Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.3% → 82.4%

Time: 7.7s

Alternatives: 17

Speedup: 0.5×

• 57.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 59.6

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

   8. Applied rewrites 59.6%

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

4. Final simplification 86.2%

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

Alternative 2: 69.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.03 \lor \neg \left(b \leq 1.32 \cdot 10^{-48}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-a, i, c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-i\right) \cdot j, y, \mathsf{fma}\left(\mathsf{fma}\left(-a, x, j \cdot c\right), t, \left(z \cdot y\right) \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -0.03) (not (<= b 1.32e-48)))
   (fma (fma (- a) t (* z y)) x (* (- b) (fma (- a) i (* c z))))
   (fma (* (- i) j) y (fma (fma (- a) x (* j c)) t (* (* z y) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((b <= -0.03) || !(b <= 1.32e-48)) {
		tmp = fma(fma(-a, t, (z * y)), x, (-b * fma(-a, i, (c * z))));
	} else {
		tmp = fma((-i * j), y, fma(fma(-a, x, (j * c)), t, ((z * y) * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -0.03) || !(b <= 1.32e-48))
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(-b) * fma(Float64(-a), i, Float64(c * z))));
	else
		tmp = fma(Float64(Float64(-i) * j), y, fma(fma(Float64(-a), x, Float64(j * c)), t, Float64(Float64(z * y) * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -0.03], N[Not[LessEqual[b, 1.32e-48]], $MachinePrecision]], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[((-a) * i + N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-i) * j), $MachinePrecision] * y + N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t + N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.029999999999999999 or 1.32e-48 < b

   1. Initial program 73.5%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 76.7%

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

   if -0.029999999999999999 < b < 1.32e-48

   1. Initial program 73.7%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 76.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.03 \lor \neg \left(b \leq 1.32 \cdot 10^{-48}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-a, i, c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-i\right) \cdot j, y, \mathsf{fma}\left(\mathsf{fma}\left(-a, x, j \cdot c\right), t, \left(z \cdot y\right) \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{+53} \lor \neg \left(c \leq 2.8 \cdot 10^{+124}\right):\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-a, i, c \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= c -4.6e+53) (not (<= c 2.8e+124)))
   (* (fma j t (* (- b) z)) c)
   (fma (fma (- a) t (* z y)) x (* (- b) (fma (- a) i (* 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 ((c <= -4.6e+53) || !(c <= 2.8e+124)) {
		tmp = fma(j, t, (-b * z)) * c;
	} else {
		tmp = fma(fma(-a, t, (z * y)), x, (-b * fma(-a, i, (c * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((c <= -4.6e+53) || !(c <= 2.8e+124))
		tmp = Float64(fma(j, t, Float64(Float64(-b) * z)) * c);
	else
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(-b) * fma(Float64(-a), i, Float64(c * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[c, -4.6e+53], N[Not[LessEqual[c, 2.8e+124]], $MachinePrecision]], N[(N[(j * t + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[((-a) * i + N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.60000000000000039e53 or 2.8e124 < c

   1. Initial program 59.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -4.60000000000000039e53 < c < 2.8e124

   1. Initial program 81.3%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 72.1%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{+53} \lor \neg \left(c \leq 2.8 \cdot 10^{+124}\right):\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-a, i, c \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 3.65:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, \frac{i}{c}, -z\right) \cdot c\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -1.05e-26)
   (* (fma i a (* (- c) z)) b)
   (if (<= b 1.85e-93)
     (* (fma (- a) t (* z y)) x)
     (if (<= b 3.65)
       (* (fma (- i) y (* c t)) j)
       (* (* (fma a (/ i c) (- z)) c) 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 (b <= -1.05e-26) {
		tmp = fma(i, a, (-c * z)) * b;
	} else if (b <= 1.85e-93) {
		tmp = fma(-a, t, (z * y)) * x;
	} else if (b <= 3.65) {
		tmp = fma(-i, y, (c * t)) * j;
	} else {
		tmp = (fma(a, (i / c), -z) * c) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -1.05e-26)
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	elseif (b <= 1.85e-93)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	elseif (b <= 3.65)
		tmp = Float64(fma(Float64(-i), y, Float64(c * t)) * j);
	else
		tmp = Float64(Float64(fma(a, Float64(i / c), Float64(-z)) * c) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -1.05e-26], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[b, 1.85e-93], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 3.65], N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], N[(N[(N[(a * N[(i / c), $MachinePrecision] + (-z)), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.05000000000000004e-26

