Data.Colour.Matrix:determinant from colour-2.3.3, A

Percentage Accurate: 72.8% → 81.1%

Time: 8.1s

Alternatives: 21

Speedup: 0.5×

• 57.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

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) - (t * i)))) + (j * ((c * a) - (y * i)))
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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

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

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(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
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[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

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) - (t * i)))) + (j * ((c * a) - (y * i)))
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) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

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

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(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
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[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.1% accurate, 0.5× speedup

Math

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

FPCore

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

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) - (t * i)))) + (j * ((c * a) - (y * i)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (-t * fma(a, x, (-b * i))) + (j * (-i * y));
	}
	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(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-t) * fma(a, x, Float64(Float64(-b) * i))) + Float64(j * Float64(Float64(-i) * y)));
	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[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[((-t) * N[(a * x + N[((-b) * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[((-i) * y), $MachinePrecision]), $MachinePrecision]), $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 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 88.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 52.2

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

   8. Applied rewrites 52.2%

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

Alternative 2: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, t, c \cdot z\right)\\ \mathbf{if}\;b \leq -13200000000000:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \left(j \cdot c\right) \cdot a\right) - t\_1 \cdot b\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\left(-i\right) \cdot j, y, \mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \left(z \cdot y\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- i) t (* c z))))
   (if (<= b -13200000000000.0)
     (- (fma (- a) (* t x) (* (* j c) a)) (* t_1 b))
     (if (<= b 2e-115)
       (fma (* (- i) j) y (fma (fma (- t) x (* j c)) a (* (* z y) x)))
       (fma (fma (- a) t (* z y)) x (* (- b) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-i, t, (c * z));
	double tmp;
	if (b <= -13200000000000.0) {
		tmp = fma(-a, (t * x), ((j * c) * a)) - (t_1 * b);
	} else if (b <= 2e-115) {
		tmp = fma((-i * j), y, fma(fma(-t, x, (j * c)), a, ((z * y) * x)));
	} else {
		tmp = fma(fma(-a, t, (z * y)), x, (-b * t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-i), t, Float64(c * z))
	tmp = 0.0
	if (b <= -13200000000000.0)
		tmp = Float64(fma(Float64(-a), Float64(t * x), Float64(Float64(j * c) * a)) - Float64(t_1 * b));
	elseif (b <= 2e-115)
		tmp = fma(Float64(Float64(-i) * j), y, fma(fma(Float64(-t), x, Float64(j * c)), a, Float64(Float64(z * y) * x)));
	else
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(-b) * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-i) * t + N[(c * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -13200000000000.0], N[(N[((-a) * N[(t * x), $MachinePrecision] + N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-115], N[(N[((-i) * j), $MachinePrecision] * y + N[(N[((-t) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.32e13

   1. Initial program 74.6%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 74.2%

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

   if -1.32e13 < b < 2.0000000000000001e-115

   1. Initial program 67.3%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

   8. Applied rewrites 71.7%

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

   if 2.0000000000000001e-115 < b

   1. Initial program 75.9%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 80.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -13200000000000:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \left(j \cdot c\right) \cdot a\right) - \mathsf{fma}\left(-i, t, c \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(\left(-i\right) \cdot j, y, \mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \left(z \cdot y\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-i, t, c \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.9 \cdot 10^{+93} \lor \neg \left(j \leq 0.0001\right):\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(a, x, \left(-b\right) \cdot i\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-i, t, c \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= j -2.9e+93) (not (<= j 0.0001)))
   (+ (* (- t) (fma a x (* (- b) i))) (* j (- (* c a) (* y i))))
   (fma (fma (- a) t (* z y)) x (* (- b) (fma (- i) t (* c z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((j <= -2.9e+93) || !(j <= 0.0001)) {
		tmp = (-t * fma(a, x, (-b * i))) + (j * ((c * a) - (y * i)));
	} else {
		tmp = fma(fma(-a, t, (z * y)), x, (-b * fma(-i, t, (c * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((j <= -2.9e+93) || !(j <= 0.0001))
		tmp = Float64(Float64(Float64(-t) * fma(a, x, Float64(Float64(-b) * i))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))));
	else
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(-b) * fma(Float64(-i), t, Float64(c * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[j, -2.9e+93], N[Not[LessEqual[j, 0.0001]], $MachinePrecision]], N[(N[((-t) * N[(a * x + N[((-b) * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[((-i) * t + N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < -2.8999999999999998e93 or 1.00000000000000005e-4 < j

   1. Initial program 70.2%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if -2.8999999999999998e93 < j < 1.00000000000000005e-4

