Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 72.7% → 81.1%

Time: 6.5s

Alternatives: 23

Speedup: 0.5×

• 58.2% 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: 72.7% 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: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \mathsf{fma}\left(j \cdot c, t, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\right) - \mathsf{fma}\left(-a, i, c \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, \frac{z}{j}, -i\right) \cdot j\right) \cdot y\\ \end{array} \end{array} \]

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 5e+292)
     t_1
     (if (<= t_1 INFINITY)
       (-
        (fma (- a) (* t x) (fma (* j c) t (* (fma (- i) j (* z x)) y)))
        (* (fma (- a) i (* c z)) b))
       (* (* (fma x (/ z j) (- i)) j) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
	double tmp;
	if (t_1 <= 5e+292) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(-a, (t * x), fma((j * c), t, (fma(-i, j, (z * x)) * y))) - (fma(-a, i, (c * z)) * b);
	} else {
		tmp = (fma(x, (z / j), -i) * j) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
	tmp = 0.0
	if (t_1 <= 5e+292)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(-a), Float64(t * x), fma(Float64(j * c), t, Float64(fma(Float64(-i), j, Float64(z * x)) * y))) - Float64(fma(Float64(-a), i, Float64(c * z)) * b));
	else
		tmp = Float64(Float64(fma(x, Float64(z / j), Float64(-i)) * j) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(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, 5e+292], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[((-a) * N[(t * x), $MachinePrecision] + N[(N[(j * c), $MachinePrecision] * t + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[((-a) * i + N[(c * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(z / j), $MachinePrecision] + (-i)), $MachinePrecision] * j), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < 4.9999999999999996e292

   1. Initial program 93.5%

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

   if 4.9999999999999996e292 < (+.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 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. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 83.0%

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

   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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 42.1

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 45.1

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

   7. Applied rewrites 45.1%

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

Alternative 2: 81.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(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(x, Float64(z / j), Float64(-i)) * j) * y);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.4%

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

   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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 42.1

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in j around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-neg.f64 45.1

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

   7. Applied rewrites 45.1%

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

Alternative 3: 68.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -9.2 \cdot 10^{+249}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y - a \cdot b\right)\\ \mathbf{elif}\;i \leq 1.35 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-a, t \cdot x, \mathsf{fma}\left(j \cdot c, t, \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\right)\right) - \left(c \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-i\right) \cdot \mathsf{fma}\left(j, y, \left(-a\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -9.2e+249)
   (* (- i) (- (* j y) (* a b)))
   (if (<= i 1.35e+34)
     (-
      (fma (- a) (* t x) (fma (* j c) t (* (fma (- i) j (* z x)) y)))
      (* (* c z) b))
     (* (- i) (fma j y (* (- a) b))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -9.2e+249) {
		tmp = -i * ((j * y) - (a * b));
	} else if (i <= 1.35e+34) {
		tmp = fma(-a, (t * x), fma((j * c), t, (fma(-i, j, (z * x)) * y))) - ((c * z) * b);
	} else {
		tmp = -i * fma(j, y, (-a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -9.2e+249)
		tmp = Float64(Float64(-i) * Float64(Float64(j * y) - Float64(a * b)));
	elseif (i <= 1.35e+34)
		tmp = Float64(fma(Float64(-a), Float64(t * x), fma(Float64(j * c), t, Float64(fma(Float64(-i), j, Float64(z * x)) * y))) - Float64(Float64(c * z) * b));
	else
		tmp = Float64(Float64(-i) * fma(j, y, Float64(Float64(-a) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -9.2e+249], N[((-i) * N[(N[(j * y), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.35e+34], N[(N[((-a) * N[(t * x), $MachinePrecision] + N[(N[(j * c), $MachinePrecision] * t + N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[((-i) * N[(j * y + N[((-a) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -9.1999999999999993e249

   1. Initial program 53.3%

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

   2. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y - a \cdot b\right)\right)} \]
   3. 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 77.6

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

   4. Applied rewrites 77.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 77.1

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

   6. Applied rewrites 77.1%

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

   if -9.1999999999999993e249 < i < 1.35e34

   1. Initial program 76.2%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in z around inf

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

      1. lift-*.f64 70.1

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

   7. Applied rewrites 70.1%

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

   if 1.35e34 < i

   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. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y - a \cdot b\right)\right)} \]
   3. 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 61.9

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

   4. Applied rewrites 61.9%

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

Alternative 4: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-a\right) \cdot \mathsf{fma}\left(t, x, \left(-b\right) \cdot i\right)\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\mathsf{fma}\left(j \cdot c, t, \left(-i\right) \cdot \left(j \cdot y\right)\right) - \left(c \cdot z\right) \cdot b\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- a) (fma t x (* (- b) i)))))
   (if (<= a -1.3e+105)
     t_1
     (if (<= a 2.35e-138)
       (- (fma (* j c) t (* (- i) (* j y))) (* (* c z) b))
       (if (<= a 9.6e+88) (+ (* (* z y) x) (* j (- (* c t) (* i 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 = -a * fma(t, x, (-b * i));
	double tmp;
	if (a <= -1.3e+105) {
		tmp = t_1;
	} else if (a <= 2.35e-138) {
		tmp = fma((j * c), t, (-i * (j * y))) - ((c * z) * b);
	} else if (a <= 9.6e+88) {
		tmp = ((z * y) * x) + (j * ((c * t) - (i * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(-a) * fma(t, x, Float64(Float64(-b) * i)))
	tmp = 0.0
	if (a <= -1.3e+105)
		tmp = t_1;
	elseif (a <= 2.35e-138)
		tmp = Float64(fma(Float64(j * c), t, Float64(Float64(-i) * Float64(j * y))) - Float64(Float64(c * z) * b));
	elseif (a <= 9.6e+88)
		tmp = Float64(Float64(Float64(z * y) * x) + Float64(j * Float64(Float64(c * t) - Float64(i * 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[((-a) * N[(t * x + N[((-b) * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.3e+105], t$95$1, If[LessEqual[a, 2.35e-138], N[(N[(N[(j * c), $MachinePrecision] * t + N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.6e+88], N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3000000000000001e105 or 9.5999999999999996e88 < a

