Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, J

Percentage Accurate: 79.7% → 93.0%

Time: 10.1s

Alternatives: 16

Speedup: 0.8×

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
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
    code = ((((x * 9.0d0) * y) - (((z * 4.0d0) * t) * a)) + b) / (z * c)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

Alternative 1: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-4 \cdot t}{c\_m}, \frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c\_m}}{z}\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 4e-10)
    (/ (fma (* -4.0 t) a (/ (fma (* y x) 9.0 b) z)) c_m)
    (fma a (/ (* -4.0 t) c_m) (/ (/ (fma (* y 9.0) x b) c_m) z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 4e-10) {
		tmp = fma((-4.0 * t), a, (fma((y * x), 9.0, b) / z)) / c_m;
	} else {
		tmp = fma(a, ((-4.0 * t) / c_m), ((fma((y * 9.0), x, b) / c_m) / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 4e-10)
		tmp = Float64(fma(Float64(-4.0 * t), a, Float64(fma(Float64(y * x), 9.0, b) / z)) / c_m);
	else
		tmp = fma(a, Float64(Float64(-4.0 * t) / c_m), Float64(Float64(fma(Float64(y * 9.0), x, b) / c_m) / z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 4e-10], N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(a * N[(N[(-4.0 * t), $MachinePrecision] / c$95$m), $MachinePrecision] + N[(N[(N[(N[(y * 9.0), $MachinePrecision] * x + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 4.00000000000000015e-10

   1. Initial program 83.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]

   if 4.00000000000000015e-10 < c

   1. Initial program 67.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-4 \cdot t}{c}}, \frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c}}{z}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{9 \cdot y}{z} \cdot \frac{x}{c\_m}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{y \cdot x}{z} \cdot 9\right)}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c\_m} \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ (* 9.0 y) z) (/ x c_m))
      (if (<= t_1 -5e-22)
        (/ (fma (* -4.0 a) t (* (/ (* y x) z) 9.0)) c_m)
        (if (<= t_1 10000.0)
          (/ (fma (* -4.0 t) a (/ b z)) c_m)
          (if (<= t_1 2e+272)
            (/ (/ (fma (* x 9.0) y b) c_m) z)
            (* 9.0 (* (/ y c_m) (/ x z))))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((9.0 * y) / z) * (x / c_m);
	} else if (t_1 <= -5e-22) {
		tmp = fma((-4.0 * a), t, (((y * x) / z) * 9.0)) / c_m;
	} else if (t_1 <= 10000.0) {
		tmp = fma((-4.0 * t), a, (b / z)) / c_m;
	} else if (t_1 <= 2e+272) {
		tmp = (fma((x * 9.0), y, b) / c_m) / z;
	} else {
		tmp = 9.0 * ((y / c_m) * (x / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(9.0 * y) / z) * Float64(x / c_m));
	elseif (t_1 <= -5e-22)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(Float64(Float64(y * x) / z) * 9.0)) / c_m);
	elseif (t_1 <= 10000.0)
		tmp = Float64(fma(Float64(-4.0 * t), a, Float64(b / z)) / c_m);
	elseif (t_1 <= 2e+272)
		tmp = Float64(Float64(fma(Float64(x * 9.0), y, b) / c_m) / z);
	else
		tmp = Float64(9.0 * Float64(Float64(y / c_m) * Float64(x / z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-22], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 10000.0], N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision], N[(9.0 * N[(N[(y / c$95$m), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 41.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 99.8

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \frac{9 \cdot y}{z} \cdot \color{blue}{\frac{x}{c}} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999954e-22

   1. Initial program 91.4%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-4 \cdot \left(a \cdot t\right) + 9 \cdot \frac{x \cdot y}{z}}{c} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{y \cdot x}{z} \cdot 9\right)}{c} \]

   if -4.99999999999999954e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1e4

   1. Initial program 83.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c} \]
   7. Step-by-step derivation

   8. Applied rewrites 84.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c} \]

   if 1e4 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.0000000000000001e272

   1. Initial program 80.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}}{c} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{z}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{z}}{c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

      9. lower-*.f64 69.4

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c}}{z}} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.6%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c}}{z} \]

   if 2.0000000000000001e272 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 47.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 88.4

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.3%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.5%

      \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m}}{z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{9 \cdot y}{z} \cdot \frac{x}{c\_m}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c\_m} \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (/ (/ (fma (* x 9.0) y b) c_m) z)))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ (* 9.0 y) z) (/ x c_m))
      (if (<= t_1 -5e+26)
        t_2
        (if (<= t_1 10000.0)
          (/ (fma (* -4.0 t) a (/ b z)) c_m)
          (if (<= t_1 2e+272) t_2 (* 9.0 (* (/ y c_m) (/ x z))))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = (fma((x * 9.0), y, b) / c_m) / z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((9.0 * y) / z) * (x / c_m);
	} else if (t_1 <= -5e+26) {
		tmp = t_2;
	} else if (t_1 <= 10000.0) {
		tmp = fma((-4.0 * t), a, (b / z)) / c_m;
	} else if (t_1 <= 2e+272) {
		tmp = t_2;
	} else {
		tmp = 9.0 * ((y / c_m) * (x / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(fma(Float64(x * 9.0), y, b) / c_m) / z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(9.0 * y) / z) * Float64(x / c_m));
	elseif (t_1 <= -5e+26)
		tmp = t_2;
	elseif (t_1 <= 10000.0)
		tmp = Float64(fma(Float64(-4.0 * t), a, Float64(b / z)) / c_m);
	elseif (t_1 <= 2e+272)
		tmp = t_2;
	else
		tmp = Float64(9.0 * Float64(Float64(y / c_m) * Float64(x / z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+26], t$95$2, If[LessEqual[t$95$1, 10000.0], N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], t$95$2, N[(9.0 * N[(N[(y / c$95$m), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 41.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 99.8

