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

Percentage Accurate: 80.2% → 90.6%

Time: 7.2s

Alternatives: 15

Speedup: 1.2×

• 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: 80.2% 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: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;c\_m \leq 9.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, a \cdot \frac{t}{c\_m}, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{c\_m \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= c_m 9.5e+68)
    (/ (fma (* -4.0 a) t (/ (fma (* 9.0 x) y b) z)) c_m)
    (fma -4.0 (* a (/ t c_m)) (/ (fma (* y x) 9.0 b) (* c_m z))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
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 <= 9.5e+68) {
		tmp = fma((-4.0 * a), t, (fma((9.0 * x), y, b) / z)) / c_m;
	} else {
		tmp = fma(-4.0, (a * (t / c_m)), (fma((y * x), 9.0, b) / (c_m * z)));
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (c_m <= 9.5e+68)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(fma(Float64(9.0 * x), y, b) / z)) / c_m);
	else
		tmp = fma(-4.0, Float64(a * Float64(t / c_m)), Float64(fma(Float64(y * x), 9.0, b) / Float64(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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[c$95$m, 9.5e+68], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(-4.0 * N[(a * N[(t / c$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 9.50000000000000069e68

   1. Initial program 89.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 36.4

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

   6. Applied rewrites 36.4%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 96.0

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

   9. Applied rewrites 96.0%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 95.9

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

   11. Applied rewrites 95.9%

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

   if 9.50000000000000069e68 < c

   1. Initial program 66.0%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.2

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

   6. Applied rewrites 82.2%

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

Alternative 2: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x \cdot 9\right) \cdot y \leq 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{c\_m} \cdot -9 - \frac{b}{c\_m \cdot y}}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= (* (* x 9.0) y) 1e+244)
    (/ (fma (* a t) -4.0 (/ (fma (* 9.0 x) y b) z)) c_m)
    (- (* (/ (- (* (/ x c_m) -9.0) (/ b (* c_m y))) z) y)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (((x * 9.0) * y) <= 1e+244) {
		tmp = fma((a * t), -4.0, (fma((9.0 * x), y, b) / z)) / c_m;
	} else {
		tmp = -(((((x / c_m) * -9.0) - (b / (c_m * y))) / z) * y);
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (Float64(Float64(x * 9.0) * y) <= 1e+244)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(fma(Float64(9.0 * x), y, b) / z)) / c_m);
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(x / c_m) * -9.0) - Float64(b / Float64(c_m * y))) / z) * y));
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision], 1e+244], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], (-N[(N[(N[(N[(N[(x / c$95$m), $MachinePrecision] * -9.0), $MachinePrecision] - N[(b / N[(c$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.00000000000000007e244

   1. Initial program 81.3%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 88.2

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

   9. Applied rewrites 88.2%

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

   if 1.00000000000000007e244 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 69.5%

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

   2. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \cdot y \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \cdot y \]

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in z around 0

      \[\leadsto -\frac{-9 \cdot \frac{x}{c} - \frac{b}{c \cdot y}}{z} \cdot y \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -\frac{-9 \cdot \frac{x}{c} - \frac{b}{c \cdot y}}{z} \cdot y \]

      2. lower--.f64 N/A

         \[\leadsto -\frac{-9 \cdot \frac{x}{c} - \frac{b}{c \cdot y}}{z} \cdot y \]

      3. *-commutative N/A

         \[\leadsto -\frac{\frac{x}{c} \cdot -9 - \frac{b}{c \cdot y}}{z} \cdot y \]

      4. lower-*.f64 N/A

         \[\leadsto -\frac{\frac{x}{c} \cdot -9 - \frac{b}{c \cdot y}}{z} \cdot y \]

      5. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{x}{c} \cdot -9 - \frac{b}{c \cdot y}}{z} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{x}{c} \cdot -9 - \frac{b}{c \cdot y}}{z} \cdot y \]

      7. lower-*.f64 83.7

         \[\leadsto -\frac{\frac{x}{c} \cdot -9 - \frac{b}{c \cdot y}}{z} \cdot y \]