   1. Initial program 77.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if -1.05000000000000004e-26 < b < 1.85000000000000001e-93

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. 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 58.0

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

   5. Applied rewrites 58.0%

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

   if 1.85000000000000001e-93 < b < 3.64999999999999991

   1. Initial program 64.5%

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

   3. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if 3.64999999999999991 < b

   1. Initial program 70.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 74.2

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

   8. Applied rewrites 74.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 3.65:\\ \;\;\;\;\mathsf{fma}\left(-i, y, c \cdot t\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, \frac{i}{c}, -z\right) \cdot c\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-10} \lor \neg \left(c \leq 2.8 \cdot 10^{+124}\right):\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= c -3e-10) (not (<= c 2.8e+124)))
   (* (fma j t (* (- b) z)) c)
   (fma (fma (- a) t (* z y)) x (* (* i b) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((c <= -3e-10) || !(c <= 2.8e+124)) {
		tmp = fma(j, t, (-b * z)) * c;
	} else {
		tmp = fma(fma(-a, t, (z * y)), x, ((i * b) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((c <= -3e-10) || !(c <= 2.8e+124))
		tmp = Float64(fma(j, t, Float64(Float64(-b) * z)) * c);
	else
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(i * b) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[c, -3e-10], N[Not[LessEqual[c, 2.8e+124]], $MachinePrecision]], N[(N[(j * t + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3e-10 or 2.8e124 < c

   1. Initial program 62.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -3e-10 < c < 2.8e124

   1. Initial program 81.3%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 69.5

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

   8. Applied rewrites 69.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-10} \lor \neg \left(c \leq 2.8 \cdot 10^{+124}\right):\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(i \cdot b\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{if}\;c \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -3.9 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-270}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+151}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- c) z) b)))
   (if (<= c -1.15e-9)
     t_1
     (if (<= c -3.9e-69)
       (* (* (- i) j) y)
       (if (<= c -4.8e-270)
         (* (* (- a) t) x)
         (if (<= c 1.65e+151) (* (* i b) a) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-c * z) * b;
	double tmp;
	if (c <= -1.15e-9) {
		tmp = t_1;
	} else if (c <= -3.9e-69) {
		tmp = (-i * j) * y;
	} else if (c <= -4.8e-270) {
		tmp = (-a * t) * x;
	} else if (c <= 1.65e+151) {
		tmp = (i * b) * 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 = (-c * z) * b
    if (c <= (-1.15d-9)) then
        tmp = t_1
    else if (c <= (-3.9d-69)) then
        tmp = (-i * j) * y
    else if (c <= (-4.8d-270)) then
        tmp = (-a * t) * x
    else if (c <= 1.65d+151) then
        tmp = (i * b) * 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 = (-c * z) * b;
	double tmp;
	if (c <= -1.15e-9) {
		tmp = t_1;
	} else if (c <= -3.9e-69) {
		tmp = (-i * j) * y;
	} else if (c <= -4.8e-270) {
		tmp = (-a * t) * x;
	} else if (c <= 1.65e+151) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-c * z) * b
	tmp = 0
	if c <= -1.15e-9:
		tmp = t_1
	elif c <= -3.9e-69:
		tmp = (-i * j) * y
	elif c <= -4.8e-270:
		tmp = (-a * t) * x
	elif c <= 1.65e+151:
		tmp = (i * b) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-c) * z) * b)
	tmp = 0.0
	if (c <= -1.15e-9)
		tmp = t_1;
	elseif (c <= -3.9e-69)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (c <= -4.8e-270)
		tmp = Float64(Float64(Float64(-a) * t) * x);
	elseif (c <= 1.65e+151)
		tmp = Float64(Float64(i * b) * 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 = (-c * z) * b;
	tmp = 0.0;
	if (c <= -1.15e-9)
		tmp = t_1;
	elseif (c <= -3.9e-69)
		tmp = (-i * j) * y;
	elseif (c <= -4.8e-270)
		tmp = (-a * t) * x;
	elseif (c <= 1.65e+151)
		tmp = (i * b) * 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[((-c) * z), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[c, -1.15e-9], t$95$1, If[LessEqual[c, -3.9e-69], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, -4.8e-270], N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[c, 1.65e+151], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.15e-9 or 1.65000000000000012e151 < c