   1. Initial program 61.4%

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

   2. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 68.6

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

   4. Applied rewrites 68.6%

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

   if -1.3000000000000001e105 < a < 2.3500000000000001e-138

   1. Initial program 78.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in z around inf

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

      1. lift-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-*.f64 54.4

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

   10. Applied rewrites 54.4%

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

   if 2.3500000000000001e-138 < a < 9.5999999999999996e88

   1. Initial program 78.4%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 52.7

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

   4. Applied rewrites 52.7%

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

Alternative 5: 60.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-a\right) \cdot \mathsf{fma}\left(t, x, \left(-b\right) \cdot i\right)\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- a) (fma t x (* (- b) i)))))
   (if (<= a -1.4e+105)
     t_1
     (if (<= a 9.6e+88) (+ (* (* z y) x) (* j (- (* c t) (* i 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 = -a * fma(t, x, (-b * i));
	double tmp;
	if (a <= -1.4e+105) {
		tmp = t_1;
	} else if (a <= 9.6e+88) {
		tmp = ((z * y) * x) + (j * ((c * t) - (i * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(-a) * fma(t, x, Float64(Float64(-b) * i)))
	tmp = 0.0
	if (a <= -1.4e+105)
		tmp = t_1;
	elseif (a <= 9.6e+88)
		tmp = Float64(Float64(Float64(z * y) * x) + Float64(j * Float64(Float64(c * t) - Float64(i * 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[((-a) * N[(t * x + N[((-b) * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.4e+105], t$95$1, If[LessEqual[a, 9.6e+88], N[(N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4000000000000001e105 or 9.5999999999999996e88 < a

   1. Initial program 61.4%

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

   2. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 68.6

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

   4. Applied rewrites 68.6%

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

   if -1.4000000000000001e105 < a < 9.5999999999999996e88

   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. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

Alternative 6: 28.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \left(a \cdot \left(t \cdot x\right)\right)\\ t_2 := \left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{if}\;i \leq -1.82 \cdot 10^{-35}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq -7.8 \cdot 10^{-287}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 7.5 \cdot 10^{-263}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{-148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 1.25 \cdot 10^{+101}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* -1.0 (* a (* t x)))) (t_2 (* (* (- i) j) y)))
   (if (<= i -1.82e-35)
     t_2
     (if (<= i -7.8e-287)
       t_1
       (if (<= i 7.5e-263)
         (* (* (- b) z) c)
         (if (<= i 9.5e-148) t_1 (if (<= i 1.25e+101) t_2 (* (* 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 t_1 = -1.0 * (a * (t * x));
	double t_2 = (-i * j) * y;
	double tmp;
	if (i <= -1.82e-35) {
		tmp = t_2;
	} else if (i <= -7.8e-287) {
		tmp = t_1;
	} else if (i <= 7.5e-263) {
		tmp = (-b * z) * c;
	} else if (i <= 9.5e-148) {
		tmp = t_1;
	} else if (i <= 1.25e+101) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-1.0d0) * (a * (t * x))
    t_2 = (-i * j) * y
    if (i <= (-1.82d-35)) then
        tmp = t_2
    else if (i <= (-7.8d-287)) then
        tmp = t_1
    else if (i <= 7.5d-263) then
        tmp = (-b * z) * c
    else if (i <= 9.5d-148) then
        tmp = t_1
    else if (i <= 1.25d+101) then
        tmp = t_2
    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 t_1 = -1.0 * (a * (t * x));
	double t_2 = (-i * j) * y;
	double tmp;
	if (i <= -1.82e-35) {
		tmp = t_2;
	} else if (i <= -7.8e-287) {
		tmp = t_1;
	} else if (i <= 7.5e-263) {
		tmp = (-b * z) * c;
	} else if (i <= 9.5e-148) {
		tmp = t_1;
	} else if (i <= 1.25e+101) {
		tmp = t_2;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = -1.0 * (a * (t * x))
	t_2 = (-i * j) * y
	tmp = 0
	if i <= -1.82e-35:
		tmp = t_2
	elif i <= -7.8e-287:
		tmp = t_1
	elif i <= 7.5e-263:
		tmp = (-b * z) * c
	elif i <= 9.5e-148:
		tmp = t_1
	elif i <= 1.25e+101:
		tmp = t_2
	else:
		tmp = (b * a) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(-1.0 * Float64(a * Float64(t * x)))
	t_2 = Float64(Float64(Float64(-i) * j) * y)
	tmp = 0.0
	if (i <= -1.82e-35)
		tmp = t_2;
	elseif (i <= -7.8e-287)
		tmp = t_1;
	elseif (i <= 7.5e-263)
		tmp = Float64(Float64(Float64(-b) * z) * c);
	elseif (i <= 9.5e-148)
		tmp = t_1;
	elseif (i <= 1.25e+101)
		tmp = t_2;
	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)
	t_1 = -1.0 * (a * (t * x));
	t_2 = (-i * j) * y;
	tmp = 0.0;
	if (i <= -1.82e-35)
		tmp = t_2;
	elseif (i <= -7.8e-287)
		tmp = t_1;
	elseif (i <= 7.5e-263)
		tmp = (-b * z) * c;
	elseif (i <= 9.5e-148)
		tmp = t_1;
	elseif (i <= 1.25e+101)
		tmp = t_2;
	else
		tmp = (b * a) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(-1.0 * N[(a * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[i, -1.82e-35], t$95$2, If[LessEqual[i, -7.8e-287], t$95$1, If[LessEqual[i, 7.5e-263], N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[i, 9.5e-148], t$95$1, If[LessEqual[i, 1.25e+101], t$95$2, N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if i < -1.82000000000000003e-35 or 9.50000000000000069e-148 < i < 1.24999999999999997e101

