((-y) * i + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[b, 3.4e-58], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.59999999999999965e172 or 3.39999999999999973e-58 < b

   1. Initial program 72.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.6

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

   5. Applied rewrites 37.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.9%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 69.9

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

   11. Applied rewrites 69.9%

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

   if -6.59999999999999965e172 < b < -8.000000000000001e-30 or -1.6e-295 < b < 3.39999999999999973e-58

   1. Initial program 73.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   if -8.000000000000001e-30 < b < -1.6e-295

   1. Initial program 71.2%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 50.5

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

   5. Applied rewrites 50.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 50.5

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 52.4

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

   7. Applied rewrites 52.4%

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

   8. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 48.4

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

   10. Applied rewrites 48.4%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+73} \lor \neg \left(z \leq 1.95 \cdot 10^{+43}\right):\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(i \cdot b\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -2.1e+73) (not (<= z 1.95e+43)))
   (* (fma (- b) c (* y x)) z)
   (fma (fma (- i) y (* c t)) j (* (* i b) a))))

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 <= -2.1e+73) || !(z <= 1.95e+43)) {
		tmp = fma(-b, c, (y * x)) * z;
	} else {
		tmp = fma(fma(-i, y, (c * t)), j, ((i * b) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -2.1e+73) || !(z <= 1.95e+43))
		tmp = Float64(fma(Float64(-b), c, Float64(y * x)) * z);
	else
		tmp = fma(fma(Float64(-i), y, Float64(c * t)), j, Float64(Float64(i * b) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -2.1e+73], N[Not[LessEqual[z, 1.95e+43]], $MachinePrecision]], N[(N[((-b) * c + N[(y * x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e73 or 1.95e43 < z

   1. Initial program 60.6%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -2.1000000000000001e73 < z < 1.95e43

   1. Initial program 79.4%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 60.4

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 61.0

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

   7. Applied rewrites 61.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+73} \lor \neg \left(z \leq 1.95 \cdot 10^{+43}\right):\\ \;\;\;\;\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-i, y, c \cdot t\right), j, \left(i \cdot b\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 28.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot x\\ \mathbf{if}\;b \leq -5.6 \cdot 10^{+172}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-101}:\\ \;\;\;\;\left(\left(-x\right) \cdot a\right) \cdot t\\ \mathbf{elif}\;b \leq -3.1 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-152}:\\ \;\;\;\;\left(-a\right) \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* z y) x)))
   (if (<= b -5.6e+172)
     (* (* a b) i)
     (if (<= b -6.5e-101)
       (* (* (- x) a) t)
       (if (<= b -3.1e-299)
         t_1
         (if (<= b 3.8e-152)
           (* (- a) (* x t))
           (if (<= b 4.9e-60) t_1 (* (* i b) a))))))))

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 * y) * x;
	double tmp;
	if (b <= -5.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= -6.5e-101) {
		tmp = (-x * a) * t;
	} else if (b <= -3.1e-299) {
		tmp = t_1;
	} else if (b <= 3.8e-152) {
		tmp = -a * (x * t);
	} else if (b <= 4.9e-60) {
		tmp = t_1;
	} else {
		tmp = (i * b) * a;
	}
	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 * y) * x
    if (b <= (-5.6d+172)) then
        tmp = (a * b) * i
    else if (b <= (-6.5d-101)) then
        tmp = (-x * a) * t
    else if (b <= (-3.1d-299)) then
        tmp = t_1
    else if (b <= 3.8d-152) then
        tmp = -a * (x * t)
    else if (b <= 4.9d-60) then
        tmp = t_1
    else
        tmp = (i * b) * a
    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 * y) * x;
	double tmp;
	if (b <= -5.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= -6.5e-101) {
		tmp = (-x * a) * t;
	} else if (b <= -3.1e-299) {
		tmp = t_1;
	} else if (b <= 3.8e-152) {
		tmp = -a * (x * t);
	} else if (b <= 4.9e-60) {
		tmp = t_1;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (z * y) * x
	tmp = 0
	if b <= -5.6e+172:
		tmp = (a * b) * i
	elif b <= -6.5e-101:
		tmp = (-x * a) * t
	elif b <= -3.1e-299:
		tmp = t_1
	elif b <= 3.8e-152:
		tmp = -a * (x * t)
	elif b <= 4.9e-60:
		tmp = t_1
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(z * y) * x)
	tmp = 0.0
	if (b <= -5.6e+172)
		tmp = Float64(Float64(a * b) * i);
	elseif (b <= -6.5e-101)
		tmp = Float64(Float64(Float64(-x) * a) * t);
	elseif (b <= -3.1e-299)
		tmp = t_1;
	elseif (b <= 3.8e-152)
		tmp = Float64(Float64(-a) * Float64(x * t));
	elseif (b <= 4.9e-60)
		tmp = t_1;
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (z * y) * x;
	tmp = 0.0;
	if (b <= -5.6e+172)
		tmp = (a * b) * i;
	elseif (b <= -6.5e-101)
		tmp = (-x * a) * t;
	elseif (b <= -3.1e-299)
		tmp = t_1;
	elseif (b <= 3.8e-152)
		tmp = -a * (x * t);
	elseif (b <= 4.9e-60)
		tmp = t_1;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -5.6e+172], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, -6.5e-101], N[(N[((-x) * a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, -3.1e-299], t$95$1, If[LessEqual[b, 3.8e-152], N[((-a) * N[(x * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.9e-60], t$95$1, N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -5.5999999999999999e172