   1. Initial program 62.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 59.5

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

   5. Applied rewrites 59.5%

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

   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 52.2

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

   8. Applied rewrites 52.2%

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

   if -1.15e-9 < c < -3.89999999999999981e-69

   1. Initial program 84.2%

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

   3. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 53.8

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

   8. Applied rewrites 53.8%

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

   if -3.89999999999999981e-69 < c < -4.80000000000000003e-270

   1. Initial program 91.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. 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 66.8

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

   5. Applied rewrites 66.8%

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

   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 49.1

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

   8. Applied rewrites 49.1%

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

   if -4.80000000000000003e-270 < c < 1.65000000000000012e151

   1. Initial program 75.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;c \leq -3.9 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-270}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+151}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{if}\;c \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -3.9 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-270}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+151}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- c) z) b)))
   (if (<= c -1.15e-9)
     t_1
     (if (<= c -3.9e-69)
       (* (* (- i) j) y)
       (if (<= c -4.8e-270)
         (* (- a) (* t x))
         (if (<= c 1.65e+151) (* (* i b) a) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-c * z) * b;
	double tmp;
	if (c <= -1.15e-9) {
		tmp = t_1;
	} else if (c <= -3.9e-69) {
		tmp = (-i * j) * y;
	} else if (c <= -4.8e-270) {
		tmp = -a * (t * x);
	} else if (c <= 1.65e+151) {
		tmp = (i * b) * 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 = (-c * z) * b
    if (c <= (-1.15d-9)) then
        tmp = t_1
    else if (c <= (-3.9d-69)) then
        tmp = (-i * j) * y
    else if (c <= (-4.8d-270)) then
        tmp = -a * (t * x)
    else if (c <= 1.65d+151) then
        tmp = (i * b) * 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 = (-c * z) * b;
	double tmp;
	if (c <= -1.15e-9) {
		tmp = t_1;
	} else if (c <= -3.9e-69) {
		tmp = (-i * j) * y;
	} else if (c <= -4.8e-270) {
		tmp = -a * (t * x);
	} else if (c <= 1.65e+151) {
		tmp = (i * b) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-c * z) * b
	tmp = 0
	if c <= -1.15e-9:
		tmp = t_1
	elif c <= -3.9e-69:
		tmp = (-i * j) * y
	elif c <= -4.8e-270:
		tmp = -a * (t * x)
	elif c <= 1.65e+151:
		tmp = (i * b) * a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-c) * z) * b)
	tmp = 0.0
	if (c <= -1.15e-9)
		tmp = t_1;
	elseif (c <= -3.9e-69)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (c <= -4.8e-270)
		tmp = Float64(Float64(-a) * Float64(t * x));
	elseif (c <= 1.65e+151)
		tmp = Float64(Float64(i * b) * 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 = (-c * z) * b;
	tmp = 0.0;
	if (c <= -1.15e-9)
		tmp = t_1;
	elseif (c <= -3.9e-69)
		tmp = (-i * j) * y;
	elseif (c <= -4.8e-270)
		tmp = -a * (t * x);
	elseif (c <= 1.65e+151)
		tmp = (i * b) * 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[((-c) * z), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[c, -1.15e-9], t$95$1, If[LessEqual[c, -3.9e-69], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, -4.8e-270], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.65e+151], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.15e-9 or 1.65000000000000012e151 < c