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \frac{9 \cdot y}{z} \cdot \color{blue}{\frac{x}{c}} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -5.0000000000000001e26 or 1e4 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.0000000000000001e272

   1. Initial program 87.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}}{c} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{z}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{z}}{c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

      9. lower-*.f64 70.0

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c}}{z}} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.3%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c}}{z} \]

   if -5.0000000000000001e26 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1e4

   1. Initial program 83.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{b}{z}\right)}{c} \]

   if 2.0000000000000001e272 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 47.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 88.4

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.3%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.5%

      \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c\_m}}{z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{9 \cdot y}{z} \cdot \frac{x}{c\_m}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c\_m} \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (/ (/ (fma (* x 9.0) y b) c_m) z)))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ (* 9.0 y) z) (/ x c_m))
      (if (<= t_1 -5e-22)
        t_2
        (if (<= t_1 5e-72)
          (/ (fma -4.0 (* (* t z) a) b) (* z c_m))
          (if (<= t_1 2e+272) t_2 (* 9.0 (* (/ y c_m) (/ x z))))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = (fma((x * 9.0), y, b) / c_m) / z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((9.0 * y) / z) * (x / c_m);
	} else if (t_1 <= -5e-22) {
		tmp = t_2;
	} else if (t_1 <= 5e-72) {
		tmp = fma(-4.0, ((t * z) * a), b) / (z * c_m);
	} else if (t_1 <= 2e+272) {
		tmp = t_2;
	} else {
		tmp = 9.0 * ((y / c_m) * (x / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(Float64(fma(Float64(x * 9.0), y, b) / c_m) / z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(9.0 * y) / z) * Float64(x / c_m));
	elseif (t_1 <= -5e-22)
		tmp = t_2;
	elseif (t_1 <= 5e-72)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / Float64(z * c_m));
	elseif (t_1 <= 2e+272)
		tmp = t_2;
	else
		tmp = Float64(9.0 * Float64(Float64(y / c_m) * Float64(x / z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-22], t$95$2, If[LessEqual[t$95$1, 5e-72], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], t$95$2, N[(9.0 * N[(N[(y / c$95$m), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 41.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 99.8

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \frac{9 \cdot y}{z} \cdot \color{blue}{\frac{x}{c}} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999954e-22 or 4.9999999999999996e-72 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.0000000000000001e272

   1. Initial program 85.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}}{c} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{z}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{z}}{c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

      9. lower-*.f64 68.7

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c}}{z}} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.1%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{c}}{z} \]

   if -4.99999999999999954e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999996e-72

   1. Initial program 84.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right) \cdot a}, b\right)}{z \cdot c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right) \cdot a}, b\right)}{z \cdot c} \]

      7. lower-*.f64 82.8

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right)} \cdot a, b\right)}{z \cdot c} \]

   5. Applied rewrites 82.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}}{z \cdot c} \]

   if 2.0000000000000001e272 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 47.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 88.4

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.3%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.5%

      \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ t_2 := \frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c\_m \cdot z}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{9 \cdot y}{z} \cdot \frac{x}{c\_m}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+192}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c\_m} \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)) (t_2 (/ (fma (* y 9.0) x b) (* c_m z))))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ (* 9.0 y) z) (/ x c_m))
      (if (<= t_1 -5e-22)
        t_2
        (if (<= t_1 5e-72)
          (/ (fma -4.0 (* (* t z) a) b) (* z c_m))
          (if (<= t_1 5e+192) t_2 (* 9.0 (* (/ y c_m) (/ x z))))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double t_2 = fma((y * 9.0), x, b) / (c_m * z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((9.0 * y) / z) * (x / c_m);
	} else if (t_1 <= -5e-22) {
		tmp = t_2;
	} else if (t_1 <= 5e-72) {
		tmp = fma(-4.0, ((t * z) * a), b) / (z * c_m);
	} else if (t_1 <= 5e+192) {
		tmp = t_2;
	} else {
		tmp = 9.0 * ((y / c_m) * (x / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	t_2 = Float64(fma(Float64(y * 9.0), x, b) / Float64(c_m * z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(9.0 * y) / z) * Float64(x / c_m));
	elseif (t_1 <= -5e-22)
		tmp = t_2;
	elseif (t_1 <= 5e-72)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), b) / Float64(z * c_m));
	elseif (t_1 <= 5e+192)
		tmp = t_2;
	else
		tmp = Float64(9.0 * Float64(Float64(y / c_m) * Float64(x / z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * 9.0), $MachinePrecision] * x + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-22], t$95$2, If[LessEqual[t$95$1, 5e-72], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+192], t$95$2, N[(9.0 * N[(N[(y / c$95$m), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 41.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 99.8

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \frac{9 \cdot y}{z} \cdot \color{blue}{\frac{x}{c}} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999954e-22 or 4.9999999999999996e-72 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.00000000000000033e192

   1. Initial program 88.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}}{c} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{z}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{z}}{c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

      9. lower-*.f64 66.3

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.3%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{\color{blue}{c \cdot z}} \]

   if -4.99999999999999954e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 4.9999999999999996e-72

   1. Initial program 84.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

         \[\leadsto \frac{\color{blue}{b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot \left(t \cdot z\right)\right)}}{z \cdot c} \]

      2. metadata-eval N/A

         \[\leadsto \frac{b + \color{blue}{-4} \cdot \left(a \cdot \left(t \cdot z\right)\right)}{z \cdot c} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-4 \cdot \left(a \cdot \left(t \cdot z\right)\right) + b}}{z \cdot c} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot \left(t \cdot z\right), b\right)}}{z \cdot c} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right) \cdot a}, b\right)}{z \cdot c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right) \cdot a}, b\right)}{z \cdot c} \]