   1. Initial program 73.5%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.9 \cdot 10^{+93} \lor \neg \left(j \leq 0.0001\right):\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(a, x, \left(-b\right) \cdot i\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-i, t, c \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.9 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-i, t, c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(a, x, \left(-b\right) \cdot i\right) + j \cdot \left(\left(-i\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.9e+93)
   (* (fma (- i) y (* c a)) j)
   (if (<= j 8.5e+138)
     (fma (fma (- a) t (* z y)) x (* (- b) (fma (- i) t (* c z))))
     (+ (* (- t) (fma a x (* (- b) i))) (* j (* (- i) 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 (j <= -2.9e+93) {
		tmp = fma(-i, y, (c * a)) * j;
	} else if (j <= 8.5e+138) {
		tmp = fma(fma(-a, t, (z * y)), x, (-b * fma(-i, t, (c * z))));
	} else {
		tmp = (-t * fma(a, x, (-b * i))) + (j * (-i * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.9e+93)
		tmp = Float64(fma(Float64(-i), y, Float64(c * a)) * j);
	elseif (j <= 8.5e+138)
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(-b) * fma(Float64(-i), t, Float64(c * z))));
	else
		tmp = Float64(Float64(Float64(-t) * fma(a, x, Float64(Float64(-b) * i))) + Float64(j * Float64(Float64(-i) * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.9e+93], N[(N[((-i) * y + N[(c * a), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], If[LessEqual[j, 8.5e+138], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[((-i) * t + N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t) * N[(a * x + N[((-b) * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[((-i) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.8999999999999998e93

   1. Initial program 67.0%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -2.8999999999999998e93 < j < 8.5000000000000006e138

   1. Initial program 72.5%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 74.4%

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

   if 8.5000000000000006e138 < j

   1. Initial program 76.1%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 75.1

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

   8. Applied rewrites 75.1%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.9 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot a\right) \cdot j\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-i, t, c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(a, x, \left(-b\right) \cdot i\right) + j \cdot \left(\left(-i\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{-55} \lor \neg \left(i \leq 1450000000000\right):\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(a, x, \left(-b\right) \cdot i\right) + j \cdot \left(\left(-i\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \left(c \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= i -4.2e-55) (not (<= i 1450000000000.0)))
   (+ (* (- t) (fma a x (* (- b) i))) (* j (* (- i) y)))
   (fma (fma (- a) t (* z y)) x (* (- b) (* c z)))))

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 <= -4.2e-55) || !(i <= 1450000000000.0)) {
		tmp = (-t * fma(a, x, (-b * i))) + (j * (-i * y));
	} else {
		tmp = fma(fma(-a, t, (z * y)), x, (-b * (c * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((i <= -4.2e-55) || !(i <= 1450000000000.0))
		tmp = Float64(Float64(Float64(-t) * fma(a, x, Float64(Float64(-b) * i))) + Float64(j * Float64(Float64(-i) * y)));
	else
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(-b) * Float64(c * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[i, -4.2e-55], N[Not[LessEqual[i, 1450000000000.0]], $MachinePrecision]], N[(N[((-t) * N[(a * x + N[((-b) * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[((-i) * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -4.2000000000000003e-55 or 1.45e12 < i

   1. Initial program 59.8%

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

   3. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if -4.2000000000000003e-55 < i < 1.45e12

   1. Initial program 83.2%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in z around inf

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

      1. lift-*.f64 71.6

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

   8. Applied rewrites 71.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{-55} \lor \neg \left(i \leq 1450000000000\right):\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(a, x, \left(-b\right) \cdot i\right) + j \cdot \left(\left(-i\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \left(c \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, t, z \cdot y\right)\\ \mathbf{if}\;i \leq -5.7 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, \left(b \cdot t\right) \cdot i\right)\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, \left(-b\right) \cdot \left(c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-i\right) \cdot \left(\mathsf{fma}\left(j, \frac{y}{t}, -b\right) \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- a) t (* z y))))
   (if (<= i -5.7e-55)
     (fma t_1 x (* (* b t) i))
     (if (<= i 3.8e+33)
       (fma t_1 x (* (- b) (* c z)))
       (* (- i) (* (fma j (/ y t) (- b)) t))))))

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));
	double tmp;
	if (i <= -5.7e-55) {
		tmp = fma(t_1, x, ((b * t) * i));
	} else if (i <= 3.8e+33) {
		tmp = fma(t_1, x, (-b * (c * z)));
	} else {
		tmp = -i * (fma(j, (y / t), -b) * t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(-a), t, Float64(z * y))
	tmp = 0.0
	if (i <= -5.7e-55)
		tmp = fma(t_1, x, Float64(Float64(b * t) * i));
	elseif (i <= 3.8e+33)
		tmp = fma(t_1, x, Float64(Float64(-b) * Float64(c * z)));
	else
		tmp = Float64(Float64(-i) * Float64(fma(j, Float64(y / t), Float64(-b)) * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -5.7e-55], N[(t$95$1 * x + N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3.8e+33], N[(t$95$1 * x + N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-i) * N[(N[(j * N[(y / t), $MachinePrecision] + (-b)), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if i < -5.7000000000000002e-55