   1. Initial program 62.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 59.5

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

   5. Applied rewrites 59.5%

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

   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 52.2

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

   8. Applied rewrites 52.2%

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

   if -1.15e-9 < c < -3.89999999999999981e-69

   1. Initial program 84.2%

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

   3. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 53.8

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

   8. Applied rewrites 53.8%

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

   if -3.89999999999999981e-69 < c < -4.80000000000000003e-270

   1. Initial program 91.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 46.3

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

   8. Applied rewrites 46.3%

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

   if -4.80000000000000003e-270 < c < 1.65000000000000012e151

   1. Initial program 75.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 34.9

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

   8. Applied rewrites 34.9%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;c \leq -3.9 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-270}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+151}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 1.46 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma i a (* (- c) z)) b)))
   (if (<= b -1.05e-26)
     t_1
     (if (<= b 8.2e-164)
       (* (fma (- a) t (* z y)) x)
       (if (<= b 1.46e-47) (* (fma (- i) j (* z x)) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(i, a, (-c * z)) * b;
	double tmp;
	if (b <= -1.05e-26) {
		tmp = t_1;
	} else if (b <= 8.2e-164) {
		tmp = fma(-a, t, (z * y)) * x;
	} else if (b <= 1.46e-47) {
		tmp = fma(-i, j, (z * x)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(i, a, Float64(Float64(-c) * z)) * b)
	tmp = 0.0
	if (b <= -1.05e-26)
		tmp = t_1;
	elseif (b <= 8.2e-164)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	elseif (b <= 1.46e-47)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.05e-26], t$95$1, If[LessEqual[b, 8.2e-164], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[b, 1.46e-47], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05000000000000004e-26 or 1.46e-47 < b

   1. Initial program 74.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if -1.05000000000000004e-26 < b < 8.1999999999999996e-164

   1. Initial program 76.8%

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

   3. Taylor expanded in x around 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 60.1

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

   5. Applied rewrites 60.1%

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

   if 8.1999999999999996e-164 < b < 1.46e-47

   1. Initial program 55.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{elif}\;b \leq 1.46 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-244}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma i a (* (- c) z)) b)))
   (if (<= b -1.05e-43)
     t_1
     (if (<= b 5.6e-244)
       (* (* y x) z)
       (if (<= b 8e-103) (* (* (- a) x) t) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(i, a, (-c * z)) * b;
	double tmp;
	if (b <= -1.05e-43) {
		tmp = t_1;
	} else if (b <= 5.6e-244) {
		tmp = (y * x) * z;
	} else if (b <= 8e-103) {
		tmp = (-a * x) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(i, a, Float64(Float64(-c) * z)) * b)
	tmp = 0.0
	if (b <= -1.05e-43)
		tmp = t_1;
	elseif (b <= 5.6e-244)
		tmp = Float64(Float64(y * x) * z);
	elseif (b <= 8e-103)
		tmp = Float64(Float64(Float64(-a) * x) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.05e-43], t$95$1, If[LessEqual[b, 5.6e-244], N[(N[(y * x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 8e-103], N[(N[((-a) * x), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e-43 or 7.99999999999999966e-103 < b