   7. Applied rewrites 83.7%

      \[\leadsto -\frac{\frac{x}{c} \cdot -9 - \frac{b}{c \cdot y}}{z} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{\mathsf{fma}\left(9 \cdot x, y, b\right)}{z}\right)}{c\_m} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (* c_s (/ (fma (* -4.0 a) t (/ (fma (* 9.0 x) y b) z)) c_m)))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	return c_s * (fma((-4.0 * a), t, (fma((9.0 * x), y, b) / z)) / c_m);
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	return Float64(c_s * Float64(fma(Float64(-4.0 * a), t, Float64(fma(Float64(9.0 * x), y, b) / z)) / c_m))
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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(N[(N[(9.0 * x), $MachinePrecision] * y + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 36.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 34.1

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

6. Applied rewrites 34.1%

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

7. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. div-add N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 87.4

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

9. Applied rewrites 87.4%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-*.f64 87.4

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

11. Applied rewrites 87.4%

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

Alternative 4: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{y \cdot x}{z} \cdot 9\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (fma (* -4.0 a) t (/ b z)) c_m)))
   (*
    c_s
    (if (<= b -2.65e+42)
      t_1
      (if (<= b 3.2e+90)
        (/ (fma (* -4.0 a) t (* (/ (* y x) z) 9.0)) c_m)
        t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma((-4.0 * a), t, (b / z)) / c_m;
	double tmp;
	if (b <= -2.65e+42) {
		tmp = t_1;
	} else if (b <= 3.2e+90) {
		tmp = fma((-4.0 * a), t, (((y * x) / z) * 9.0)) / c_m;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m)
	tmp = 0.0
	if (b <= -2.65e+42)
		tmp = t_1;
	elseif (b <= 3.2e+90)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(Float64(Float64(y * x) / z) * 9.0)) / c_m);
	else
		tmp = t_1;
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[b, -2.65e+42], t$95$1, If[LessEqual[b, 3.2e+90], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.65000000000000014e42 or 3.19999999999999998e90 < b

   1. Initial program 79.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 84.0

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

   9. Applied rewrites 84.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 73.2%

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

   if -2.65000000000000014e42 < b < 3.19999999999999998e90

   1. Initial program 81.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 22.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 21.1

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

   6. Applied rewrites 21.1%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 89.7

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

   9. Applied rewrites 89.7%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 77.0

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

   12. Applied rewrites 77.0%

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

Alternative 5: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{\left(y \cdot x\right) \cdot 9}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (fma (* -4.0 a) t (/ b z)) c_m)))
   (*
    c_s
    (if (<= b -2.65e+42)
      t_1
      (if (<= b 3.2e+90)
        (/ (fma (* a t) -4.0 (/ (* (* y x) 9.0) z)) c_m)
        t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma((-4.0 * a), t, (b / z)) / c_m;
	double tmp;
	if (b <= -2.65e+42) {
		tmp = t_1;
	} else if (b <= 3.2e+90) {
		tmp = fma((a * t), -4.0, (((y * x) * 9.0) / z)) / c_m;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m)
	tmp = 0.0
	if (b <= -2.65e+42)
		tmp = t_1;
	elseif (b <= 3.2e+90)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(Float64(Float64(y * x) * 9.0) / z)) / c_m);
	else
		tmp = t_1;
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[b, -2.65e+42], t$95$1, If[LessEqual[b, 3.2e+90], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.65000000000000014e42 or 3.19999999999999998e90 < b