   1. Initial program 72.6%

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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.3%

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

   if -5.5999999999999999e172 < b < -6.4999999999999996e-101

   1. Initial program 69.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 37.0%

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

   if -6.4999999999999996e-101 < b < -3.1e-299 or 3.80000000000000012e-152 < b < 4.89999999999999988e-60

   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 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.4%

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

   if -3.1e-299 < b < 3.80000000000000012e-152

   1. Initial program 71.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.0%

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

   if 4.89999999999999988e-60 < b

   1. Initial program 72.6%

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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.7%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -3.3 \cdot 10^{-271}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-t, a, y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma i a (* (- z) c)) b)))
   (if (<= b -2.3e+153)
     t_1
     (if (<= b -3.3e-271)
       (* (fma (- i) j (* z x)) y)
       (if (<= b 3.15e-58) (* (fma (- t) a (* y z)) x) 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(i, a, (-z * c)) * b;
	double tmp;
	if (b <= -2.3e+153) {
		tmp = t_1;
	} else if (b <= -3.3e-271) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (b <= 3.15e-58) {
		tmp = fma(-t, a, (y * z)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(i, a, Float64(Float64(-z) * c)) * b)
	tmp = 0.0
	if (b <= -2.3e+153)
		tmp = t_1;
	elseif (b <= -3.3e-271)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (b <= 3.15e-58)
		tmp = Float64(fma(Float64(-t), a, Float64(y * z)) * x);
	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[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -2.3e+153], t$95$1, If[LessEqual[b, -3.3e-271], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[b, 3.15e-58], N[(N[((-t) * a + N[(y * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3000000000000001e153 or 3.14999999999999999e-58 < b

   1. Initial program 72.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 22.1%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 67.4

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

   11. Applied rewrites 67.4%

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

   if -2.3000000000000001e153 < b < -3.3000000000000002e-271

   1. Initial program 70.0%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   if -3.3000000000000002e-271 < b < 3.14999999999999999e-58

   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 t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

Alternative 13: 47.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{+163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-165}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-65}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma i a (* (- z) c)) b)))
   (if (<= b -2.4e+163)
     t_1
     (if (<= b 4.2e-165)
       (* (fma (- y) i (* c t)) j)
       (if (<= b 2.3e-65) (* (* z y) x) 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(i, a, (-z * c)) * b;
	double tmp;
	if (b <= -2.4e+163) {
		tmp = t_1;
	} else if (b <= 4.2e-165) {
		tmp = fma(-y, i, (c * t)) * j;
	} else if (b <= 2.3e-65) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(i, a, Float64(Float64(-z) * c)) * b)
	tmp = 0.0
	if (b <= -2.4e+163)
		tmp = t_1;
	elseif (b <= 4.2e-165)
		tmp = Float64(fma(Float64(-y), i, Float64(c * t)) * j);
	elseif (b <= 2.3e-65)
		tmp = Float64(Float64(z * y) * x);
	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[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -2.4e+163], t$95$1, If[LessEqual[b, 4.2e-165], N[(N[((-y) * i + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[b, 2.3e-65], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.3999999999999999e163 or 2.3e-65 < b

   1. Initial program 72.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 22.1%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 66.6