   1. Initial program 64.6%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 60.6

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

   8. Applied rewrites 60.6%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 59.0

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

   11. Applied rewrites 59.0%

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

   if -5.7000000000000002e-55 < i < 3.80000000000000002e33

   1. Initial program 83.0%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in z around inf

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

      1. lift-*.f64 71.7

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

   8. Applied rewrites 71.7%

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

   if 3.80000000000000002e33 < i

   1. Initial program 53.4%

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

   3. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-neg.f64 62.7

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

   8. Applied rewrites 62.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -5.7 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(b \cdot t\right) \cdot i\right)\\ \mathbf{elif}\;i \leq 3.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \left(c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-i\right) \cdot \left(\mathsf{fma}\left(j, \frac{y}{t}, -b\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.9 \cdot 10^{+123} \lor \neg \left(c \leq 1.9 \cdot 10^{+148}\right):\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(b \cdot t\right) \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= c -2.9e+123) (not (<= c 1.9e+148)))
   (* (fma j a (* (- b) z)) c)
   (fma (fma (- a) t (* z y)) x (* (* b t) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((c <= -2.9e+123) || !(c <= 1.9e+148)) {
		tmp = fma(j, a, (-b * z)) * c;
	} else {
		tmp = fma(fma(-a, t, (z * y)), x, ((b * t) * i));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((c <= -2.9e+123) || !(c <= 1.9e+148))
		tmp = Float64(fma(j, a, Float64(Float64(-b) * z)) * c);
	else
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(b * t) * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[c, -2.9e+123], N[Not[LessEqual[c, 1.9e+148]], $MachinePrecision]], N[(N[(j * a + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(b * t), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.9000000000000001e123 or 1.8999999999999999e148 < c

   1. Initial program 64.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -2.9000000000000001e123 < c < 1.8999999999999999e148

   1. Initial program 74.9%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 57.0

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

   8. Applied rewrites 57.0%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 55.4

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

   11. Applied rewrites 55.4%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.9 \cdot 10^{+123} \lor \neg \left(c \leq 1.9 \cdot 10^{+148}\right):\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(b \cdot t\right) \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+97}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -2.2e+97)
   (* (- i) (* j y))
   (if (<= y -1.1e-212)
     (* (fma j a (* (- b) z)) c)
     (if (<= y 3.15e+90) (* (fma i t (* (- c) z)) b) (* (* y x) z)))))

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 <= -2.2e+97) {
		tmp = -i * (j * y);
	} else if (y <= -1.1e-212) {
		tmp = fma(j, a, (-b * z)) * c;
	} else if (y <= 3.15e+90) {
		tmp = fma(i, t, (-c * z)) * b;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -2.2e+97)
		tmp = Float64(Float64(-i) * Float64(j * y));
	elseif (y <= -1.1e-212)
		tmp = Float64(fma(j, a, Float64(Float64(-b) * z)) * c);
	elseif (y <= 3.15e+90)
		tmp = Float64(fma(i, t, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -2.2e+97], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.1e-212], N[(N[(j * a + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 3.15e+90], N[(N[(i * t + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2000000000000001e97

   1. Initial program 56.5%

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

   3. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 48.0

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

   8. Applied rewrites 48.0%

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

   if -2.2000000000000001e97 < y < -1.10000000000000002e-212

   1. Initial program 77.4%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 55.2

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

   5. Applied rewrites 55.2%

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

   if -1.10000000000000002e-212 < y < 3.15e90

   1. Initial program 78.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if 3.15e90 < y

   1. Initial program 61.7%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 55.3

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lift-*.f64 47.7

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

   11. Applied rewrites 47.7%

       \[\leadsto \left(y \cdot x\right) \cdot z \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+97}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(j, a, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot x\right) \cdot z\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-184}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;y \leq 4400000000000:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+85}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* y x) z)))
   (if (<= y -2.6e+25)
     t_1
     (if (<= y -2.05e-184)
       (* (* j a) c)
       (if (<= y 4400000000000.0)
         (* (* (- c) z) b)
         (if (<= y 1.9e+85) (* (* i t) b) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (y * x) * z;
	double tmp;
	if (y <= -2.6e+25) {
		tmp = t_1;
	} else if (y <= -2.05e-184) {
		tmp = (j * a) * c;
	} else if (y <= 4400000000000.0) {
		tmp = (-c * z) * b;
	} else if (y <= 1.9e+85) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * x) * z
    if (y <= (-2.6d+25)) then
        tmp = t_1
    else if (y <= (-2.05d-184)) then
        tmp = (j * a) * c
    else if (y <= 4400000000000.0d0) then
        tmp = (-c * z) * b
    else if (y <= 1.9d+85) then
        tmp = (i * t) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (y * x) * z;
	double tmp;
	if (y <= -2.6e+25) {
		tmp = t_1;
	} else if (y <= -2.05e-184) {
		tmp = (j * a) * c;
	} else if (y <= 4400000000000.0) {
		tmp = (-c * z) * b;
	} else if (y <= 1.9e+85) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (y * x) * z
	tmp = 0
	if y <= -2.6e+25:
		tmp = t_1
	elif y <= -2.05e-184:
		tmp = (j * a) * c
	elif y <= 4400000000000.0:
		tmp = (-c * z) * b
	elif y <= 1.9e+85:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(y * x) * z)
	tmp = 0.0
	if (y <= -2.6e+25)
		tmp = t_1;
	elseif (y <= -2.05e-184)
		tmp = Float64(Float64(j * a) * c);
	elseif (y <= 4400000000000.0)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (y <= 1.9e+85)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (y * x) * z;
	tmp = 0.0;
	if (y <= -2.6e+25)
		tmp = t_1;
	elseif (y <= -2.05e-184)
		tmp = (j * a) * c;
	elseif (y <= 4400000000000.0)
		tmp = (-c * z) * b;
	elseif (y <= 1.9e+85)
		tmp = (i * t) * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -2.6e+25], t$95$1, If[LessEqual[y, -2.05e-184], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 4400000000000.0], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[y, 1.9e+85], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.5999999999999998e25 or 1.89999999999999996e85 < y