      7. lower-*.f64 82.8

         \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right)} \cdot a, b\right)}{z \cdot c} \]

   5. Applied rewrites 82.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, b\right)}}{z \cdot c} \]

   if 5.00000000000000033e192 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 51.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 83.6

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.7%

      \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 50.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot y\right) \cdot \frac{x}{z \cdot c\_m}\\ t_2 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-190}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-217}:\\ \;\;\;\;\frac{t \cdot -4}{c\_m} \cdot a\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* 9.0 y) (/ x (* z c_m)))) (t_2 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_2 -5e-22)
      t_1
      (if (<= t_2 -1e-190)
        (/ b (* c_m z))
        (if (<= t_2 5e-217)
          (* (/ (* t -4.0) c_m) a)
          (if (<= t_2 2e+182) (/ (/ b c_m) z) t_1)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * y) * (x / (z * c_m));
	double t_2 = (x * 9.0) * y;
	double tmp;
	if (t_2 <= -5e-22) {
		tmp = t_1;
	} else if (t_2 <= -1e-190) {
		tmp = b / (c_m * z);
	} else if (t_2 <= 5e-217) {
		tmp = ((t * -4.0) / c_m) * a;
	} else if (t_2 <= 2e+182) {
		tmp = (b / c_m) / z;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    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_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (9.0d0 * y) * (x / (z * c_m))
    t_2 = (x * 9.0d0) * y
    if (t_2 <= (-5d-22)) then
        tmp = t_1
    else if (t_2 <= (-1d-190)) then
        tmp = b / (c_m * z)
    else if (t_2 <= 5d-217) then
        tmp = ((t * (-4.0d0)) / c_m) * a
    else if (t_2 <= 2d+182) then
        tmp = (b / c_m) / z
    else
        tmp = t_1
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (9.0 * y) * (x / (z * c_m));
	double t_2 = (x * 9.0) * y;
	double tmp;
	if (t_2 <= -5e-22) {
		tmp = t_1;
	} else if (t_2 <= -1e-190) {
		tmp = b / (c_m * z);
	} else if (t_2 <= 5e-217) {
		tmp = ((t * -4.0) / c_m) * a;
	} else if (t_2 <= 2e+182) {
		tmp = (b / c_m) / z;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = (9.0 * y) * (x / (z * c_m))
	t_2 = (x * 9.0) * y
	tmp = 0
	if t_2 <= -5e-22:
		tmp = t_1
	elif t_2 <= -1e-190:
		tmp = b / (c_m * z)
	elif t_2 <= 5e-217:
		tmp = ((t * -4.0) / c_m) * a
	elif t_2 <= 2e+182:
		tmp = (b / c_m) / z
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(9.0 * y) * Float64(x / Float64(z * c_m)))
	t_2 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_2 <= -5e-22)
		tmp = t_1;
	elseif (t_2 <= -1e-190)
		tmp = Float64(b / Float64(c_m * z));
	elseif (t_2 <= 5e-217)
		tmp = Float64(Float64(Float64(t * -4.0) / c_m) * a);
	elseif (t_2 <= 2e+182)
		tmp = Float64(Float64(b / c_m) / z);
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = (9.0 * y) * (x / (z * c_m));
	t_2 = (x * 9.0) * y;
	tmp = 0.0;
	if (t_2 <= -5e-22)
		tmp = t_1;
	elseif (t_2 <= -1e-190)
		tmp = b / (c_m * z);
	elseif (t_2 <= 5e-217)
		tmp = ((t * -4.0) / c_m) * a;
	elseif (t_2 <= 2e+182)
		tmp = (b / c_m) / z;
	else
		tmp = t_1;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(9.0 * y), $MachinePrecision] * N[(x / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -5e-22], t$95$1, If[LessEqual[t$95$2, -1e-190], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-217], N[(N[(N[(t * -4.0), $MachinePrecision] / c$95$m), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$2, 2e+182], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -4.99999999999999954e-22 or 2.0000000000000001e182 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 71.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 70.7

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.2%

      \[\leadsto \left(9 \cdot y\right) \cdot \color{blue}{\frac{x}{z \cdot c}} \]

   if -4.99999999999999954e-22 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -1e-190

   1. Initial program 88.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 61.5

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   if -1e-190 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.0000000000000002e-217

   1. Initial program 78.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 56.1

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.0%

      \[\leadsto \frac{t \cdot -4}{c} \cdot \color{blue}{a} \]

   if 5.0000000000000002e-217 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 2.0000000000000001e182

   1. Initial program 87.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   4. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(y \cdot 9, x, b\right)\right)}{c}}{z}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 58.2

         \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]

   7. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{9 \cdot y}{z} \cdot \frac{x}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c\_m} \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ (* 9.0 y) z) (/ x c_m))
      (if (<= t_1 1e+300)
        (/ (fma (* -4.0 t) a (/ (fma (* y x) 9.0 b) z)) c_m)
        (* 9.0 (* (/ y c_m) (/ x z))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((9.0 * y) / z) * (x / c_m);
	} else if (t_1 <= 1e+300) {
		tmp = fma((-4.0 * t), a, (fma((y * x), 9.0, b) / z)) / c_m;
	} else {
		tmp = 9.0 * ((y / c_m) * (x / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(9.0 * y) / z) * Float64(x / c_m));
	elseif (t_1 <= 1e+300)
		tmp = Float64(fma(Float64(-4.0 * t), a, Float64(fma(Float64(y * x), 9.0, b) / z)) / c_m);
	else
		tmp = Float64(9.0 * Float64(Float64(y / c_m) * Float64(x / z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+300], N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(9.0 * N[(N[(y / c$95$m), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 41.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 99.8

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \frac{9 \cdot y}{z} \cdot \color{blue}{\frac{x}{c}} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.0000000000000001e300