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

   11. Applied rewrites 66.6%

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

   if -2.3999999999999999e163 < b < 4.1999999999999999e-165

   1. Initial program 71.9%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 44.4

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 45.3

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

   7. Applied rewrites 45.3%

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

   8. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 42.7

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

   10. Applied rewrites 42.7%

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

   if 4.1999999999999999e-165 < b < 2.3e-65

   1. Initial program 72.0%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.9%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-165}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-65}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{if}\;j \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.85 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(-x\right) \cdot a\right) \cdot t\\ \mathbf{elif}\;j \leq 1.15 \cdot 10^{+19}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- y) i (* c t)) j)))
   (if (<= j -1.05e-22)
     t_1
     (if (<= j 1.85e-208)
       (* (* (- x) a) t)
       (if (<= j 1.15e+19) (* (* i b) a) 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(-y, i, (c * t)) * j;
	double tmp;
	if (j <= -1.05e-22) {
		tmp = t_1;
	} else if (j <= 1.85e-208) {
		tmp = (-x * a) * t;
	} else if (j <= 1.15e+19) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-y), i, Float64(c * t)) * j)
	tmp = 0.0
	if (j <= -1.05e-22)
		tmp = t_1;
	elseif (j <= 1.85e-208)
		tmp = Float64(Float64(Float64(-x) * a) * t);
	elseif (j <= 1.15e+19)
		tmp = Float64(Float64(i * b) * a);
	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[((-y) * i + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.05e-22], t$95$1, If[LessEqual[j, 1.85e-208], N[(N[((-x) * a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 1.15e+19], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.05000000000000004e-22 or 1.15e19 < j

   1. Initial program 74.8%

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

   3. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 63.7

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 54.8

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

   10. Applied rewrites 54.8%

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

   if -1.05000000000000004e-22 < j < 1.8500000000000001e-208

   1. Initial program 67.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 40.0%

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

   if 1.8500000000000001e-208 < j < 1.15e19

   1. Initial program 77.2%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.3%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \mathbf{elif}\;j \leq 1.85 \cdot 10^{-208}:\\ \;\;\;\;\left(\left(-x\right) \cdot a\right) \cdot t\\ \mathbf{elif}\;j \leq 1.15 \cdot 10^{+19}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, i, c \cdot t\right) \cdot j\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+172}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-35}:\\ \;\;\;\;\left(\left(-x\right) \cdot a\right) \cdot t\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-165}:\\ \;\;\;\;\left(\left(-j\right) \cdot y\right) \cdot i\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-60}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -5.6e+172)
   (* (* a b) i)
   (if (<= b -2.2e-35)
     (* (* (- x) a) t)
     (if (<= b 3.9e-165)
       (* (* (- j) y) i)
       (if (<= b 4.9e-60) (* (* z y) x) (* (* i b) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -5.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= -2.2e-35) {
		tmp = (-x * a) * t;
	} else if (b <= 3.9e-165) {
		tmp = (-j * y) * i;
	} else if (b <= 4.9e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	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 (b <= (-5.6d+172)) then
        tmp = (a * b) * i
    else if (b <= (-2.2d-35)) then
        tmp = (-x * a) * t
    else if (b <= 3.9d-165) then
        tmp = (-j * y) * i
    else if (b <= 4.9d-60) then
        tmp = (z * y) * x
    else
        tmp = (i * b) * a
    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 (b <= -5.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= -2.2e-35) {
		tmp = (-x * a) * t;
	} else if (b <= 3.9e-165) {
		tmp = (-j * y) * i;
	} else if (b <= 4.9e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -5.6e+172:
		tmp = (a * b) * i
	elif b <= -2.2e-35:
		tmp = (-x * a) * t
	elif b <= 3.9e-165:
		tmp = (-j * y) * i
	elif b <= 4.9e-60:
		tmp = (z * y) * x
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -5.6e+172)
		tmp = Float64(Float64(a * b) * i);
	elseif (b <= -2.2e-35)
		tmp = Float64(Float64(Float64(-x) * a) * t);
	elseif (b <= 3.9e-165)
		tmp = Float64(Float64(Float64(-j) * y) * i);
	elseif (b <= 4.9e-60)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -5.6e+172)
		tmp = (a * b) * i;
	elseif (b <= -2.2e-35)
		tmp = (-x * a) * t;
	elseif (b <= 3.9e-165)
		tmp = (-j * y) * i;
	elseif (b <= 4.9e-60)
		tmp = (z * y) * x;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -5.6e+172], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, -2.2e-35], N[(N[((-x) * a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 3.9e-165], N[(N[((-j) * y), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 4.9e-60], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -5.5999999999999999e172