   1. Initial program 60.7%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.8

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

   8. Applied rewrites 51.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lift-*.f64 43.3

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

   11. Applied rewrites 43.3%

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

   if -2.5999999999999998e25 < y < -2.05e-184

   1. Initial program 78.2%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \left(j \cdot a\right) \cdot c \]

      2. lower-*.f64 39.6

         \[\leadsto \left(j \cdot a\right) \cdot c \]

   8. Applied rewrites 39.6%

      \[\leadsto \left(j \cdot a\right) \cdot c \]

   if -2.05e-184 < y < 4.4e12

   1. Initial program 80.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 34.5

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

   8. Applied rewrites 34.5%

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

   if 4.4e12 < y < 1.89999999999999996e85

   1. Initial program 72.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 39.1

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

   5. Applied rewrites 39.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 35.1

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

   8. Applied rewrites 35.1%

      \[\leadsto \left(i \cdot t\right) \cdot b \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+25}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-184}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \mathbf{elif}\;y \leq 4400000000000:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+85}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+38} \lor \neg \left(z \leq 1.38 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(-b, i, a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -6.4e+38) (not (<= z 1.38e-6)))
   (* (fma y x (* (- b) c)) z)
   (* (- t) (fma (- b) i (* a 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 ((z <= -6.4e+38) || !(z <= 1.38e-6)) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = -t * fma(-b, i, (a * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -6.4e+38) || !(z <= 1.38e-6))
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	else
		tmp = Float64(Float64(-t) * fma(Float64(-b), i, Float64(a * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -6.4e+38], N[Not[LessEqual[z, 1.38e-6]], $MachinePrecision]], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[((-t) * N[((-b) * i + N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.3999999999999997e38 or 1.3799999999999999e-6 < z

   1. Initial program 59.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -6.3999999999999997e38 < z < 1.3799999999999999e-6

   1. Initial program 81.7%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. mul-1-neg N/A

         \[\leadsto \left(-t\right) \cdot \left(a \cdot x + \left(-1 \cdot b\right) \cdot i\right) \]

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lower-*.f64 52.9

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

   8. Applied rewrites 52.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+38} \lor \neg \left(z \leq 1.38 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(-b, i, a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+38} \lor \neg \left(z \leq 1.38 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -6.4e+38) (not (<= z 1.38e-6)))
   (* (fma y x (* (- b) c)) z)
   (* (fma (- a) x (* i b)) t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -6.4e+38) || !(z <= 1.38e-6)) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = fma(-a, x, (i * b)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -6.4e+38) || !(z <= 1.38e-6))
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	else
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -6.4e+38], N[Not[LessEqual[z, 1.38e-6]], $MachinePrecision]], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.3999999999999997e38 or 1.3799999999999999e-6 < z

   1. Initial program 59.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -6.3999999999999997e38 < z < 1.3799999999999999e-6

   1. Initial program 81.7%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right) \cdot t \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot x + b \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 52.9

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

   8. Applied rewrites 52.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+38} \lor \neg \left(z \leq 1.38 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.235 \lor \neg \left(x \leq 3.4 \cdot 10^{+52}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= x -0.235) (not (<= x 3.4e+52)))
   (* (fma (- a) t (* z y)) x)
   (* (fma i t (* (- c) z)) b)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((x <= -0.235) || !(x <= 3.4e+52)) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = fma(i, t, (-c * z)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((x <= -0.235) || !(x <= 3.4e+52))
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = Float64(fma(i, t, Float64(Float64(-c) * z)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[x, -0.235], N[Not[LessEqual[x, 3.4e+52]], $MachinePrecision]], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * t + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.23499999999999999 or 3.4e52 < x