   1. Initial program 84.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]

   if 1.0000000000000001e300 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 47.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 95.7

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.5%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 95.9%

      \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{9 \cdot y}{z} \cdot \frac{x}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(-4 \cdot z, a \cdot t, b\right)\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c\_m} \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ (* 9.0 y) z) (/ x c_m))
      (if (<= t_1 5e+192)
        (/ (fma (* y 9.0) x (fma (* -4.0 z) (* a t) b)) (* z c_m))
        (* 9.0 (* (/ y c_m) (/ x z))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((9.0 * y) / z) * (x / c_m);
	} else if (t_1 <= 5e+192) {
		tmp = fma((y * 9.0), x, fma((-4.0 * z), (a * t), b)) / (z * c_m);
	} else {
		tmp = 9.0 * ((y / c_m) * (x / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(9.0 * y) / z) * Float64(x / c_m));
	elseif (t_1 <= 5e+192)
		tmp = Float64(fma(Float64(y * 9.0), x, fma(Float64(-4.0 * z), Float64(a * t), b)) / Float64(z * c_m));
	else
		tmp = Float64(9.0 * Float64(Float64(y / c_m) * Float64(x / z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+192], N[(N[(N[(y * 9.0), $MachinePrecision] * x + N[(N[(-4.0 * z), $MachinePrecision] * N[(a * t), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(N[(y / c$95$m), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 41.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 99.8

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \frac{9 \cdot y}{z} \cdot \color{blue}{\frac{x}{c}} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.00000000000000033e192

   1. Initial program 86.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right) + b}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a\right)} + b}{z \cdot c} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + \left(\left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}}{z \cdot c} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right) \cdot y} + \left(\left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}{z \cdot c} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot 9\right)} \cdot y + \left(\left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}{z \cdot c} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(9 \cdot y\right)} + \left(\left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}{z \cdot c} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x} + \left(\left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}{z \cdot c} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9 \cdot y, x, \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot 9}, x, \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}{z \cdot c} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot 9}, x, \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + b\right)}{z \cdot c} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot 9, x, \left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot t}\right)\right) \cdot a + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot 9, x, \color{blue}{\left(\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot t\right)} \cdot a + b\right)}{z \cdot c} \]

      15. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot 9, x, \color{blue}{\left(\mathsf{neg}\left(z \cdot 4\right)\right) \cdot \left(t \cdot a\right)} + b\right)}{z \cdot c} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot 9, x, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 4\right), t \cdot a, b\right)}\right)}{z \cdot c} \]

   4. Applied rewrites 86.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot 9, x, \mathsf{fma}\left(-4 \cdot z, a \cdot t, b\right)\right)}}{z \cdot c} \]

   if 5.00000000000000033e192 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 51.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 83.6

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.7%

      \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ \begin{array}{l} t_1 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{9 \cdot y}{z} \cdot \frac{x}{c\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+192}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, \mathsf{fma}\left(y \cdot 9, x, b\right)\right)}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c\_m} \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_1 (- INFINITY))
      (* (/ (* 9.0 y) z) (/ x c_m))
      (if (<= t_1 5e+192)
        (/ (fma -4.0 (* (* t z) a) (fma (* y 9.0) x b)) (* z c_m))
        (* 9.0 (* (/ y c_m) (/ x z))))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = (x * 9.0) * y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((9.0 * y) / z) * (x / c_m);
	} else if (t_1 <= 5e+192) {
		tmp = fma(-4.0, ((t * z) * a), fma((y * 9.0), x, b)) / (z * c_m);
	} else {
		tmp = 9.0 * ((y / c_m) * (x / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(9.0 * y) / z) * Float64(x / c_m));
	elseif (t_1 <= 5e+192)
		tmp = Float64(fma(-4.0, Float64(Float64(t * z) * a), fma(Float64(y * 9.0), x, b)) / Float64(z * c_m));
	else
		tmp = Float64(9.0 * Float64(Float64(y / c_m) * Float64(x / z)));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(9.0 * y), $MachinePrecision] / z), $MachinePrecision] * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+192], N[(N[(-4.0 * N[(N[(t * z), $MachinePrecision] * a), $MachinePrecision] + N[(N[(y * 9.0), $MachinePrecision] * x + b), $MachinePrecision]), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(N[(y / c$95$m), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -inf.0

   1. Initial program 41.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 99.8

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \frac{9 \cdot y}{z} \cdot \color{blue}{\frac{x}{c}} \]

   if -inf.0 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 5.00000000000000033e192

   1. Initial program 86.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}}{z \cdot c} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b}{z \cdot c} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \color{blue}{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}\right) + b}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot 9\right) \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a\right)} + b}{z \cdot c} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + \left(x \cdot 9\right) \cdot y\right)} + b}{z \cdot c} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z \cdot 4\right) \cdot t\right)\right) \cdot a + \left(\left(x \cdot 9\right) \cdot y + b\right)}}{z \cdot c} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(\left(z \cdot 4\right) \cdot t\right)} \cdot a\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right) \cdot \left(t \cdot a\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 4\right)} \cdot \left(t \cdot a\right)\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot z\right)} \cdot \left(t \cdot a\right)\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(z \cdot \left(t \cdot a\right)\right)}\right)\right) + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

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

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot \left(t \cdot a\right)\right)} + \left(\left(x \cdot 9\right) \cdot y + b\right)}{z \cdot c} \]

      14. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(z \cdot \left(t \cdot a\right)\right) + \color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right)}}{z \cdot c} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(4\right), z \cdot \left(t \cdot a\right), b + \left(x \cdot 9\right) \cdot y\right)}}{z \cdot c} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-4}, z \cdot \left(t \cdot a\right), b + \left(x \cdot 9\right) \cdot y\right)}{z \cdot c} \]