   1. Initial program 70.3%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 40.9

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 26.0

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

   7. Applied rewrites 26.0%

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

   if -1.82000000000000003e-35 < i < -7.7999999999999999e-287 or 7.50000000000000044e-263 < i < 9.50000000000000069e-148

   1. Initial program 81.2%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 27.1

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

   7. Applied rewrites 27.1%

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

   if -7.7999999999999999e-287 < i < 7.50000000000000044e-263

   1. Initial program 79.6%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   3. 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 48.4

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 26.0

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

   7. Applied rewrites 26.0%

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

   if 1.24999999999999997e101 < i

   1. Initial program 62.3%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 39.3

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

   7. Applied rewrites 39.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 39.7

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

   9. Applied rewrites 39.7%

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

Alternative 7: 30.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.9 \cdot 10^{+72}:\\ \;\;\;\;\left(c \cdot j\right) \cdot t\\ \mathbf{elif}\;j \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;j \leq 6 \cdot 10^{-309}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;j \leq 3.5 \cdot 10^{-232}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;j \leq 3.4 \cdot 10^{-66}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.9e+72)
   (* (* c j) t)
   (if (<= j -7.4e-9)
     (* (* (- i) j) y)
     (if (<= j 6e-309)
       (* (* z y) x)
       (if (<= j 3.5e-232)
         (* (* b a) i)
         (if (<= j 3.4e-66) (* (* z x) y) (* (- i) (* j y))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -2.9e+72) {
		tmp = (c * j) * t;
	} else if (j <= -7.4e-9) {
		tmp = (-i * j) * y;
	} else if (j <= 6e-309) {
		tmp = (z * y) * x;
	} else if (j <= 3.5e-232) {
		tmp = (b * a) * i;
	} else if (j <= 3.4e-66) {
		tmp = (z * x) * y;
	} else {
		tmp = -i * (j * y);
	}
	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 (j <= (-2.9d+72)) then
        tmp = (c * j) * t
    else if (j <= (-7.4d-9)) then
        tmp = (-i * j) * y
    else if (j <= 6d-309) then
        tmp = (z * y) * x
    else if (j <= 3.5d-232) then
        tmp = (b * a) * i
    else if (j <= 3.4d-66) then
        tmp = (z * x) * y
    else
        tmp = -i * (j * y)
    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 (j <= -2.9e+72) {
		tmp = (c * j) * t;
	} else if (j <= -7.4e-9) {
		tmp = (-i * j) * y;
	} else if (j <= 6e-309) {
		tmp = (z * y) * x;
	} else if (j <= 3.5e-232) {
		tmp = (b * a) * i;
	} else if (j <= 3.4e-66) {
		tmp = (z * x) * y;
	} else {
		tmp = -i * (j * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if j <= -2.9e+72:
		tmp = (c * j) * t
	elif j <= -7.4e-9:
		tmp = (-i * j) * y
	elif j <= 6e-309:
		tmp = (z * y) * x
	elif j <= 3.5e-232:
		tmp = (b * a) * i
	elif j <= 3.4e-66:
		tmp = (z * x) * y
	else:
		tmp = -i * (j * y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.9e+72)
		tmp = Float64(Float64(c * j) * t);
	elseif (j <= -7.4e-9)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (j <= 6e-309)
		tmp = Float64(Float64(z * y) * x);
	elseif (j <= 3.5e-232)
		tmp = Float64(Float64(b * a) * i);
	elseif (j <= 3.4e-66)
		tmp = Float64(Float64(z * x) * y);
	else
		tmp = Float64(Float64(-i) * Float64(j * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (j <= -2.9e+72)
		tmp = (c * j) * t;
	elseif (j <= -7.4e-9)
		tmp = (-i * j) * y;
	elseif (j <= 6e-309)
		tmp = (z * y) * x;
	elseif (j <= 3.5e-232)
		tmp = (b * a) * i;
	elseif (j <= 3.4e-66)
		tmp = (z * x) * y;
	else
		tmp = -i * (j * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.9e+72], N[(N[(c * j), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, -7.4e-9], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[j, 6e-309], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[j, 3.5e-232], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[j, 3.4e-66], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if j < -2.90000000000000017e72

   1. Initial program 70.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 38.8

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

   7. Applied rewrites 38.8%

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

   if -2.90000000000000017e72 < j < -7.4e-9

   1. Initial program 76.9%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 41.9

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 23.9

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

   7. Applied rewrites 23.9%

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

   if -7.4e-9 < j < 6.000000000000001e-309

   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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 26.0