   1. Initial program 71.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if -0.23499999999999999 < x < 3.4e52

   1. Initial program 73.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 48.6

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

   5. Applied rewrites 48.6%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.235 \lor \neg \left(x \leq 3.4 \cdot 10^{+52}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98} \lor \neg \left(z \leq 1.2 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -1.5e+98) (not (<= z 1.2e-15)))
   (* (fma y x (* (- b) c)) z)
   (* (fma i t (* (- c) z)) b)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -1.5e+98) || !(z <= 1.2e-15)) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = fma(i, t, (-c * z)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -1.5e+98) || !(z <= 1.2e-15))
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	else
		tmp = Float64(fma(i, t, Float64(Float64(-c) * z)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -1.5e+98], N[Not[LessEqual[z, 1.2e-15]], $MachinePrecision]], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(i * t + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5000000000000001e98 or 1.19999999999999997e-15 < z

   1. Initial program 59.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -1.5000000000000001e98 < z < 1.19999999999999997e-15

   1. Initial program 80.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 40.6

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

   5. Applied rewrites 40.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98} \lor \neg \left(z \leq 1.2 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+75}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+34}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -5.1e+75)
   (* (* (- a) t) x)
   (if (<= t 2.9e-11)
     (* (* (- c) z) b)
     (if (<= t 7.5e+34) (* (* j c) a) (* (* (- t) x) 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 (t <= -5.1e+75) {
		tmp = (-a * t) * x;
	} else if (t <= 2.9e-11) {
		tmp = (-c * z) * b;
	} else if (t <= 7.5e+34) {
		tmp = (j * c) * a;
	} else {
		tmp = (-t * x) * 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 (t <= (-5.1d+75)) then
        tmp = (-a * t) * x
    else if (t <= 2.9d-11) then
        tmp = (-c * z) * b
    else if (t <= 7.5d+34) then
        tmp = (j * c) * a
    else
        tmp = (-t * x) * 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 (t <= -5.1e+75) {
		tmp = (-a * t) * x;
	} else if (t <= 2.9e-11) {
		tmp = (-c * z) * b;
	} else if (t <= 7.5e+34) {
		tmp = (j * c) * a;
	} else {
		tmp = (-t * x) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -5.1e+75:
		tmp = (-a * t) * x
	elif t <= 2.9e-11:
		tmp = (-c * z) * b
	elif t <= 7.5e+34:
		tmp = (j * c) * a
	else:
		tmp = (-t * x) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -5.1e+75)
		tmp = Float64(Float64(Float64(-a) * t) * x);
	elseif (t <= 2.9e-11)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (t <= 7.5e+34)
		tmp = Float64(Float64(j * c) * a);
	else
		tmp = Float64(Float64(Float64(-t) * x) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= -5.1e+75)
		tmp = (-a * t) * x;
	elseif (t <= 2.9e-11)
		tmp = (-c * z) * b;
	elseif (t <= 7.5e+34)
		tmp = (j * c) * a;
	else
		tmp = (-t * x) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -5.1e+75], N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 2.9e-11], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t, 7.5e+34], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.10000000000000037e75

   1. Initial program 58.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if -5.10000000000000037e75 < t < 2.9e-11

   1. Initial program 81.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 46.6

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 34.7

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

   8. Applied rewrites 34.7%

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

   if 2.9e-11 < t < 7.49999999999999976e34

   1. Initial program 56.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(j \cdot c\right) \cdot a \]

      2. lift-*.f64 70.6

         \[\leadsto \left(j \cdot c\right) \cdot a \]

   8. Applied rewrites 70.6%

      \[\leadsto \left(j \cdot c\right) \cdot a \]

   if 7.49999999999999976e34 < t

   1. Initial program 63.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 41.8

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

   8. Applied rewrites 41.8%

      \[\leadsto \left(\left(-t\right) \cdot x\right) \cdot a \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+75}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+34}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{if}\;t \leq -5.1 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+108}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- a) t) x)))
   (if (<= t -5.1e+75)
     t_1
     (if (<= t 2.9e-11)
       (* (* (- c) z) b)
       (if (<= t 8e+108) (* (* j c) 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 = (-a * t) * x;
	double tmp;
	if (t <= -5.1e+75) {
		tmp = t_1;
	} else if (t <= 2.9e-11) {
		tmp = (-c * z) * b;
	} else if (t <= 8e+108) {
		tmp = (j * c) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-a * t) * x
    if (t <= (-5.1d+75)) then
        tmp = t_1
    else if (t <= 2.9d-11) then
        tmp = (-c * z) * b
    else if (t <= 8d+108) then
        tmp = (j * c) * a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-a * t) * x;
	double tmp;
	if (t <= -5.1e+75) {
		tmp = t_1;
	} else if (t <= 2.9e-11) {
		tmp = (-c * z) * b;
	} else if (t <= 8e+108) {
		tmp = (j * c) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-a * t) * x
	tmp = 0
	if t <= -5.1e+75:
		tmp = t_1
	elif t <= 2.9e-11:
		tmp = (-c * z) * b
	elif t <= 8e+108:
		tmp = (j * c) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-a) * t) * x)
	tmp = 0.0
	if (t <= -5.1e+75)
		tmp = t_1;
	elseif (t <= 2.9e-11)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (t <= 8e+108)
		tmp = Float64(Float64(j * c) * a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-a * t) * x;
	tmp = 0.0;
	if (t <= -5.1e+75)
		tmp = t_1;
	elseif (t <= 2.9e-11)
		tmp = (-c * z) * b;
	elseif (t <= 8e+108)
		tmp = (j * c) * a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -5.1e+75], t$95$1, If[LessEqual[t, 2.9e-11], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[t, 8e+108], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.10000000000000037e75 or 8.0000000000000003e108 < t