   1. Initial program 72.6%

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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.3%

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

   if -5.5999999999999999e172 < b < -2.19999999999999994e-35

   1. Initial program 68.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 49.0

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 40.8%

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

   if -2.19999999999999994e-35 < b < 3.8999999999999999e-165

   1. Initial program 73.7%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(-1 \cdot \left(j \cdot y\right)\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 34.5%

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

   if 3.8999999999999999e-165 < b < 4.89999999999999988e-60

   1. Initial program 73.1%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.8%

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

   if 4.89999999999999988e-60 < b

   1. Initial program 72.6%

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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.7%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 16: 28.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+172}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-100}:\\ \;\;\;\;\left(\left(-x\right) \cdot a\right) \cdot t\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-165}:\\ \;\;\;\;\left(\left(-j\right) \cdot i\right) \cdot y\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-60}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -5.6e+172)
   (* (* a b) i)
   (if (<= b -1e-100)
     (* (* (- x) a) t)
     (if (<= b 2.25e-165)
       (* (* (- j) i) y)
       (if (<= b 4.9e-60) (* (* z y) x) (* (* i b) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -5.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= -1e-100) {
		tmp = (-x * a) * t;
	} else if (b <= 2.25e-165) {
		tmp = (-j * i) * y;
	} else if (b <= 4.9e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	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 (b <= (-5.6d+172)) then
        tmp = (a * b) * i
    else if (b <= (-1d-100)) then
        tmp = (-x * a) * t
    else if (b <= 2.25d-165) then
        tmp = (-j * i) * y
    else if (b <= 4.9d-60) then
        tmp = (z * y) * x
    else
        tmp = (i * b) * a
    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 (b <= -5.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= -1e-100) {
		tmp = (-x * a) * t;
	} else if (b <= 2.25e-165) {
		tmp = (-j * i) * y;
	} else if (b <= 4.9e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -5.6e+172:
		tmp = (a * b) * i
	elif b <= -1e-100:
		tmp = (-x * a) * t
	elif b <= 2.25e-165:
		tmp = (-j * i) * y
	elif b <= 4.9e-60:
		tmp = (z * y) * x
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -5.6e+172)
		tmp = Float64(Float64(a * b) * i);
	elseif (b <= -1e-100)
		tmp = Float64(Float64(Float64(-x) * a) * t);
	elseif (b <= 2.25e-165)
		tmp = Float64(Float64(Float64(-j) * i) * y);
	elseif (b <= 4.9e-60)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -5.6e+172)
		tmp = (a * b) * i;
	elseif (b <= -1e-100)
		tmp = (-x * a) * t;
	elseif (b <= 2.25e-165)
		tmp = (-j * i) * y;
	elseif (b <= 4.9e-60)
		tmp = (z * y) * x;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -5.6e+172], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, -1e-100], N[(N[((-x) * a), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 2.25e-165], N[(N[((-j) * i), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[b, 4.9e-60], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -5.5999999999999999e172

   1. Initial program 72.6%

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

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.3%

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

   if -5.5999999999999999e172 < b < -1e-100

   1. Initial program 69.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 37.0%

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

   if -1e-100 < b < 2.24999999999999996e-165

   1. Initial program 74.6%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 36.3%

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

   if 2.24999999999999996e-165 < b < 4.89999999999999988e-60

   1. Initial program 73.1%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.8%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if 4.89999999999999988e-60 < b

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      7. *-lft-identity N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 49.5

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.7%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-48} \lor \neg \left(y \leq 1.55 \cdot 10^{+81}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= y -1.9e-48) (not (<= y 1.55e+81)))
   (* (fma (- i) j (* z x)) y)
   (* (fma (- x) t (* i b)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((y <= -1.9e-48) || !(y <= 1.55e+81)) {
		tmp = fma(-i, j, (z * x)) * y;
	} else {
		tmp = fma(-x, t, (i * b)) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((y <= -1.9e-48) || !(y <= 1.55e+81))
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = Float64(fma(Float64(-x), t, Float64(i * b)) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[y, -1.9e-48], N[Not[LessEqual[y, 1.55e+81]], $MachinePrecision]], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[((-x) * t + N[(i * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.90000000000000001e-48 or 1.55e81 < y