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

   7. Applied rewrites 26.0%

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

   if 6.000000000000001e-309 < j < 3.4999999999999998e-232

   1. Initial program 70.1%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 29.2

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

   7. Applied rewrites 29.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.6

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

   9. Applied rewrites 31.6%

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

   if 3.4999999999999998e-232 < j < 3.39999999999999997e-66

   1. Initial program 72.1%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 34.4

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

   4. Applied rewrites 34.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 28.0

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

   7. Applied rewrites 28.0%

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

   if 3.39999999999999997e-66 < j

   1. Initial program 73.4%

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

   2. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y - a \cdot b\right)\right)} \]
   3. 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 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 31.2

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

   7. Applied rewrites 31.2%

      \[\leadsto \left(-i\right) \cdot \left(j \cdot \color{blue}{y}\right) \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 8: 30.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{if}\;j \leq -2.9 \cdot 10^{+72}:\\ \;\;\;\;\left(c \cdot j\right) \cdot t\\ \mathbf{elif}\;j \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 6 \cdot 10^{-309}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;j \leq 3.5 \cdot 10^{-232}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;j \leq 3.4 \cdot 10^{-66}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- i) j) y)))
   (if (<= j -2.9e+72)
     (* (* c j) t)
     (if (<= j -7.4e-9)
       t_1
       (if (<= j 6e-309)
         (* (* z y) x)
         (if (<= j 3.5e-232)
           (* (* b a) i)
           (if (<= j 3.4e-66) (* (* 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 = (-i * j) * y;
	double tmp;
	if (j <= -2.9e+72) {
		tmp = (c * j) * t;
	} else if (j <= -7.4e-9) {
		tmp = t_1;
	} else if (j <= 6e-309) {
		tmp = (z * y) * x;
	} else if (j <= 3.5e-232) {
		tmp = (b * a) * i;
	} else if (j <= 3.4e-66) {
		tmp = (z * x) * y;
	} 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 = (-i * j) * y
    if (j <= (-2.9d+72)) then
        tmp = (c * j) * t
    else if (j <= (-7.4d-9)) then
        tmp = t_1
    else if (j <= 6d-309) then
        tmp = (z * y) * x
    else if (j <= 3.5d-232) then
        tmp = (b * a) * i
    else if (j <= 3.4d-66) then
        tmp = (z * x) * y
    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 = (-i * j) * y;
	double tmp;
	if (j <= -2.9e+72) {
		tmp = (c * j) * t;
	} else if (j <= -7.4e-9) {
		tmp = t_1;
	} else if (j <= 6e-309) {
		tmp = (z * y) * x;
	} else if (j <= 3.5e-232) {
		tmp = (b * a) * i;
	} else if (j <= 3.4e-66) {
		tmp = (z * x) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-i * j) * y
	tmp = 0
	if j <= -2.9e+72:
		tmp = (c * j) * t
	elif j <= -7.4e-9:
		tmp = t_1
	elif j <= 6e-309:
		tmp = (z * y) * x
	elif j <= 3.5e-232:
		tmp = (b * a) * i
	elif j <= 3.4e-66:
		tmp = (z * x) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-i) * j) * y)
	tmp = 0.0
	if (j <= -2.9e+72)
		tmp = Float64(Float64(c * j) * t);
	elseif (j <= -7.4e-9)
		tmp = t_1;
	elseif (j <= 6e-309)
		tmp = Float64(Float64(z * y) * x);
	elseif (j <= 3.5e-232)
		tmp = Float64(Float64(b * a) * i);
	elseif (j <= 3.4e-66)
		tmp = Float64(Float64(z * x) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-i * j) * y;
	tmp = 0.0;
	if (j <= -2.9e+72)
		tmp = (c * j) * t;
	elseif (j <= -7.4e-9)
		tmp = t_1;
	elseif (j <= 6e-309)
		tmp = (z * y) * x;
	elseif (j <= 3.5e-232)
		tmp = (b * a) * i;
	elseif (j <= 3.4e-66)
		tmp = (z * x) * y;
	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[((-i) * j), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[j, -2.9e+72], N[(N[(c * j), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, -7.4e-9], t$95$1, If[LessEqual[j, 6e-309], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[j, 3.5e-232], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[j, 3.4e-66], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if j < -2.90000000000000017e72

   1. Initial program 70.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 38.8

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

   7. Applied rewrites 38.8%

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

   if -2.90000000000000017e72 < j < -7.4e-9 or 3.39999999999999997e-66 < j

   1. Initial program 74.0%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 43.3

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 29.9

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

   7. Applied rewrites 29.9%

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

   if -7.4e-9 < j < 6.000000000000001e-309

   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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 26.0