   1. Initial program 58.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if -5.10000000000000037e75 < t < 2.9e-11

   1. Initial program 81.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 46.6

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 34.7

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

   8. Applied rewrites 34.7%

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

   if 2.9e-11 < t < 8.0000000000000003e108

   1. Initial program 68.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(j \cdot c\right) \cdot a \]

      2. lift-*.f64 42.6

         \[\leadsto \left(j \cdot c\right) \cdot a \]

   8. Applied rewrites 42.6%

      \[\leadsto \left(j \cdot c\right) \cdot a \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+75}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-11}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+108}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \left(a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -6.2e+150)
   (* (* (- t) x) a)
   (if (<= x 4.2e+179) (* (fma i t (* (- c) z)) b) (* (- t) (* a 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 (x <= -6.2e+150) {
		tmp = (-t * x) * a;
	} else if (x <= 4.2e+179) {
		tmp = fma(i, t, (-c * z)) * b;
	} else {
		tmp = -t * (a * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -6.2e+150)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	elseif (x <= 4.2e+179)
		tmp = Float64(fma(i, t, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(Float64(-t) * Float64(a * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -6.2e+150], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[x, 4.2e+179], N[(N[(i * t + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[((-t) * N[(a * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.20000000000000028e150

   1. Initial program 63.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 50.0

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

   8. Applied rewrites 50.0%

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

   if -6.20000000000000028e150 < x < 4.1999999999999997e179

   1. Initial program 72.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if 4.1999999999999997e179 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. mul-1-neg N/A

         \[\leadsto \left(-t\right) \cdot \left(a \cdot x + \left(-1 \cdot b\right) \cdot i\right) \]

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in x around inf

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

       1. lift-*.f64 59.9

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

   11. Applied rewrites 59.9%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \left(a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+89}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1.05e+93)
   (* (* (- b) z) c)
   (if (<= c 2.05e+89) (* (* y x) z) (* (* (- c) z) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.05e+93) {
		tmp = (-b * z) * c;
	} else if (c <= 2.05e+89) {
		tmp = (y * x) * z;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (c <= (-1.05d+93)) then
        tmp = (-b * z) * c
    else if (c <= 2.05d+89) then
        tmp = (y * x) * z
    else
        tmp = (-c * z) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.05e+93) {
		tmp = (-b * z) * c;
	} else if (c <= 2.05e+89) {
		tmp = (y * x) * z;
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if c <= -1.05e+93:
		tmp = (-b * z) * c
	elif c <= 2.05e+89:
		tmp = (y * x) * z
	else:
		tmp = (-c * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1.05e+93)
		tmp = Float64(Float64(Float64(-b) * z) * c);
	elseif (c <= 2.05e+89)
		tmp = Float64(Float64(y * x) * z);
	else
		tmp = Float64(Float64(Float64(-c) * z) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (c <= -1.05e+93)
		tmp = (-b * z) * c;
	elseif (c <= 2.05e+89)
		tmp = (y * x) * z;
	else
		tmp = (-c * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1.05e+93], N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 2.05e+89], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.0499999999999999e93

   1. Initial program 65.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if -1.0499999999999999e93 < c < 2.04999999999999993e89

   1. Initial program 73.7%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 36.4

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

   8. Applied rewrites 36.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lift-*.f64 26.9

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

   11. Applied rewrites 26.9%

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

   if 2.04999999999999993e89 < c

   1. Initial program 74.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      8. lower-neg.f64 65.4

         \[\leadsto \mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b \]

   5. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(c \cdot z\right)\right) \cdot b \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      4. lift-neg.f64 51.1

         \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot b \]