   1. Initial program 79.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 84.0

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

   9. Applied rewrites 84.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 73.2%

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

   if -2.65000000000000014e42 < b < 3.19999999999999998e90

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 76.6

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

   10. Applied rewrites 76.6%

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

Alternative 6: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{y \cdot x}{c\_m}, 9, \frac{b}{c\_m}\right)}{z}\\ t_2 := \frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t \cdot z, b\right)}{z \cdot c\_m}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ (fma (/ (* y x) c_m) 9.0 (/ b c_m)) z))
        (t_2 (/ (fma (* -4.0 a) t (/ b z)) c_m)))
   (*
    c_s
    (if (<= z -5.5e+84)
      t_2
      (if (<= z 5e-143)
        t_1
        (if (<= z 7.8e+37)
          (/ (fma (* -4.0 a) (* t z) b) (* z c_m))
          (if (<= z 2.3e+94) t_1 t_2)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = fma(((y * x) / c_m), 9.0, (b / c_m)) / z;
	double t_2 = fma((-4.0 * a), t, (b / z)) / c_m;
	double tmp;
	if (z <= -5.5e+84) {
		tmp = t_2;
	} else if (z <= 5e-143) {
		tmp = t_1;
	} else if (z <= 7.8e+37) {
		tmp = fma((-4.0 * a), (t * z), b) / (z * c_m);
	} else if (z <= 2.3e+94) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(fma(Float64(Float64(y * x) / c_m), 9.0, Float64(b / c_m)) / z)
	t_2 = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m)
	tmp = 0.0
	if (z <= -5.5e+84)
		tmp = t_2;
	elseif (z <= 5e-143)
		tmp = t_1;
	elseif (z <= 7.8e+37)
		tmp = Float64(fma(Float64(-4.0 * a), Float64(t * z), b) / Float64(z * c_m));
	elseif (z <= 2.3e+94)
		tmp = t_1;
	else
		tmp = t_2;
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(N[(y * x), $MachinePrecision] / c$95$m), $MachinePrecision] * 9.0 + N[(b / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(c$95$s * If[LessEqual[z, -5.5e+84], t$95$2, If[LessEqual[z, 5e-143], t$95$1, If[LessEqual[z, 7.8e+37], N[(N[(N[(-4.0 * a), $MachinePrecision] * N[(t * z), $MachinePrecision] + b), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e+94], t$95$1, t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5000000000000004e84 or 2.3e94 < z

   1. Initial program 56.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 25.4

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

   6. Applied rewrites 25.4%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 87.9

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

   9. Applied rewrites 87.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 74.5%

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

   if -5.5000000000000004e84 < z < 5.0000000000000002e-143 or 7.7999999999999997e37 < z < 2.3e94

   1. Initial program 93.0%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{x \cdot y}{c} \cdot 9 + \frac{b}{c}}{z} \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if 5.0000000000000002e-143 < z < 7.7999999999999997e37

   1. Initial program 92.3%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 61.5

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

   4. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 61.5

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

   6. Applied rewrites 61.5%

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

Alternative 7: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ c\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot \frac{x}{c\_m}, 9, \frac{b}{c\_m}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (*
  c_s
  (if (<= x -1.55e+79)
    (/ (fma (* y (/ x c_m)) 9.0 (/ b c_m)) z)
    (/ (fma (* -4.0 a) t (/ b z)) c_m))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double tmp;
	if (x <= -1.55e+79) {
		tmp = fma((y * (x / c_m)), 9.0, (b / c_m)) / z;
	} else {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	tmp = 0.0
	if (x <= -1.55e+79)
		tmp = Float64(fma(Float64(y * Float64(x / c_m)), 9.0, Float64(b / c_m)) / z);
	else
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * If[LessEqual[x, -1.55e+79], N[(N[(N[(y * N[(x / c$95$m), $MachinePrecision]), $MachinePrecision] * 9.0 + N[(b / c$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < -1.5499999999999999e79

   1. Initial program 76.1%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{x \cdot y}{c} \cdot 9 + \frac{b}{c}}{z} \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 64.8

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

   7. Applied rewrites 64.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 68.0

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

   9. Applied rewrites 68.0%

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

   if -1.5499999999999999e79 < x

   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. Taylor expanded in b around inf

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

   4. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 36.3

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 88.7

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

   9. Applied rewrites 88.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 68.3%

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

Alternative 8: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := -\left(\frac{\frac{x}{c\_m}}{z} \cdot -9\right) \cdot y\\ t_2 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -7.1 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, t, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (- (* (* (/ (/ x c_m) z) -9.0) y))) (t_2 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_2 -7.1e+112)
      t_1
      (if (<= t_2 2e+112) (/ (fma (* -4.0 a) t (/ b z)) c_m) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = -((((x / c_m) / z) * -9.0) * y);
	double t_2 = (x * 9.0) * y;
	double tmp;
	if (t_2 <= -7.1e+112) {
		tmp = t_1;
	} else if (t_2 <= 2e+112) {
		tmp = fma((-4.0 * a), t, (b / z)) / c_m;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(-Float64(Float64(Float64(Float64(x / c_m) / z) * -9.0) * y))
	t_2 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_2 <= -7.1e+112)
		tmp = t_1;
	elseif (t_2 <= 2e+112)
		tmp = Float64(fma(Float64(-4.0 * a), t, Float64(b / z)) / c_m);
	else
		tmp = t_1;
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = (-N[(N[(N[(N[(x / c$95$m), $MachinePrecision] / z), $MachinePrecision] * -9.0), $MachinePrecision] * y), $MachinePrecision])}, Block[{t$95$2 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -7.1e+112], t$95$1, If[LessEqual[t$95$2, 2e+112], N[(N[(N[(-4.0 * a), $MachinePrecision] * t + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -7.1e112 or 1.9999999999999999e112 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 75.1%