      17. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(z \cdot t\right) \cdot a}, b + \left(x \cdot 9\right) \cdot y\right)}{z \cdot c} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(z \cdot t\right) \cdot a}, b + \left(x \cdot 9\right) \cdot y\right)}{z \cdot c} \]

      19. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right)} \cdot a, b + \left(x \cdot 9\right) \cdot y\right)}{z \cdot c} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-4, \color{blue}{\left(t \cdot z\right)} \cdot a, b + \left(x \cdot 9\right) \cdot y\right)}{z \cdot c} \]

      21. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, \color{blue}{\left(x \cdot 9\right) \cdot y + b}\right)}{z \cdot c} \]

   4. Applied rewrites 86.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \left(t \cdot z\right) \cdot a, \mathsf{fma}\left(y \cdot 9, x, b\right)\right)}}{z \cdot c} \]

   if 5.00000000000000033e192 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 51.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{9 \cdot \frac{x \cdot y}{c \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{9 \cdot \color{blue}{\left(y \cdot x\right)}}{c \cdot z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(9 \cdot y\right) \cdot x}}{c \cdot z} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{9 \cdot y}{c} \cdot \frac{x}{z}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot 9}}{c} \cdot \frac{x}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right)} \cdot \frac{x}{z} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{y}{c}} \cdot 9\right) \cdot \frac{x}{z} \]

      10. lower-/.f64 83.6

          \[\leadsto \left(\frac{y}{c} \cdot 9\right) \cdot \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\frac{y}{c} \cdot 9\right) \cdot \frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.2%

      \[\leadsto \frac{\left(x \cdot 9\right) \cdot y}{\color{blue}{z \cdot c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.7%

      \[\leadsto 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 93.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t \cdot \frac{-4}{c\_m}, \frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c\_m}}{z}\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 5e-10)
    (/ (fma (* -4.0 t) a (/ (fma (* y x) 9.0 b) z)) c_m)
    (fma a (* t (/ -4.0 c_m)) (/ (/ (fma (* y 9.0) x b) c_m) z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (c_m <= 5e-10) {
		tmp = fma((-4.0 * t), a, (fma((y * x), 9.0, b) / z)) / c_m;
	} else {
		tmp = fma(a, (t * (-4.0 / c_m)), ((fma((y * 9.0), x, b) / c_m) / z));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 5e-10)
		tmp = Float64(fma(Float64(-4.0 * t), a, Float64(fma(Float64(y * x), 9.0, b) / z)) / c_m);
	else
		tmp = fma(a, Float64(t * Float64(-4.0 / c_m)), Float64(Float64(fma(Float64(y * 9.0), x, b) / c_m) / z));
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 5e-10], N[(N[(N[(-4.0 * t), $MachinePrecision] * a + N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(a * N[(t * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(y * 9.0), $MachinePrecision] * x + b), $MachinePrecision] / c$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 5.00000000000000031e-10

   1. Initial program 83.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]

   if 5.00000000000000031e-10 < c

   1. Initial program 67.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(9 \cdot \frac{x \cdot y}{c \cdot z} + \frac{b}{c \cdot z}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b}{c \cdot z} + 9 \cdot \frac{x \cdot y}{c \cdot z}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\frac{b}{c \cdot z} + \color{blue}{\frac{9 \cdot \left(x \cdot y\right)}{c \cdot z}}\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{a \cdot t}{c} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{-4} \cdot \frac{a \cdot t}{c} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c} + \color{blue}{\frac{-4 \cdot \left(a \cdot t\right)}{c}} \]

      9. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{b}{z} + \frac{9 \cdot \left(x \cdot y\right)}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\left(\frac{b}{z} + \color{blue}{9 \cdot \frac{x \cdot y}{z}}\right) + -4 \cdot \left(a \cdot t\right)}{c} \]

      12. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} + -4 \cdot \left(a \cdot t\right)}{c} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(4\right)\right)} \cdot \left(a \cdot t\right)}{c} \]

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

          \[\leadsto \frac{\color{blue}{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}}{c} \]

      15. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(9 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right) - 4 \cdot \left(a \cdot t\right)}{c}} \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot t, a, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-4 \cdot t}{c}}, \frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c}}{z}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 82.4%

      \[\leadsto \mathsf{fma}\left(a, t \cdot \color{blue}{\frac{-4}{c}}, \frac{\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c}}{z}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\left(t \cdot -4\right) \cdot \frac{a}{c\_m}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{-4}{c\_m}\right) \cdot t\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -1e+102)
    (* (* t -4.0) (/ a c_m))
    (if (<= z 4e+113)
      (/ (fma (* y 9.0) x b) (* c_m z))
      (* (* a (/ -4.0 c_m)) t)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1e+102) {
		tmp = (t * -4.0) * (a / c_m);
	} else if (z <= 4e+113) {
		tmp = fma((y * 9.0), x, b) / (c_m * z);
	} else {
		tmp = (a * (-4.0 / c_m)) * t;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -1e+102)
		tmp = Float64(Float64(t * -4.0) * Float64(a / c_m));
	elseif (z <= 4e+113)
		tmp = Float64(fma(Float64(y * 9.0), x, b) / Float64(c_m * z));
	else
		tmp = Float64(Float64(a * Float64(-4.0 / c_m)) * t);
	end
	return Float64(c_s * tmp)
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -1e+102], N[(N[(t * -4.0), $MachinePrecision] * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+113], N[(N[(N[(y * 9.0), $MachinePrecision] * x + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999977e101

   1. Initial program 53.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 51.9

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

      \[\leadsto \left(t \cdot -4\right) \cdot \color{blue}{\frac{a}{c}} \]

   if -9.99999999999999977e101 < z < 4e113

   1. Initial program 93.5%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{c \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{b + 9 \cdot \left(x \cdot y\right)}{\color{blue}{z \cdot c}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}}{c} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{9 \cdot \left(x \cdot y\right) + b}}{z}}{c} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot y\right) \cdot 9} + b}{z}}{c} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot y, 9, b\right)}}{z}}{c} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