   1. Initial program 66.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right)} \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 61.7

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   if -1.90000000000000001e-48 < y < 1.55e81

   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 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) - \color{blue}{\left(-1 \cdot b\right) \cdot i}\right) \cdot a \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot b\right)\right) \cdot i\right)} \cdot a \]

      5. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot b\right)\right) \cdot i\right) \cdot a \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot t}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot b\right)\right) \cdot i\right) \cdot a \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot t} + \left(\mathsf{neg}\left(-1 \cdot b\right)\right) \cdot i\right) \cdot a \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right) \cdot i\right) \cdot a \]

      9. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot b}\right)\right) \cdot i\right) \cdot a \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) \cdot b\right)} \cdot i\right) \cdot a \]

      11. associate-*r* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot i\right)}\right) \cdot a \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \color{blue}{1} \cdot \left(b \cdot i\right)\right) \cdot a \]

      13. *-lft-identity N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(x\right)\right) \cdot t + \color{blue}{b \cdot i}\right) \cdot a \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), t, b \cdot i\right)} \cdot a \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-x}, t, b \cdot i\right) \cdot a \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

      17. lower-*.f64 61.1

          \[\leadsto \mathsf{fma}\left(-x, t, \color{blue}{i \cdot b}\right) \cdot a \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-48} \lor \neg \left(y \leq 1.55 \cdot 10^{+81}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, t, i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153} \lor \neg \left(b \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -2.3e+153) (not (<= b 8.2e-63)))
   (* (fma i a (* (- z) c)) b)
   (* (fma (- i) j (* z x)) y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((b <= -2.3e+153) || !(b <= 8.2e-63)) {
		tmp = fma(i, a, (-z * c)) * b;
	} else {
		tmp = fma(-i, j, (z * x)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -2.3e+153) || !(b <= 8.2e-63))
		tmp = Float64(fma(i, a, Float64(Float64(-z) * c)) * b);
	else
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -2.3e+153], N[Not[LessEqual[b, 8.2e-63]], $MachinePrecision]], N[(N[(i * a + N[((-z) * c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.3000000000000001e153 or 8.1999999999999995e-63 < b

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

      8. lower-*.f64 40.0

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 22.6%

      \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot i - c \cdot z\right) \cdot b} \]

       3. *-commutative N/A

          \[\leadsto \left(a \cdot i - \color{blue}{z \cdot c}\right) \cdot b \]

       4. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(a \cdot i + \left(\mathsf{neg}\left(z\right)\right) \cdot c\right)} \cdot b \]

       5. *-commutative N/A

          \[\leadsto \left(\color{blue}{i \cdot a} + \left(\mathsf{neg}\left(z\right)\right) \cdot c\right) \cdot b \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(z\right)\right) \cdot c\right)} \cdot b \]

       7. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, a, \color{blue}{\left(-1 \cdot z\right)} \cdot c\right) \cdot b \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, a, \color{blue}{\left(-1 \cdot z\right) \cdot c}\right) \cdot b \]

       9. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(i, a, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot c\right) \cdot b \]

       10. lower-neg.f64 66.3

           \[\leadsto \mathsf{fma}\left(i, a, \color{blue}{\left(-z\right)} \cdot c\right) \cdot b \]

   11. Applied rewrites 66.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b} \]

   if -2.3000000000000001e153 < b < 8.1999999999999995e-63

   1. Initial program 72.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right)} \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 53.0