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

   7. Applied rewrites 26.0%

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

   if 6.000000000000001e-309 < j < 3.4999999999999998e-232

   1. Initial program 70.1%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 29.2

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

   7. Applied rewrites 29.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.6

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

   9. Applied rewrites 31.6%

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

   if 3.4999999999999998e-232 < j < 3.39999999999999997e-66

   1. Initial program 72.1%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 34.4

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

   4. Applied rewrites 34.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 28.0

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

   7. Applied rewrites 28.0%

      \[\leadsto \left(z \cdot x\right) \cdot y \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\left(c \cdot j\right) \cdot t\\ \mathbf{elif}\;j \leq 1.5 \cdot 10^{-236}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;j \leq 2.05 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -3.1e+72)
   (* (* c j) t)
   (if (<= j 1.5e-236)
     (* (fma i a (* (- c) z)) b)
     (if (<= j 2.05e+174) (* (fma y x (* (- b) c)) z) (* (- i) (* j y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -3.1e+72) {
		tmp = (c * j) * t;
	} else if (j <= 1.5e-236) {
		tmp = fma(i, a, (-c * z)) * b;
	} else if (j <= 2.05e+174) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = -i * (j * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -3.1e+72)
		tmp = Float64(Float64(c * j) * t);
	elseif (j <= 1.5e-236)
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	elseif (j <= 2.05e+174)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	else
		tmp = Float64(Float64(-i) * Float64(j * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -3.1e+72], N[(N[(c * j), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 1.5e-236], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[j, 2.05e+174], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -3.09999999999999988e72

   1. Initial program 70.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 38.8

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

   7. Applied rewrites 38.8%

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

   if -3.09999999999999988e72 < j < 1.50000000000000007e-236

   1. Initial program 73.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 45.3

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

   4. Applied rewrites 45.3%

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

   if 1.50000000000000007e-236 < j < 2.05000000000000015e174

   1. Initial program 73.9%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. 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 40.6

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

   4. Applied rewrites 40.6%

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

   if 2.05000000000000015e174 < j

   1. Initial program 69.9%

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

   2. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y - a \cdot b\right)\right)} \]
   3. 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 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 42.7

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

   7. Applied rewrites 42.7%

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

Alternative 10: 30.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot j\right) \cdot t\\ \mathbf{if}\;j \leq -24000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 6 \cdot 10^{-309}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{elif}\;j \leq 3.5 \cdot 10^{-232}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;j \leq 3.6 \cdot 10^{-66}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* c j) t)))
   (if (<= j -24000000000.0)
     t_1
     (if (<= j 6e-309)
       (* (* z y) x)
       (if (<= j 3.5e-232)
         (* (* b a) i)
         (if (<= j 3.6e-66) (* (* 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 = (c * j) * t;
	double tmp;
	if (j <= -24000000000.0) {
		tmp = t_1;
	} else if (j <= 6e-309) {
		tmp = (z * y) * x;
	} else if (j <= 3.5e-232) {
		tmp = (b * a) * i;
	} else if (j <= 3.6e-66) {
		tmp = (z * x) * y;
	} 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 * j) * t
    if (j <= (-24000000000.0d0)) then
        tmp = t_1
    else if (j <= 6d-309) then
        tmp = (z * y) * x
    else if (j <= 3.5d-232) then
        tmp = (b * a) * i
    else if (j <= 3.6d-66) then
        tmp = (z * x) * y
    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 * j) * t;
	double tmp;
	if (j <= -24000000000.0) {
		tmp = t_1;
	} else if (j <= 6e-309) {
		tmp = (z * y) * x;
	} else if (j <= 3.5e-232) {
		tmp = (b * a) * i;
	} else if (j <= 3.6e-66) {
		tmp = (z * x) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (c * j) * t
	tmp = 0
	if j <= -24000000000.0:
		tmp = t_1
	elif j <= 6e-309:
		tmp = (z * y) * x
	elif j <= 3.5e-232:
		tmp = (b * a) * i
	elif j <= 3.6e-66:
		tmp = (z * x) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * j) * t)
	tmp = 0.0
	if (j <= -24000000000.0)
		tmp = t_1;
	elseif (j <= 6e-309)
		tmp = Float64(Float64(z * y) * x);
	elseif (j <= 3.5e-232)
		tmp = Float64(Float64(b * a) * i);
	elseif (j <= 3.6e-66)
		tmp = Float64(Float64(z * x) * y);
	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 * j) * t;
	tmp = 0.0;
	if (j <= -24000000000.0)
		tmp = t_1;
	elseif (j <= 6e-309)
		tmp = (z * y) * x;
	elseif (j <= 3.5e-232)
		tmp = (b * a) * i;
	elseif (j <= 3.6e-66)
		tmp = (z * x) * y;
	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 * j), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[j, -24000000000.0], t$95$1, If[LessEqual[j, 6e-309], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[j, 3.5e-232], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[j, 3.6e-66], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if j < -2.4e10 or 3.60000000000000012e-66 < j

   1. Initial program 72.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 45.4

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

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 33.0

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

   7. Applied rewrites 33.0%

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

   if -2.4e10 < j < 6.000000000000001e-309

   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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 33.3

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

   4. Applied rewrites 33.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 25.7

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

   7. Applied rewrites 25.7%

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

   if 6.000000000000001e-309 < j < 3.4999999999999998e-232

   1. Initial program 70.1%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 29.2

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

   7. Applied rewrites 29.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.6

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

   9. Applied rewrites 31.6%

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

   if 3.4999999999999998e-232 < j < 3.60000000000000012e-66

   1. Initial program 72.1%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 34.4

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

   4. Applied rewrites 34.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 28.0

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

   7. Applied rewrites 28.0%

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

Alternative 11: 52.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) x (* j c)) t)))
   (if (<= t -3.1e-12)
     t_1
     (if (<= t 1.3e-42) (* (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(-a, x, (j * c)) * t;
	double tmp;
	if (t <= -3.1e-12) {
		tmp = t_1;
	} else if (t <= 1.3e-42) {
		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(Float64(-a), x, Float64(j * c)) * t)
	tmp = 0.0
	if (t <= -3.1e-12)
		tmp = t_1;
	elseif (t <= 1.3e-42)
		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[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -3.1e-12], t$95$1, If[LessEqual[t, 1.3e-42], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1000000000000001e-12 or 1.3e-42 < t