      9. lower-*.f64 71.0

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{y \cdot x}, 9, b\right)}{z}}{c} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.8%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot 9, x, b\right)}{\color{blue}{c \cdot z}} \]

   if 4e113 < z

   1. Initial program 46.7%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 43.4

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.9%

      \[\leadsto \left(t \cdot -4\right) \cdot \color{blue}{\frac{a}{c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

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

Alternative 12: 51.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -3300000:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+132}:\\ \;\;\;\;\left(a \cdot \frac{-4}{c\_m}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c\_m}}{z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= b -3300000.0)
    (/ b (* c_m z))
    (if (<= b 6.5e+132) (* (* a (/ -4.0 c_m)) t) (/ (/ b c_m) z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -3300000.0) {
		tmp = b / (c_m * z);
	} else if (b <= 6.5e+132) {
		tmp = (a * (-4.0 / c_m)) * t;
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    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_m
    real(8) :: tmp
    if (b <= (-3300000.0d0)) then
        tmp = b / (c_m * z)
    else if (b <= 6.5d+132) then
        tmp = (a * ((-4.0d0) / c_m)) * t
    else
        tmp = (b / c_m) / z
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (b <= -3300000.0) {
		tmp = b / (c_m * z);
	} else if (b <= 6.5e+132) {
		tmp = (a * (-4.0 / c_m)) * t;
	} else {
		tmp = (b / c_m) / z;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if b <= -3300000.0:
		tmp = b / (c_m * z)
	elif b <= 6.5e+132:
		tmp = (a * (-4.0 / c_m)) * t
	else:
		tmp = (b / c_m) / z
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (b <= -3300000.0)
		tmp = Float64(b / Float64(c_m * z));
	elseif (b <= 6.5e+132)
		tmp = Float64(Float64(a * Float64(-4.0 / c_m)) * t);
	else
		tmp = Float64(Float64(b / c_m) / z);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (b <= -3300000.0)
		tmp = b / (c_m * z);
	elseif (b <= 6.5e+132)
		tmp = (a * (-4.0 / c_m)) * t;
	else
		tmp = (b / c_m) / z;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[b, -3300000.0], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+132], N[(N[(a * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(b / c$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if b < -3.3e6

   1. Initial program 81.2%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 53.6

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   if -3.3e6 < b < 6.4999999999999994e132

   1. Initial program 76.6%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 40.8

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 40.8%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.2%

      \[\leadsto \left(t \cdot -4\right) \cdot \color{blue}{\frac{a}{c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 43.2%

      \[\leadsto \left(a \cdot \frac{-4}{c}\right) \cdot \color{blue}{t} \]

   if 6.4999999999999994e132 < b

   1. Initial program 84.0%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{z \cdot c}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{\color{blue}{c \cdot z}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}}{z}} \]

   4. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(-4 \cdot z\right) \cdot a, t, \mathsf{fma}\left(y \cdot 9, x, b\right)\right)}{c}}{z}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
   6. Step-by-step derivation

      1. lower-/.f64 86.9

         \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]

   7. Applied rewrites 86.9%

      \[\leadsto \frac{\color{blue}{\frac{b}{c}}}{z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 49.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+102} \lor \neg \left(z \leq 1.95 \cdot 10^{+71}\right):\\ \;\;\;\;\left(a \cdot \frac{-4}{c\_m}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (or (<= z -1e+102) (not (<= z 1.95e+71)))
    (* (* a (/ -4.0 c_m)) t)
    (/ b (* c_m z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((z <= -1e+102) || !(z <= 1.95e+71)) {
		tmp = (a * (-4.0 / c_m)) * t;
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    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_m
    real(8) :: tmp
    if ((z <= (-1d+102)) .or. (.not. (z <= 1.95d+71))) then
        tmp = (a * ((-4.0d0) / c_m)) * t
    else
        tmp = b / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((z <= -1e+102) || !(z <= 1.95e+71)) {
		tmp = (a * (-4.0 / c_m)) * t;
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if (z <= -1e+102) or not (z <= 1.95e+71):
		tmp = (a * (-4.0 / c_m)) * t
	else:
		tmp = b / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if ((z <= -1e+102) || !(z <= 1.95e+71))
		tmp = Float64(Float64(a * Float64(-4.0 / c_m)) * t);
	else
		tmp = Float64(b / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if ((z <= -1e+102) || ~((z <= 1.95e+71)))
		tmp = (a * (-4.0 / c_m)) * t;
	else
		tmp = b / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[Or[LessEqual[z, -1e+102], N[Not[LessEqual[z, 1.95e+71]], $MachinePrecision]], N[(N[(a * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999977e101 or 1.9500000000000001e71 < z

   1. Initial program 52.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 47.5

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.5%

      \[\leadsto \left(t \cdot -4\right) \cdot \color{blue}{\frac{a}{c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.5%

      \[\leadsto \left(a \cdot \frac{-4}{c}\right) \cdot \color{blue}{t} \]

   if -9.99999999999999977e101 < z < 1.9500000000000001e71

   1. Initial program 93.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+102} \lor \neg \left(z \leq 1.95 \cdot 10^{+71}\right):\\ \;\;\;\;\left(a \cdot \frac{-4}{c}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+102} \lor \neg \left(z \leq 1.95 \cdot 10^{+71}\right):\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (or (<= z -1e+102) (not (<= z 1.95e+71)))
    (* -4.0 (/ (* a t) c_m))
    (/ b (* c_m z)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((z <= -1e+102) || !(z <= 1.95e+71)) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    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_m
    real(8) :: tmp
    if ((z <= (-1d+102)) .or. (.not. (z <= 1.95d+71))) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    else
        tmp = b / (c_m * z)
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if ((z <= -1e+102) || !(z <= 1.95e+71)) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = b / (c_m * z);
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if (z <= -1e+102) or not (z <= 1.95e+71):
		tmp = -4.0 * ((a * t) / c_m)
	else:
		tmp = b / (c_m * z)
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if ((z <= -1e+102) || !(z <= 1.95e+71))
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	else
		tmp = Float64(b / Float64(c_m * z));
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if ((z <= -1e+102) || ~((z <= 1.95e+71)))
		tmp = -4.0 * ((a * t) / c_m);
	else
		tmp = b / (c_m * z);
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[Or[LessEqual[z, -1e+102], N[Not[LessEqual[z, 1.95e+71]], $MachinePrecision]], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999977e101 or 1.9500000000000001e71 < z