   1. Initial program 73.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 64.2

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

   5. Applied rewrites 64.2%

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

   if -1.05e-43 < b < 5.60000000000000026e-244

   1. Initial program 75.7%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 56.5%

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

   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 40.8

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

   8. Applied rewrites 40.8%

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

   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 37.1

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

   11. Applied rewrites 37.1%

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

   if 5.60000000000000026e-244 < b < 7.99999999999999966e-103

   1. Initial program 68.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 49.1

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

   8. Applied rewrites 49.1%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-244}:\\ \;\;\;\;\left(y \cdot x\right) \cdot z\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-103}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-26} \lor \neg \left(b \leq 1.3 \cdot 10^{-92}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -1.05e-26) (not (<= b 1.3e-92)))
   (* (fma i a (* (- c) z)) b)
   (* (fma (- a) t (* z y)) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((b <= -1.05e-26) || !(b <= 1.3e-92)) {
		tmp = fma(i, a, (-c * z)) * b;
	} else {
		tmp = fma(-a, t, (z * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -1.05e-26) || !(b <= 1.3e-92))
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -1.05e-26], N[Not[LessEqual[b, 1.3e-92]], $MachinePrecision]], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05000000000000004e-26 or 1.3e-92 < b

   1. Initial program 72.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -1.05000000000000004e-26 < b < 1.3e-92

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. 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 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-26} \lor \neg \left(b \leq 1.3 \cdot 10^{-92}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-41} \lor \neg \left(b \leq 6.5 \cdot 10^{-62}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= b -1.45e-41) (not (<= b 6.5e-62)))
   (* (fma i a (* (- c) z)) b)
   (* (fma 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 ((b <= -1.45e-41) || !(b <= 6.5e-62)) {
		tmp = fma(i, a, (-c * z)) * b;
	} else {
		tmp = fma(y, x, (-b * c)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((b <= -1.45e-41) || !(b <= 6.5e-62))
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[b, -1.45e-41], N[Not[LessEqual[b, 6.5e-62]], $MachinePrecision]], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.44999999999999989e-41 or 6.50000000000000026e-62 < b

   1. Initial program 73.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -1.44999999999999989e-41 < b < 6.50000000000000026e-62

   1. Initial program 73.2%

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

   3. Taylor expanded in 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 38.6

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

   5. Applied rewrites 38.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-41} \lor \neg \left(b \leq 6.5 \cdot 10^{-62}\right):\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+160}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-174}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+27}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -5.4e+160)
   (* (* i b) a)
   (if (<= a 5.5e-174)
     (* (* z y) x)
     (if (<= a 1.15e+27) (* (* j c) t) (* (- a) (* t 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 (a <= -5.4e+160) {
		tmp = (i * b) * a;
	} else if (a <= 5.5e-174) {
		tmp = (z * y) * x;
	} else if (a <= 1.15e+27) {
		tmp = (j * c) * t;
	} else {
		tmp = -a * (t * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-5.4d+160)) then
        tmp = (i * b) * a
    else if (a <= 5.5d-174) then
        tmp = (z * y) * x
    else if (a <= 1.15d+27) then
        tmp = (j * c) * t
    else
        tmp = -a * (t * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -5.4e+160) {
		tmp = (i * b) * a;
	} else if (a <= 5.5e-174) {
		tmp = (z * y) * x;
	} else if (a <= 1.15e+27) {
		tmp = (j * c) * t;
	} else {
		tmp = -a * (t * x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -5.4e+160:
		tmp = (i * b) * a
	elif a <= 5.5e-174:
		tmp = (z * y) * x
	elif a <= 1.15e+27:
		tmp = (j * c) * t
	else:
		tmp = -a * (t * x)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -5.4e+160)
		tmp = Float64(Float64(i * b) * a);
	elseif (a <= 5.5e-174)
		tmp = Float64(Float64(z * y) * x);
	elseif (a <= 1.15e+27)
		tmp = Float64(Float64(j * c) * t);
	else
		tmp = Float64(Float64(-a) * Float64(t * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -5.4e+160)
		tmp = (i * b) * a;
	elseif (a <= 5.5e-174)
		tmp = (z * y) * x;
	elseif (a <= 1.15e+27)
		tmp = (j * c) * t;
	else
		tmp = -a * (t * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -5.4e+160], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 5.5e-174], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[a, 1.15e+27], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.4e160