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

   2. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \cdot y \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \cdot y \]

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.9

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

   9. Applied rewrites 71.9%

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

   if -7.1e112 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.9999999999999999e112

   1. Initial program 82.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 45.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 43.3

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

   6. Applied rewrites 43.3%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 91.7

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

   9. Applied rewrites 91.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 79.6%

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

Alternative 9: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := -\left(\frac{\frac{x}{c\_m}}{z} \cdot -9\right) \cdot y\\ t_2 := \left(x \cdot 9\right) \cdot y\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -7.1 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{b}{z}\right)}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (- (* (* (/ (/ x c_m) z) -9.0) y))) (t_2 (* (* x 9.0) y)))
   (*
    c_s
    (if (<= t_2 -7.1e+112)
      t_1
      (if (<= t_2 2e+112) (/ (fma (* a t) -4.0 (/ b z)) c_m) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = -((((x / c_m) / z) * -9.0) * y);
	double t_2 = (x * 9.0) * y;
	double tmp;
	if (t_2 <= -7.1e+112) {
		tmp = t_1;
	} else if (t_2 <= 2e+112) {
		tmp = fma((a * t), -4.0, (b / z)) / c_m;
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(-Float64(Float64(Float64(Float64(x / c_m) / z) * -9.0) * y))
	t_2 = Float64(Float64(x * 9.0) * y)
	tmp = 0.0
	if (t_2 <= -7.1e+112)
		tmp = t_1;
	elseif (t_2 <= 2e+112)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(b / z)) / c_m);
	else
		tmp = t_1;
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = (-N[(N[(N[(N[(x / c$95$m), $MachinePrecision] / z), $MachinePrecision] * -9.0), $MachinePrecision] * y), $MachinePrecision])}, Block[{t$95$2 = N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision]}, N[(c$95$s * If[LessEqual[t$95$2, -7.1e+112], t$95$1, If[LessEqual[t$95$2, 2e+112], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(b / z), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 9 binary64)) y) < -7.1e112 or 1.9999999999999999e112 < (*.f64 (*.f64 x #s(literal 9 binary64)) y)

   1. Initial program 75.1%

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

   2. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -y \cdot \left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \cdot y \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-9 \cdot \frac{x}{c \cdot z} + -1 \cdot \frac{\frac{b}{c \cdot z} - 4 \cdot \frac{a \cdot t}{c}}{y}\right) \cdot y \]

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.9

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

   9. Applied rewrites 71.9%

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

   if -7.1e112 < (*.f64 (*.f64 x #s(literal 9 binary64)) y) < 1.9999999999999999e112

   1. Initial program 82.9%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. div-add N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.3%