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+153} \lor \neg \left(b \leq 8.2 \cdot 10^{-63}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-z\right) \cdot c\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+163}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-152}:\\ \;\;\;\;\left(-a\right) \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-60}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -2.5e+163)
   (* (* a b) i)
   (if (<= b 3.8e-152)
     (* (- a) (* x t))
     (if (<= b 4.9e-60) (* (* z y) x) (* (* i b) a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -2.5e+163) {
		tmp = (a * b) * i;
	} else if (b <= 3.8e-152) {
		tmp = -a * (x * t);
	} else if (b <= 4.9e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	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 (b <= (-2.5d+163)) then
        tmp = (a * b) * i
    else if (b <= 3.8d-152) then
        tmp = -a * (x * t)
    else if (b <= 4.9d-60) then
        tmp = (z * y) * x
    else
        tmp = (i * b) * a
    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 (b <= -2.5e+163) {
		tmp = (a * b) * i;
	} else if (b <= 3.8e-152) {
		tmp = -a * (x * t);
	} else if (b <= 4.9e-60) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -2.5e+163:
		tmp = (a * b) * i
	elif b <= 3.8e-152:
		tmp = -a * (x * t)
	elif b <= 4.9e-60:
		tmp = (z * y) * x
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -2.5e+163)
		tmp = Float64(Float64(a * b) * i);
	elseif (b <= 3.8e-152)
		tmp = Float64(Float64(-a) * Float64(x * t));
	elseif (b <= 4.9e-60)
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -2.5e+163)
		tmp = (a * b) * i;
	elseif (b <= 3.8e-152)
		tmp = -a * (x * t);
	elseif (b <= 4.9e-60)
		tmp = (z * y) * x;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -2.5e+163], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 3.8e-152], N[((-a) * N[(x * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.9e-60], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.5e163

   1. Initial program 72.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      7. *-lft-identity N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 59.9

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   if -2.5e163 < b < 3.80000000000000012e-152

   1. Initial program 70.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

      8. lower-*.f64 44.3

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.7%

      \[\leadsto \left(-a\right) \cdot \color{blue}{\left(x \cdot t\right)} \]

   if 3.80000000000000012e-152 < b < 4.89999999999999988e-60

   1. Initial program 78.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right)} \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 65.7

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.4%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if 4.89999999999999988e-60 < b

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      7. *-lft-identity N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 49.5

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.7%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+27} \lor \neg \left(b \leq 4.9 \cdot 10^{-60}\right):\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -1.6e+27) (not (<= b 4.9e-60))) (* (* i b) a) (* (* 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 tmp;
	if ((b <= -1.6e+27) || !(b <= 4.9e-60)) {
		tmp = (i * b) * a;
	} else {
		tmp = (z * y) * x;
	}
	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 ((b <= (-1.6d+27)) .or. (.not. (b <= 4.9d-60))) then
        tmp = (i * b) * a
    else
        tmp = (z * y) * x
    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 ((b <= -1.6e+27) || !(b <= 4.9e-60)) {
		tmp = (i * b) * a;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (b <= -1.6e+27) or not (b <= 4.9e-60):
		tmp = (i * b) * a
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -1.6e+27) || !(b <= 4.9e-60))
		tmp = Float64(Float64(i * b) * a);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((b <= -1.6e+27) || ~((b <= 4.9e-60)))
		tmp = (i * b) * a;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -1.6e+27], N[Not[LessEqual[b, 4.9e-60]], $MachinePrecision]], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.60000000000000008e27 or 4.89999999999999988e-60 < b

   1. Initial program 71.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      7. *-lft-identity N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 48.1

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]

   if -1.60000000000000008e27 < b < 4.89999999999999988e-60

   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 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right)} \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 54.2

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.5%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+27} \lor \neg \left(b \leq 4.9 \cdot 10^{-60}\right):\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 31.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+58} \lor \neg \left(z \leq 4.6 \cdot 10^{+33}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -8.6e+58) (not (<= z 4.6e+33))) (* (* z y) 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 ((z <= -8.6e+58) || !(z <= 4.6e+33)) {
		tmp = (z * y) * 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 ((z <= (-8.6d+58)) .or. (.not. (z <= 4.6d+33))) then
        tmp = (z * y) * 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 ((z <= -8.6e+58) || !(z <= 4.6e+33)) {
		tmp = (z * y) * x;
	} else {
		tmp = (j * t) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (z <= -8.6e+58) or not (z <= 4.6e+33):
		tmp = (z * y) * x
	else:
		tmp = (j * t) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -8.6e+58) || !(z <= 4.6e+33))
		tmp = Float64(Float64(z * y) * 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 ((z <= -8.6e+58) || ~((z <= 4.6e+33)))
		tmp = (z * y) * x;
	else
		tmp = (j * t) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -8.6e+58], N[Not[LessEqual[z, 4.6e+33]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.59999999999999982e58 or 4.60000000000000021e33 < z

   1. Initial program 62.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right)} \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 55.8

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.9%

      \[\leadsto \left(z \cdot y\right) \cdot \color{blue}{x} \]

   if -8.59999999999999982e58 < z < 4.60000000000000021e33

   1. Initial program 78.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

      8. lower-*.f64 43.7

         \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   5. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.9%