   1. Initial program 63.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.0

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

   8. Applied rewrites 67.0%

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

   if -5.4e160 < a < 5.4999999999999999e-174

   1. Initial program 78.7%

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

   3. Taylor expanded in 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 46.8

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

   5. Applied rewrites 46.8%

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

   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 32.4

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

   8. Applied rewrites 32.4%

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

   if 5.4999999999999999e-174 < a < 1.15e27

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 36.6

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

   8. Applied rewrites 36.6%

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

   if 1.15e27 < a

   1. Initial program 61.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-*.f64 43.9

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

   8. Applied rewrites 43.9%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+160}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-174}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+27}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+160}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-174}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -5.4e+160)
   (* (* i b) a)
   (if (<= a 5.5e-174)
     (* (* z y) x)
     (if (<= a 2.5e-11) (* (* j c) t) (* (* b a) i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -5.4e+160) {
		tmp = (i * b) * a;
	} else if (a <= 5.5e-174) {
		tmp = (z * y) * x;
	} else if (a <= 2.5e-11) {
		tmp = (j * c) * t;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-5.4d+160)) then
        tmp = (i * b) * a
    else if (a <= 5.5d-174) then
        tmp = (z * y) * x
    else if (a <= 2.5d-11) then
        tmp = (j * c) * t
    else
        tmp = (b * a) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -5.4e+160) {
		tmp = (i * b) * a;
	} else if (a <= 5.5e-174) {
		tmp = (z * y) * x;
	} else if (a <= 2.5e-11) {
		tmp = (j * c) * t;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -5.4e+160:
		tmp = (i * b) * a
	elif a <= 5.5e-174:
		tmp = (z * y) * x
	elif a <= 2.5e-11:
		tmp = (j * c) * t
	else:
		tmp = (b * a) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -5.4e+160)
		tmp = Float64(Float64(i * b) * a);
	elseif (a <= 5.5e-174)
		tmp = Float64(Float64(z * y) * x);
	elseif (a <= 2.5e-11)
		tmp = Float64(Float64(j * c) * t);
	else
		tmp = Float64(Float64(b * a) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -5.4e+160)
		tmp = (i * b) * a;
	elseif (a <= 5.5e-174)
		tmp = (z * y) * x;
	elseif (a <= 2.5e-11)
		tmp = (j * c) * t;
	else
		tmp = (b * a) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -5.4e+160], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 5.5e-174], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[a, 2.5e-11], N[(N[(j * c), $MachinePrecision] * t), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.4e160

   1. Initial program 63.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.0

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

   8. Applied rewrites 67.0%

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

   if -5.4e160 < a < 5.4999999999999999e-174

   1. Initial program 78.7%

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

   3. Taylor expanded in 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 46.8

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

   5. Applied rewrites 46.8%

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

   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 32.4

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

   8. Applied rewrites 32.4%

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

   if 5.4999999999999999e-174 < a < 2.50000000000000009e-11

   1. Initial program 86.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 36.9

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

   8. Applied rewrites 36.9%

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

   if 2.50000000000000009e-11 < a

   1. Initial program 61.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 37.7

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

   8. Applied rewrites 37.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 40.2

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

   10. Applied rewrites 40.2%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+160}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-174}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\left(j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+160}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-215}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= a -5.4e+160)
   (* (* i b) a)
   (if (<= a 2.25e-215)
     (* (* z y) x)
     (if (<= a 2.7e-11) (* (* j t) c) (* (* b a) i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -5.4e+160) {
		tmp = (i * b) * a;
	} else if (a <= 2.25e-215) {
		tmp = (z * y) * x;
	} else if (a <= 2.7e-11) {
		tmp = (j * t) * c;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (a <= (-5.4d+160)) then
        tmp = (i * b) * a
    else if (a <= 2.25d-215) then
        tmp = (z * y) * x
    else if (a <= 2.7d-11) then
        tmp = (j * t) * c
    else
        tmp = (b * a) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (a <= -5.4e+160) {
		tmp = (i * b) * a;
	} else if (a <= 2.25e-215) {
		tmp = (z * y) * x;
	} else if (a <= 2.7e-11) {
		tmp = (j * t) * c;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if a <= -5.4e+160:
		tmp = (i * b) * a
	elif a <= 2.25e-215:
		tmp = (z * y) * x
	elif a <= 2.7e-11:
		tmp = (j * t) * c
	else:
		tmp = (b * a) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (a <= -5.4e+160)
		tmp = Float64(Float64(i * b) * a);
	elseif (a <= 2.25e-215)
		tmp = Float64(Float64(z * y) * x);
	elseif (a <= 2.7e-11)
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = Float64(Float64(b * a) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (a <= -5.4e+160)
		tmp = (i * b) * a;
	elseif (a <= 2.25e-215)
		tmp = (z * y) * x;
	elseif (a <= 2.7e-11)
		tmp = (j * t) * c;
	else
		tmp = (b * a) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[a, -5.4e+160], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 2.25e-215], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[a, 2.7e-11], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.4e160