   8. Applied rewrites 51.1%

      \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot b \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+89}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98} \lor \neg \left(z \leq 1.1 \cdot 10^{-15}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= z -1.5e+98) (not (<= z 1.1e-15))) (* (* z y) x) (* (* i t) b)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((z <= -1.5e+98) || !(z <= 1.1e-15)) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * t) * b;
	}
	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 <= (-1.5d+98)) .or. (.not. (z <= 1.1d-15))) then
        tmp = (z * y) * x
    else
        tmp = (i * t) * b
    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 <= -1.5e+98) || !(z <= 1.1e-15)) {
		tmp = (z * y) * x;
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (z <= -1.5e+98) or not (z <= 1.1e-15):
		tmp = (z * y) * x
	else:
		tmp = (i * t) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((z <= -1.5e+98) || !(z <= 1.1e-15))
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(i * t) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((z <= -1.5e+98) || ~((z <= 1.1e-15)))
		tmp = (z * y) * x;
	else
		tmp = (i * t) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[z, -1.5e+98], N[Not[LessEqual[z, 1.1e-15]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5000000000000001e98 or 1.09999999999999993e-15 < z

   1. Initial program 59.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot x \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(y \cdot z + \left(\mathsf{neg}\left(a\right)\right) \cdot t\right) \cdot x \]

      6. mul-1-neg N/A

         \[\leadsto \left(y \cdot z + \left(-1 \cdot a\right) \cdot t\right) \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \left(y \cdot z + -1 \cdot \left(a \cdot t\right)\right) \cdot x \]

      8. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right) \cdot x \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot t + y \cdot z\right) \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(a\right)\right) \cdot t + y \cdot z\right) \cdot x \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, y \cdot z\right) \cdot x \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, t, y \cdot z\right) \cdot x \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x \]

      14. lower-*.f64 54.1

          \[\leadsto \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x \]

   5. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z \cdot y\right) \cdot x \]

      2. lift-*.f64 40.8

         \[\leadsto \left(z \cdot y\right) \cdot x \]

   8. Applied rewrites 40.8%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -1.5000000000000001e98 < z < 1.09999999999999993e-15

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(i \cdot t + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      8. lower-neg.f64 40.6

         \[\leadsto \mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b \]

   5. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

      1. lower-*.f64 27.3

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   8. Applied rewrites 27.3%

      \[\leadsto \left(i \cdot t\right) \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98} \lor \neg \left(z \leq 1.1 \cdot 10^{-15}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+163} \lor \neg \left(b \leq 5.2 \cdot 10^{+63}\right):\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -1.45e+163) (not (<= b 5.2e+63))) (* (* i t) b) (* (* j a) 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 ((b <= -1.45e+163) || !(b <= 5.2e+63)) {
		tmp = (i * t) * b;
	} else {
		tmp = (j * a) * 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 ((b <= (-1.45d+163)) .or. (.not. (b <= 5.2d+63))) then
        tmp = (i * t) * b
    else
        tmp = (j * a) * 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 ((b <= -1.45e+163) || !(b <= 5.2e+63)) {
		tmp = (i * t) * b;
	} else {
		tmp = (j * a) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (b <= -1.45e+163) or not (b <= 5.2e+63):
		tmp = (i * t) * b
	else:
		tmp = (j * a) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -1.45e+163) || !(b <= 5.2e+63))
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = Float64(Float64(j * a) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((b <= -1.45e+163) || ~((b <= 5.2e+63)))
		tmp = (i * t) * b;
	else
		tmp = (j * a) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -1.45e+163], N[Not[LessEqual[b, 5.2e+63]], $MachinePrecision]], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], N[(N[(j * a), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.44999999999999999e163 or 5.2000000000000002e63 < b

   1. Initial program 73.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(i \cdot t + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      8. lower-neg.f64 67.5

         \[\leadsto \mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b \]

   5. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

      1. lower-*.f64 36.7

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   8. Applied rewrites 36.7%

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   if -1.44999999999999999e163 < b < 5.2000000000000002e63

   1. Initial program 71.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(a \cdot j + \left(\mathsf{neg}\left(b\right)\right) \cdot z\right) \cdot c \]

      4. *-commutative N/A

         \[\leadsto \left(j \cdot a + \left(\mathsf{neg}\left(b\right)\right) \cdot z\right) \cdot c \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(j, a, \left(\mathsf{neg}\left(b\right)\right) \cdot z\right) \cdot c \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(j, a, \left(\mathsf{neg}\left(b\right)\right) \cdot z\right) \cdot c \]

      7. lower-neg.f64 32.4

         \[\leadsto \mathsf{fma}\left(j, a, \left(-b\right) \cdot z\right) \cdot c \]

   5. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, a, \left(-b\right) \cdot z\right) \cdot c} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot j\right) \cdot c \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(j \cdot a\right) \cdot c \]

      2. lower-*.f64 20.9

         \[\leadsto \left(j \cdot a\right) \cdot c \]

   8. Applied rewrites 20.9%

      \[\leadsto \left(j \cdot a\right) \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+163} \lor \neg \left(b \leq 5.2 \cdot 10^{+63}\right):\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot a\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.5e+98)
   (* (* z y) x)
   (if (<= z 1.1e-15) (* (* i t) b) (* (* y x) z))))