   1. Initial program 66.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 57.1

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

   4. Applied rewrites 57.1%

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

   if -3.1000000000000001e-12 < t < 1.3e-42

   1. Initial program 80.5%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 46.2

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

   4. Applied rewrites 46.2%

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

Alternative 12: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- a) x (* j c)) t)))
   (if (<= t -4.3e-12)
     t_1
     (if (<= t 8.5e-31) (* (fma y x (* (- b) c)) z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-a, x, (j * c)) * t;
	double tmp;
	if (t <= -4.3e-12) {
		tmp = t_1;
	} else if (t <= 8.5e-31) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-a), x, Float64(j * c)) * t)
	tmp = 0.0
	if (t <= -4.3e-12)
		tmp = t_1;
	elseif (t <= 8.5e-31)
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	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[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -4.3e-12], t$95$1, If[LessEqual[t, 8.5e-31], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.29999999999999985e-12 or 8.5000000000000007e-31 < t

   1. Initial program 66.3%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 57.5

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

   4. Applied rewrites 57.5%

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

   if -4.29999999999999985e-12 < t < 8.5000000000000007e-31

   1. Initial program 80.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. 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 46.8

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

   4. Applied rewrites 46.8%

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

Alternative 13: 52.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{if}\;c \leq -9 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma j t (* (- b) z)) c)))
   (if (<= c -9e+79) t_1 (if (<= c 7.6e-33) (* (fma (- a) t (* z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(j, t, (-b * z)) * c;
	double tmp;
	if (c <= -9e+79) {
		tmp = t_1;
	} else if (c <= 7.6e-33) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(j, t, Float64(Float64(-b) * z)) * c)
	tmp = 0.0
	if (c <= -9e+79)
		tmp = t_1;
	elseif (c <= 7.6e-33)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(j * t + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -9e+79], t$95$1, If[LessEqual[c, 7.6e-33], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -8.99999999999999987e79 or 7.59999999999999988e-33 < c

   1. Initial program 65.0%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   3. 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 61.0

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

   4. Applied rewrites 61.0%

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

   if -8.99999999999999987e79 < c < 7.59999999999999988e-33

   1. Initial program 79.2%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   3. 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 46.0

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

   4. Applied rewrites 46.0%

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

Alternative 14: 43.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.4 \cdot 10^{+117}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;i \leq 265000000000:\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -4.4e+117)
   (* (* (- i) j) y)
   (if (<= i 265000000000.0)
     (* (fma j t (* (- b) z)) c)
     (* (fma i a (* (- c) z)) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -4.4e+117) {
		tmp = (-i * j) * y;
	} else if (i <= 265000000000.0) {
		tmp = fma(j, t, (-b * z)) * c;
	} else {
		tmp = fma(i, a, (-c * z)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -4.4e+117)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (i <= 265000000000.0)
		tmp = Float64(fma(j, t, Float64(Float64(-b) * z)) * c);
	else
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -4.4e+117], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[i, 265000000000.0], N[(N[(j * t + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -4.40000000000000028e117

   1. Initial program 58.8%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 47.7

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 40.3

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

   7. Applied rewrites 40.3%

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

   if -4.40000000000000028e117 < i < 2.65e11

   1. Initial program 78.9%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   3. 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 44.4

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

   4. Applied rewrites 44.4%

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

   if 2.65e11 < i

   1. Initial program 66.0%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 45.0

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

   4. Applied rewrites 45.0%

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

Alternative 15: 42.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\left(c \cdot j\right) \cdot t\\ \mathbf{elif}\;j \leq 2.8 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(-i\right) \cdot \left(j \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -3.1e+72)
   (* (* c j) t)
   (if (<= j 2.8e+144) (* (fma i a (* (- c) z)) b) (* (- i) (* j y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -3.1e+72) {
		tmp = (c * j) * t;
	} else if (j <= 2.8e+144) {
		tmp = fma(i, a, (-c * z)) * b;
	} else {
		tmp = -i * (j * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -3.1e+72)
		tmp = Float64(Float64(c * j) * t);
	elseif (j <= 2.8e+144)
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(Float64(-i) * Float64(j * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -3.1e+72], N[(N[(c * j), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[j, 2.8e+144], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[((-i) * N[(j * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.09999999999999988e72

   1. Initial program 70.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 38.8

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

   7. Applied rewrites 38.8%

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

   if -3.09999999999999988e72 < j < 2.80000000000000007e144

   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. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 43.6

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

   4. Applied rewrites 43.6%

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

   if 2.80000000000000007e144 < j

   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. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y - a \cdot b\right)\right)} \]
   3. 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 48.5