   1. Initial program 63.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.0

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

   8. Applied rewrites 67.0%

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

   if -5.4e160 < a < 2.25e-215

   1. Initial program 78.3%

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

   3. Taylor expanded in 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 47.8

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

   5. Applied rewrites 47.8%

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

   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 32.6

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

   8. Applied rewrites 32.6%

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

   if 2.25e-215 < a < 2.70000000000000005e-11

   1. Initial program 86.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 33.7

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

   8. Applied rewrites 33.7%

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

   if 2.70000000000000005e-11 < a

   1. Initial program 61.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 37.7

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

   8. Applied rewrites 37.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 40.2

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

   10. Applied rewrites 40.2%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+160}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-215}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+48} \lor \neg \left(t \leq 1.5 \cdot 10^{-77}\right):\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -1.25e+48) (not (<= t 1.5e-77))) (* (* j t) c) (* (* i b) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -1.25e+48) || !(t <= 1.5e-77)) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if ((t <= (-1.25d+48)) .or. (.not. (t <= 1.5d-77))) then
        tmp = (j * t) * c
    else
        tmp = (i * b) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -1.25e+48) || !(t <= 1.5e-77)) {
		tmp = (j * t) * c;
	} else {
		tmp = (i * b) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (t <= -1.25e+48) or not (t <= 1.5e-77):
		tmp = (j * t) * c
	else:
		tmp = (i * b) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -1.25e+48) || !(t <= 1.5e-77))
		tmp = Float64(Float64(j * t) * c);
	else
		tmp = Float64(Float64(i * b) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((t <= -1.25e+48) || ~((t <= 1.5e-77)))
		tmp = (j * t) * c;
	else
		tmp = (i * b) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -1.25e+48], N[Not[LessEqual[t, 1.5e-77]], $MachinePrecision]], N[(N[(j * t), $MachinePrecision] * c), $MachinePrecision], N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999993e48 or 1.50000000000000008e-77 < t

   1. Initial program 65.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 33.5

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

   8. Applied rewrites 33.5%

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

   if -1.24999999999999993e48 < t < 1.50000000000000008e-77

   1. Initial program 81.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 33.3

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

   8. Applied rewrites 33.3%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+48} \lor \neg \left(t \leq 1.5 \cdot 10^{-77}\right):\\ \;\;\;\;\left(j \cdot t\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 22.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot b\right) \cdot a \end{array} \]

FPCore

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

C

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

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 * b) * a
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 * b) * a;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * b), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 44.1

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

5. Applied rewrites 44.1%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 25.2

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

8. Applied rewrites 25.2%

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

9. Final simplification 25.2%

   \[\leadsto \left(i \cdot b\right) \cdot a \]
10. Add Preprocessing

Alternative 17: 21.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot a\right) \cdot i \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 44.1

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

5. Applied rewrites 44.1%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 25.2

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

8. Applied rewrites 25.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 24.8

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

10. Applied rewrites 24.8%

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

11. Final simplification 24.8%

    \[\leadsto \left(b \cdot a\right) \cdot i \]
12. Add Preprocessing

Developer Target 1: 68.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))