   1. Initial program 52.8%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 47.5

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

   if -9.99999999999999977e101 < z < 1.9500000000000001e71

   1. Initial program 93.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+102} \lor \neg \left(z \leq 1.95 \cdot 10^{+71}\right):\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\left(t \cdot -4\right) \cdot \frac{a}{c\_m}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\frac{b}{c\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \frac{-4}{c\_m}\right) \cdot t\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= z -1e+102)
    (* (* t -4.0) (/ a c_m))
    (if (<= z 1.95e+71) (/ b (* c_m z)) (* (* a (/ -4.0 c_m)) t)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1e+102) {
		tmp = (t * -4.0) * (a / c_m);
	} else if (z <= 1.95e+71) {
		tmp = b / (c_m * z);
	} else {
		tmp = (a * (-4.0 / c_m)) * t;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    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_m
    real(8) :: tmp
    if (z <= (-1d+102)) then
        tmp = (t * (-4.0d0)) * (a / c_m)
    else if (z <= 1.95d+71) then
        tmp = b / (c_m * z)
    else
        tmp = (a * ((-4.0d0) / c_m)) * t
    end if
    code = c_s * tmp
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (z <= -1e+102) {
		tmp = (t * -4.0) * (a / c_m);
	} else if (z <= 1.95e+71) {
		tmp = b / (c_m * z);
	} else {
		tmp = (a * (-4.0 / c_m)) * t;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	tmp = 0
	if z <= -1e+102:
		tmp = (t * -4.0) * (a / c_m)
	elif z <= 1.95e+71:
		tmp = b / (c_m * z)
	else:
		tmp = (a * (-4.0 / c_m)) * t
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (z <= -1e+102)
		tmp = Float64(Float64(t * -4.0) * Float64(a / c_m));
	elseif (z <= 1.95e+71)
		tmp = Float64(b / Float64(c_m * z));
	else
		tmp = Float64(Float64(a * Float64(-4.0 / c_m)) * t);
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0;
	if (z <= -1e+102)
		tmp = (t * -4.0) * (a / c_m);
	elseif (z <= 1.95e+71)
		tmp = b / (c_m * z);
	else
		tmp = (a * (-4.0 / c_m)) * t;
	end
	tmp_2 = c_s * tmp;
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[z, -1e+102], N[(N[(t * -4.0), $MachinePrecision] * N[(a / c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e+71], N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(-4.0 / c$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999977e101

   1. Initial program 53.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 51.9

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

      \[\leadsto \left(t \cdot -4\right) \cdot \color{blue}{\frac{a}{c}} \]

   if -9.99999999999999977e101 < z < 1.9500000000000001e71

   1. Initial program 93.9%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   if 1.9500000000000001e71 < z

   1. Initial program 52.3%

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{a \cdot t}{c}} \]

      3. lower-*.f64 43.5

         \[\leadsto -4 \cdot \frac{\color{blue}{a \cdot t}}{c} \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.1%

      \[\leadsto \left(t \cdot -4\right) \cdot \color{blue}{\frac{a}{c}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.1%

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

Alternative 16: 35.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ [x, y, z, t, a, b, c_m] = \mathsf{sort}([x, y, z, t, a, b, c_m])\\ \\ c\_s \cdot \frac{b}{c\_m \cdot z} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* c_m z))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
assert(x < y && y < z && z < t && t < a && a < b && b < c_m);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (c_m * z));
}

Fortran

c\_m =     private
c\_s =     private
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
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(c_s, x, y, z, t, a, b, c_m)
use fmin_fmax_functions
    real(8), intent (in) :: c_s
    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_m
    code = c_s * (b / (c_m * z))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
assert x < y && y < z && z < t && t < a && a < b && b < c_m;
public static double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (b / (c_m * z));
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
[x, y, z, t, a, b, c_m] = sort([x, y, z, t, a, b, c_m])
def code(c_s, x, y, z, t, a, b, c_m):
	return c_s * (b / (c_m * z))

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
x, y, z, t, a, b, c_m = sort([x, y, z, t, a, b, c_m])
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(b / Float64(c_m * z)))
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
x, y, z, t, a, b, c_m = num2cell(sort([x, y, z, t, a, b, c_m])){:}
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (c_m * z));
end

Wolfram

c\_m = N[Abs[c], $MachinePrecision]
c\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x, y, z, t, a, b, and c_m should be sorted in increasing order before calling this function.
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(b / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

   \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]

   2. lower-*.f64 36.9

      \[\leadsto \frac{b}{\color{blue}{c \cdot z}} \]