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 <= -1.5e+98) {
		tmp = (z * y) * x;
	} else if (z <= 1.1e-15) {
		tmp = (i * t) * b;
	} else {
		tmp = (y * x) * z;
	}
	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 <= (-1.5d+98)) then
        tmp = (z * y) * x
    else if (z <= 1.1d-15) then
        tmp = (i * t) * b
    else
        tmp = (y * x) * z
    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 <= -1.5e+98) {
		tmp = (z * y) * x;
	} else if (z <= 1.1e-15) {
		tmp = (i * t) * b;
	} else {
		tmp = (y * x) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -1.5e+98:
		tmp = (z * y) * x
	elif z <= 1.1e-15:
		tmp = (i * t) * b
	else:
		tmp = (y * x) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.5e+98)
		tmp = Float64(Float64(z * y) * x);
	elseif (z <= 1.1e-15)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = Float64(Float64(y * x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -1.5e+98)
		tmp = (z * y) * x;
	elseif (z <= 1.1e-15)
		tmp = (i * t) * b;
	else
		tmp = (y * x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.5e+98], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.1e-15], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5000000000000001e98

   1. Initial program 69.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot x \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(y \cdot z + \left(\mathsf{neg}\left(a\right)\right) \cdot t\right) \cdot x \]

      6. mul-1-neg N/A

         \[\leadsto \left(y \cdot z + \left(-1 \cdot a\right) \cdot t\right) \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \left(y \cdot z + -1 \cdot \left(a \cdot t\right)\right) \cdot x \]

      8. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right) \cdot x \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot t + y \cdot z\right) \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(a\right)\right) \cdot t + y \cdot z\right) \cdot x \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, y \cdot z\right) \cdot x \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, t, y \cdot z\right) \cdot x \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x \]

      14. lower-*.f64 62.8

          \[\leadsto \mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x \]

   5. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(y \cdot z\right) \cdot x \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(z \cdot y\right) \cdot x \]

      2. lift-*.f64 48.7

         \[\leadsto \left(z \cdot y\right) \cdot x \]

   8. Applied rewrites 48.7%

      \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -1.5000000000000001e98 < z < 1.09999999999999993e-15

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(i \cdot t + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      8. lower-neg.f64 40.6

         \[\leadsto \mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b \]

   5. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto \left(i \cdot t\right) \cdot b \]
   7. Step-by-step derivation

      1. lower-*.f64 27.3

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   8. Applied rewrites 27.3%

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   if 1.09999999999999993e-15 < z

   1. Initial program 52.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) + x \cdot \left(y \cdot z\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + \left(a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) + x \cdot \left(y \cdot z\right)\right)\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot t\right)} \]

   5. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(-t, x, j \cdot c\right), a, \left(z \cdot y\right) \cdot x\right)\right) - \mathsf{fma}\left(-i, t, c \cdot z\right) \cdot b} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x \cdot y + \left(\mathsf{neg}\left(b\right)\right) \cdot c\right) \cdot z \]

      4. mul-1-neg N/A

         \[\leadsto \left(x \cdot y + \left(-1 \cdot b\right) \cdot c\right) \cdot z \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot y + -1 \cdot \left(b \cdot c\right)\right) \cdot z \]

      6. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot b\right) \cdot c + x \cdot y\right) \cdot z \]

      8. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c + x \cdot y\right) \cdot z \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(b\right), c, x \cdot y\right) \cdot z \]

      10. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, c, x \cdot y\right) \cdot z \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z \]

      12. lower-*.f64 69.7

          \[\leadsto \mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z \]

   8. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, c, y \cdot x\right) \cdot z} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot y\right) \cdot z \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(y \cdot x\right) \cdot z \]

       2. lift-*.f64 38.4

          \[\leadsto \left(y \cdot x\right) \cdot z \]

   11. Applied rewrites 38.4%

       \[\leadsto \left(y \cdot x\right) \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 22.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot t\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* i t) b))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * t) * b;
}

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 = (i * t) * b
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 (i * t) * b;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (i * t) * b

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(i * t) * b)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (i * t) * b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 72.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \left(i \cdot t + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(i, t, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(i, t, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   8. lower-neg.f64 41.4

      \[\leadsto \mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b \]

5. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, t, \left(-c\right) \cdot z\right) \cdot b} \]

6. Taylor expanded in z around 0

   \[\leadsto \left(i \cdot t\right) \cdot b \]
7. Step-by-step derivation

   1. lower-*.f64 19.7

      \[\leadsto \left(i \cdot t\right) \cdot b \]

8. Applied rewrites 19.7%

   \[\leadsto \left(i \cdot t\right) \cdot b \]

9. Final simplification 19.7%

   \[\leadsto \left(i \cdot t\right) \cdot b \]
10. Add Preprocessing

Developer Target 1: 59.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) 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 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - 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 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - 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 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - 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 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - 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[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))