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 40.4

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

   7. Applied rewrites 40.4%

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

Alternative 16: 29.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot j\right) \cdot t\\ \mathbf{if}\;j \leq -2.2 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 3.25 \cdot 10^{+100}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* c j) t)))
   (if (<= j -2.2e+45) t_1 (if (<= j 3.25e+100) (* (* (- b) z) c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (c * j) * t;
	double tmp;
	if (j <= -2.2e+45) {
		tmp = t_1;
	} else if (j <= 3.25e+100) {
		tmp = (-b * z) * c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (c * j) * t
    if (j <= (-2.2d+45)) then
        tmp = t_1
    else if (j <= 3.25d+100) then
        tmp = (-b * z) * c
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (c * j) * t;
	double tmp;
	if (j <= -2.2e+45) {
		tmp = t_1;
	} else if (j <= 3.25e+100) {
		tmp = (-b * z) * c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (c * j) * t
	tmp = 0
	if j <= -2.2e+45:
		tmp = t_1
	elif j <= 3.25e+100:
		tmp = (-b * z) * c
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * j) * t)
	tmp = 0.0
	if (j <= -2.2e+45)
		tmp = t_1;
	elseif (j <= 3.25e+100)
		tmp = Float64(Float64(Float64(-b) * z) * c);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (c * j) * t;
	tmp = 0.0;
	if (j <= -2.2e+45)
		tmp = t_1;
	elseif (j <= 3.25e+100)
		tmp = (-b * z) * c;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * j), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[j, -2.2e+45], t$95$1, If[LessEqual[j, 3.25e+100], N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.2e45 or 3.25e100 < j

   1. Initial program 71.4%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 38.1

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

   7. Applied rewrites 38.1%

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

   if -2.2e45 < j < 3.25e100

   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. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   3. 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 33.4

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

   4. Applied rewrites 33.4%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 24.5

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

   7. Applied rewrites 24.5%

      \[\leadsto \left(\left(-b\right) \cdot z\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot j\right) \cdot t\\ \mathbf{if}\;j \leq -2.2 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.1 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* c j) t)))
   (if (<= j -2.2e+45) t_1 (if (<= j 2.1e+90) (* (* (- c) z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (c * j) * t;
	double tmp;
	if (j <= -2.2e+45) {
		tmp = t_1;
	} else if (j <= 2.1e+90) {
		tmp = (-c * z) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (c * j) * t
    if (j <= (-2.2d+45)) then
        tmp = t_1
    else if (j <= 2.1d+90) then
        tmp = (-c * z) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (c * j) * t;
	double tmp;
	if (j <= -2.2e+45) {
		tmp = t_1;
	} else if (j <= 2.1e+90) {
		tmp = (-c * z) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (c * j) * t
	tmp = 0
	if j <= -2.2e+45:
		tmp = t_1
	elif j <= 2.1e+90:
		tmp = (-c * z) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * j) * t)
	tmp = 0.0
	if (j <= -2.2e+45)
		tmp = t_1;
	elseif (j <= 2.1e+90)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (c * j) * t;
	tmp = 0.0;
	if (j <= -2.2e+45)
		tmp = t_1;
	elseif (j <= 2.1e+90)
		tmp = (-c * z) * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * j), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[j, -2.2e+45], t$95$1, If[LessEqual[j, 2.1e+90], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.2e45 or 2.09999999999999981e90 < j

   1. Initial program 71.3%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 37.9

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

   7. Applied rewrites 37.9%

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

   if -2.2e45 < j < 2.09999999999999981e90

   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. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 23.8

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

   7. Applied rewrites 23.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 23.8

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

   9. Applied rewrites 23.8%

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

Alternative 18: 29.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot j\right) \cdot t\\ \mathbf{if}\;j \leq -2.55 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 3.4 \cdot 10^{+100}:\\ \;\;\;\;-\left(c \cdot b\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* c j) t)))
   (if (<= j -2.55e+45) t_1 (if (<= j 3.4e+100) (- (* (* c b) z)) 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 * j) * t;
	double tmp;
	if (j <= -2.55e+45) {
		tmp = t_1;
	} else if (j <= 3.4e+100) {
		tmp = -((c * b) * z);
	} 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 * j) * t
    if (j <= (-2.55d+45)) then
        tmp = t_1
    else if (j <= 3.4d+100) then
        tmp = -((c * b) * z)
    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 * j) * t;
	double tmp;
	if (j <= -2.55e+45) {
		tmp = t_1;
	} else if (j <= 3.4e+100) {
		tmp = -((c * b) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (c * j) * t
	tmp = 0
	if j <= -2.55e+45:
		tmp = t_1
	elif j <= 3.4e+100:
		tmp = -((c * b) * z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * j) * t)
	tmp = 0.0
	if (j <= -2.55e+45)
		tmp = t_1;
	elseif (j <= 3.4e+100)
		tmp = Float64(-Float64(Float64(c * b) * z));
	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 * j) * t;
	tmp = 0.0;
	if (j <= -2.55e+45)
		tmp = t_1;
	elseif (j <= 3.4e+100)
		tmp = -((c * b) * z);
	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 * j), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[j, -2.55e+45], t$95$1, If[LessEqual[j, 3.4e+100], (-N[(N[(c * b), $MachinePrecision] * z), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.5499999999999999e45 or 3.39999999999999994e100 < j

   1. Initial program 71.4%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 48.2

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

   4. Applied rewrites 48.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 38.1

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

   7. Applied rewrites 38.1%

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

   if -2.5499999999999999e45 < j < 3.39999999999999994e100

   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. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.0