5. Applied rewrites 36.9%

   \[\leadsto \color{blue}{\frac{b}{c \cdot z}} \]
6. Add Preprocessing

Developer Target 1: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := 4 \cdot \frac{a \cdot t}{c}\\ t_3 := \left(x \cdot 9\right) \cdot y\\ t_4 := \left(t\_3 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b\\ t_5 := \frac{t\_4}{z \cdot c}\\ t_6 := \frac{\left(t\_3 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t\_5 < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 0:\\ \;\;\;\;\frac{\frac{t\_4}{z}}{c}\\ \mathbf{elif}\;t\_5 < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + t\_1\right) - t\_2\\ \mathbf{elif}\;t\_5 < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + t\_1\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* c z)))
        (t_2 (* 4.0 (/ (* a t) c)))
        (t_3 (* (* x 9.0) y))
        (t_4 (+ (- t_3 (* (* (* z 4.0) t) a)) b))
        (t_5 (/ t_4 (* z c)))
        (t_6 (/ (+ (- t_3 (* (* z 4.0) (* t a))) b) (* z c))))
   (if (< t_5 -1.100156740804105e-171)
     t_6
     (if (< t_5 0.0)
       (/ (/ t_4 z) c)
       (if (< t_5 1.1708877911747488e-53)
         t_6
         (if (< t_5 2.876823679546137e+130)
           (- (+ (* (* 9.0 (/ y c)) (/ x z)) t_1) t_2)
           (if (< t_5 1.3838515042456319e+158)
             t_6
             (- (+ (* 9.0 (* (/ y (* c z)) x)) t_1) t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - 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)
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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = b / (c * z)
    t_2 = 4.0d0 * ((a * t) / c)
    t_3 = (x * 9.0d0) * y
    t_4 = (t_3 - (((z * 4.0d0) * t) * a)) + b
    t_5 = t_4 / (z * c)
    t_6 = ((t_3 - ((z * 4.0d0) * (t * a))) + b) / (z * c)
    if (t_5 < (-1.100156740804105d-171)) then
        tmp = t_6
    else if (t_5 < 0.0d0) then
        tmp = (t_4 / z) / c
    else if (t_5 < 1.1708877911747488d-53) then
        tmp = t_6
    else if (t_5 < 2.876823679546137d+130) then
        tmp = (((9.0d0 * (y / c)) * (x / z)) + t_1) - t_2
    else if (t_5 < 1.3838515042456319d+158) then
        tmp = t_6
    else
        tmp = ((9.0d0 * ((y / (c * z)) * x)) + t_1) - 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 t_1 = b / (c * z);
	double t_2 = 4.0 * ((a * t) / c);
	double t_3 = (x * 9.0) * y;
	double t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	double t_5 = t_4 / (z * c);
	double t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	double tmp;
	if (t_5 < -1.100156740804105e-171) {
		tmp = t_6;
	} else if (t_5 < 0.0) {
		tmp = (t_4 / z) / c;
	} else if (t_5 < 1.1708877911747488e-53) {
		tmp = t_6;
	} else if (t_5 < 2.876823679546137e+130) {
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	} else if (t_5 < 1.3838515042456319e+158) {
		tmp = t_6;
	} else {
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = b / (c * z)
	t_2 = 4.0 * ((a * t) / c)
	t_3 = (x * 9.0) * y
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b
	t_5 = t_4 / (z * c)
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c)
	tmp = 0
	if t_5 < -1.100156740804105e-171:
		tmp = t_6
	elif t_5 < 0.0:
		tmp = (t_4 / z) / c
	elif t_5 < 1.1708877911747488e-53:
		tmp = t_6
	elif t_5 < 2.876823679546137e+130:
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2
	elif t_5 < 1.3838515042456319e+158:
		tmp = t_6
	else:
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(c * z))
	t_2 = Float64(4.0 * Float64(Float64(a * t) / c))
	t_3 = Float64(Float64(x * 9.0) * y)
	t_4 = Float64(Float64(t_3 - Float64(Float64(Float64(z * 4.0) * t) * a)) + b)
	t_5 = Float64(t_4 / Float64(z * c))
	t_6 = Float64(Float64(Float64(t_3 - Float64(Float64(z * 4.0) * Float64(t * a))) + b) / Float64(z * c))
	tmp = 0.0
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = Float64(Float64(t_4 / z) / c);
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = Float64(Float64(Float64(Float64(9.0 * Float64(y / c)) * Float64(x / z)) + t_1) - t_2);
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(9.0 * Float64(Float64(y / Float64(c * z)) * x)) + t_1) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = b / (c * z);
	t_2 = 4.0 * ((a * t) / c);
	t_3 = (x * 9.0) * y;
	t_4 = (t_3 - (((z * 4.0) * t) * a)) + b;
	t_5 = t_4 / (z * c);
	t_6 = ((t_3 - ((z * 4.0) * (t * a))) + b) / (z * c);
	tmp = 0.0;
	if (t_5 < -1.100156740804105e-171)
		tmp = t_6;
	elseif (t_5 < 0.0)
		tmp = (t_4 / z) / c;
	elseif (t_5 < 1.1708877911747488e-53)
		tmp = t_6;
	elseif (t_5 < 2.876823679546137e+130)
		tmp = (((9.0 * (y / c)) * (x / z)) + t_1) - t_2;
	elseif (t_5 < 1.3838515042456319e+158)
		tmp = t_6;
	else
		tmp = ((9.0 * ((y / (c * z)) * x)) + t_1) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 - N[(N[(z * 4.0), $MachinePrecision] * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$5, -1.100156740804105e-171], t$95$6, If[Less[t$95$5, 0.0], N[(N[(t$95$4 / z), $MachinePrecision] / c), $MachinePrecision], If[Less[t$95$5, 1.1708877911747488e-53], t$95$6, If[Less[t$95$5, 2.876823679546137e+130], N[(N[(N[(N[(9.0 * N[(y / c), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[t$95$5, 1.3838515042456319e+158], t$95$6, N[(N[(N[(9.0 * N[(N[(y / N[(c * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]]

Reproduce

herbie shell --seed 2024350 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -220031348160821/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 365902434742109/31250000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 28768236795461370000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 138385150424563190000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c)))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))