      \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+58} \lor \neg \left(z \leq 4.6 \cdot 10^{+33}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 28.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+172}:\\ \;\;\;\;\left(a \cdot b\right) \cdot i\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-60}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -6.6e+172)
   (* (* a b) i)
   (if (<= b 4.9e-60) (* (* z x) y) (* (* i b) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -6.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= 4.9e-60) {
		tmp = (z * x) * y;
	} else {
		tmp = (i * b) * a;
	}
	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 (b <= (-6.6d+172)) then
        tmp = (a * b) * i
    else if (b <= 4.9d-60) then
        tmp = (z * x) * y
    else
        tmp = (i * b) * a
    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 (b <= -6.6e+172) {
		tmp = (a * b) * i;
	} else if (b <= 4.9e-60) {
		tmp = (z * x) * y;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -6.6e+172:
		tmp = (a * b) * i
	elif b <= 4.9e-60:
		tmp = (z * x) * y
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -6.6e+172)
		tmp = Float64(Float64(a * b) * i);
	elseif (b <= 4.9e-60)
		tmp = Float64(Float64(z * x) * y);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -6.6e+172)
		tmp = (a * b) * i;
	elseif (b <= 4.9e-60)
		tmp = (z * x) * y;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -6.6e+172], N[(N[(a * b), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[b, 4.9e-60], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.59999999999999965e172

   1. Initial program 71.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      7. *-lft-identity N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 66.0

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(a \cdot b\right) \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 57.0%

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   if -6.59999999999999965e172 < b < 4.89999999999999988e-60

   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 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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot y} \]

      3. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i \cdot j\right)\right)} + x \cdot z\right) \cdot y \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot j} + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right)} \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-i}, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

      8. lower-*.f64 51.1

         \[\leadsto \mathsf{fma}\left(-i, j, \color{blue}{z \cdot x}\right) \cdot y \]

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 30.4%

      \[\leadsto \left(z \cdot x\right) \cdot y \]

   if 4.89999999999999988e-60 < b

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   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

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) \cdot i} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right)} \cdot i \]

      4. *-commutative N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot j\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right) \cdot j} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot b\right)\right) \cdot i \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{1} \cdot \left(a \cdot b\right)\right) \cdot i \]

      7. *-lft-identity N/A

         \[\leadsto \left(\left(-1 \cdot y\right) \cdot j + \color{blue}{a \cdot b}\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot y, j, a \cdot b\right)} \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, j, a \cdot b\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, j, a \cdot b\right) \cdot i \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

      12. lower-*.f64 49.5

          \[\leadsto \mathsf{fma}\left(-y, j, \color{blue}{b \cdot a}\right) \cdot i \]

   5. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, j, b \cdot a\right) \cdot i} \]

   6. Taylor expanded in y around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.7%

      \[\leadsto \left(i \cdot b\right) \cdot \color{blue}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 22.5% 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 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 t around inf

   \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

   6. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   8. lower-*.f64 41.4

      \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

5. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

6. Taylor expanded in x around 0

   \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
7. Step-by-step derivation

8. Applied rewrites 19.5%

   \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]
9. Add Preprocessing

Alternative 24: 22.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(c \cdot t\right) \cdot j \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* c t) j))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (c * t) * j;
}

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 = (c * t) * j
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 (c * t) * j;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (c * t) * j

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(c * t) * j)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (c * t) * j;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(c * t), $MachinePrecision] * j), $MachinePrecision]

Derivation

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 t around inf

   \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot a\right) \cdot x} + c \cdot j\right) \cdot t \]

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot x + c \cdot j\right) \cdot t \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right)} \cdot t \]

   6. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{-a}, x, c \cdot j\right) \cdot t \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

   8. lower-*.f64 41.4

      \[\leadsto \mathsf{fma}\left(-a, x, \color{blue}{j \cdot c}\right) \cdot t \]

5. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

6. Taylor expanded in x around 0

   \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
7. Step-by-step derivation

8. Applied rewrites 19.5%

   \[\leadsto \left(j \cdot t\right) \cdot \color{blue}{c} \]
9. Step-by-step derivation

10. Applied rewrites 18.4%

    \[\leadsto \left(c \cdot t\right) \cdot j \]
11. Add Preprocessing

Developer Target 1: 69.6% 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 2024364 
(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)))))