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

   10. Applied rewrites 24.0%

       \[\leadsto -\left(c \cdot b\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot j\right) \cdot t\\ \mathbf{if}\;j \leq -24000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.18 \cdot 10^{+100}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* c j) t)))
   (if (<= j -24000000000.0) t_1 (if (<= j 1.18e+100) (* (* z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (c * j) * t;
	double tmp;
	if (j <= -24000000000.0) {
		tmp = t_1;
	} else if (j <= 1.18e+100) {
		tmp = (z * y) * x;
	} 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 * j) * t
    if (j <= (-24000000000.0d0)) then
        tmp = t_1
    else if (j <= 1.18d+100) then
        tmp = (z * y) * x
    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 * j) * t;
	double tmp;
	if (j <= -24000000000.0) {
		tmp = t_1;
	} else if (j <= 1.18e+100) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (c * j) * t
	tmp = 0
	if j <= -24000000000.0:
		tmp = t_1
	elif j <= 1.18e+100:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * j) * t)
	tmp = 0.0
	if (j <= -24000000000.0)
		tmp = t_1;
	elseif (j <= 1.18e+100)
		tmp = Float64(Float64(z * y) * x);
	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 * j) * t;
	tmp = 0.0;
	if (j <= -24000000000.0)
		tmp = t_1;
	elseif (j <= 1.18e+100)
		tmp = (z * y) * x;
	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 * j), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[j, -24000000000.0], t$95$1, If[LessEqual[j, 1.18e+100], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.4e10 or 1.18e100 < j

   1. Initial program 71.7%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 36.8

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

   7. Applied rewrites 36.8%

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

   if -2.4e10 < j < 1.18e100

   1. Initial program 73.4%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 34.4

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

   4. Applied rewrites 34.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 25.2

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

   7. Applied rewrites 25.2%

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

Alternative 20: 30.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(j \cdot t\right)\\ \mathbf{if}\;j \leq -24000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.95 \cdot 10^{+99}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* j t))))
   (if (<= j -24000000000.0) t_1 (if (<= j 2.95e+99) (* (* z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (j * t);
	double tmp;
	if (j <= -24000000000.0) {
		tmp = t_1;
	} else if (j <= 2.95e+99) {
		tmp = (z * y) * x;
	} 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 * (j * t)
    if (j <= (-24000000000.0d0)) then
        tmp = t_1
    else if (j <= 2.95d+99) then
        tmp = (z * y) * x
    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 * (j * t);
	double tmp;
	if (j <= -24000000000.0) {
		tmp = t_1;
	} else if (j <= 2.95e+99) {
		tmp = (z * y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (j * t)
	tmp = 0
	if j <= -24000000000.0:
		tmp = t_1
	elif j <= 2.95e+99:
		tmp = (z * y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(j * t))
	tmp = 0.0
	if (j <= -24000000000.0)
		tmp = t_1;
	elseif (j <= 2.95e+99)
		tmp = Float64(Float64(z * y) * x);
	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 * (j * t);
	tmp = 0.0;
	if (j <= -24000000000.0)
		tmp = t_1;
	elseif (j <= 2.95e+99)
		tmp = (z * y) * x;
	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[(c * N[(j * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -24000000000.0], t$95$1, If[LessEqual[j, 2.95e+99], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2.4e10 or 2.9499999999999999e99 < j

   1. Initial program 71.7%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 36.7

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

   7. Applied rewrites 36.7%

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

   if -2.4e10 < j < 2.9499999999999999e99

   1. Initial program 73.4%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. 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 34.4

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

   4. Applied rewrites 34.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 25.3

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

   7. Applied rewrites 25.3%

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

Alternative 21: 30.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -5.2 \cdot 10^{+87}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{elif}\;i \leq 4 \cdot 10^{+100}:\\ \;\;\;\;c \cdot \left(j \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -5.2e+87)
   (* a (* b i))
   (if (<= i 4e+100) (* c (* j t)) (* (* b a) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -5.2e+87) {
		tmp = a * (b * i);
	} else if (i <= 4e+100) {
		tmp = c * (j * 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 (i <= (-5.2d+87)) then
        tmp = a * (b * i)
    else if (i <= 4d+100) then
        tmp = c * (j * 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 (i <= -5.2e+87) {
		tmp = a * (b * i);
	} else if (i <= 4e+100) {
		tmp = c * (j * t);
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if i <= -5.2e+87:
		tmp = a * (b * i)
	elif i <= 4e+100:
		tmp = c * (j * t)
	else:
		tmp = (b * a) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -5.2e+87)
		tmp = Float64(a * Float64(b * i));
	elseif (i <= 4e+100)
		tmp = Float64(c * Float64(j * 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 (i <= -5.2e+87)
		tmp = a * (b * i);
	elseif (i <= 4e+100)
		tmp = c * (j * t);
	else
		tmp = (b * a) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -5.2e+87], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 4e+100], N[(c * N[(j * t), $MachinePrecision]), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -5.19999999999999997e87

   1. Initial program 60.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.3

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

   7. Applied rewrites 41.3%

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

   if -5.19999999999999997e87 < i < 4.00000000000000006e100

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   3. 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 44.3

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 24.9

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

   7. Applied rewrites 24.9%

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

   if 4.00000000000000006e100 < i

   1. Initial program 62.3%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   3. 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 49.3

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 39.4

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

   7. Applied rewrites 39.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 39.8

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

   9. Applied rewrites 39.8%

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

Alternative 22: 22.7% 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 72.7%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
3. 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 38.9

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

4. Applied rewrites 38.9%

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

5. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 22.2

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

7. Applied rewrites 22.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 22.7

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

9. Applied rewrites 22.7%

   \[\leadsto \left(b \cdot a\right) \cdot i \]
10. Add Preprocessing

Alternative 23: 22.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(b \cdot i\right) \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
3. 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 38.9

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

4. Applied rewrites 38.9%

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

5. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 22.2

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

7. Applied rewrites 22.2%

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

Developer Target 1: 68.3% 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 2025093 
(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)))))