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

Alternative 10: 50.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-76}:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{y \cdot x}{c\_m} \cdot 9}{z}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+196}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ b (* z c_m))))
   (*
    c_s
    (if (<= b -1.25e+45)
      t_1
      (if (<= b -4.4e-76)
        (* (* (/ a c_m) -4.0) t)
        (if (<= b 5.1e+45)
          (/ (* (/ (* y x) c_m) 9.0) z)
          (if (<= b 5.8e+196) (* -4.0 (/ (* a t) c_m)) t_1)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = b / (z * c_m);
	double tmp;
	if (b <= -1.25e+45) {
		tmp = t_1;
	} else if (b <= -4.4e-76) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (b <= 5.1e+45) {
		tmp = (((y * x) / c_m) * 9.0) / z;
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
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) :: tmp
    t_1 = b / (z * c_m)
    if (b <= (-1.25d+45)) then
        tmp = t_1
    else if (b <= (-4.4d-76)) then
        tmp = ((a / c_m) * (-4.0d0)) * t
    else if (b <= 5.1d+45) then
        tmp = (((y * x) / c_m) * 9.0d0) / z
    else if (b <= 5.8d+196) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    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);
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 = b / (z * c_m);
	double tmp;
	if (b <= -1.25e+45) {
		tmp = t_1;
	} else if (b <= -4.4e-76) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (b <= 5.1e+45) {
		tmp = (((y * x) / c_m) * 9.0) / z;
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = b / (z * c_m)
	tmp = 0
	if b <= -1.25e+45:
		tmp = t_1
	elif b <= -4.4e-76:
		tmp = ((a / c_m) * -4.0) * t
	elif b <= 5.1e+45:
		tmp = (((y * x) / c_m) * 9.0) / z
	elif b <= 5.8e+196:
		tmp = -4.0 * ((a * t) / c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(b / Float64(z * c_m))
	tmp = 0.0
	if (b <= -1.25e+45)
		tmp = t_1;
	elseif (b <= -4.4e-76)
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	elseif (b <= 5.1e+45)
		tmp = Float64(Float64(Float64(Float64(y * x) / c_m) * 9.0) / z);
	elseif (b <= 5.8e+196)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = b / (z * c_m);
	tmp = 0.0;
	if (b <= -1.25e+45)
		tmp = t_1;
	elseif (b <= -4.4e-76)
		tmp = ((a / c_m) * -4.0) * t;
	elseif (b <= 5.1e+45)
		tmp = (((y * x) / c_m) * 9.0) / z;
	elseif (b <= 5.8e+196)
		tmp = -4.0 * ((a * t) / c_m);
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[b, -1.25e+45], t$95$1, If[LessEqual[b, -4.4e-76], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 5.1e+45], N[(N[(N[(N[(y * x), $MachinePrecision] / c$95$m), $MachinePrecision] * 9.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[b, 5.8e+196], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if b < -1.25e45 or 5.8e196 < b

   1. Initial program 79.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 57.5%

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

   if -1.25e45 < b < -4.39999999999999999e-76

   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. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 42.2

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

   7. Applied rewrites 42.2%

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

   if -4.39999999999999999e-76 < b < 5.0999999999999997e45

   1. Initial program 81.1%

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

   2. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{9 \cdot \frac{x \cdot y}{c} + \frac{b}{c}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{x \cdot y}{c} \cdot 9 + \frac{b}{c}}{z} \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 53.7

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

   7. Applied rewrites 53.7%

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

   8. Taylor expanded in x around 0

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

      1. lift-/.f64 16.9

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

   10. Applied rewrites 16.9%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 45.2

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

   13. Applied rewrites 45.2%

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

   if 5.0999999999999997e45 < b < 5.8e196

   1. Initial program 79.6%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 33.5

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

   4. Applied rewrites 33.5%

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

Alternative 11: 47.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-79}:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \mathbf{elif}\;b \leq 1.86 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(y \cdot x\right) \cdot 9}{z \cdot c\_m}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+196}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ b (* z c_m))))
   (*
    c_s
    (if (<= b -1.25e+45)
      t_1
      (if (<= b -1.2e-79)
        (* (* (/ a c_m) -4.0) t)
        (if (<= b 1.86e+46)
          (/ (* (* y x) 9.0) (* z c_m))
          (if (<= b 5.8e+196) (* -4.0 (/ (* a t) c_m)) t_1)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = b / (z * c_m);
	double tmp;
	if (b <= -1.25e+45) {
		tmp = t_1;
	} else if (b <= -1.2e-79) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (b <= 1.86e+46) {
		tmp = ((y * x) * 9.0) / (z * c_m);
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
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) :: tmp
    t_1 = b / (z * c_m)
    if (b <= (-1.25d+45)) then
        tmp = t_1
    else if (b <= (-1.2d-79)) then
        tmp = ((a / c_m) * (-4.0d0)) * t
    else if (b <= 1.86d+46) then
        tmp = ((y * x) * 9.0d0) / (z * c_m)
    else if (b <= 5.8d+196) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    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);
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 = b / (z * c_m);
	double tmp;
	if (b <= -1.25e+45) {
		tmp = t_1;
	} else if (b <= -1.2e-79) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (b <= 1.86e+46) {
		tmp = ((y * x) * 9.0) / (z * c_m);
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = b / (z * c_m)
	tmp = 0
	if b <= -1.25e+45:
		tmp = t_1
	elif b <= -1.2e-79:
		tmp = ((a / c_m) * -4.0) * t
	elif b <= 1.86e+46:
		tmp = ((y * x) * 9.0) / (z * c_m)
	elif b <= 5.8e+196:
		tmp = -4.0 * ((a * t) / c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(b / Float64(z * c_m))
	tmp = 0.0
	if (b <= -1.25e+45)
		tmp = t_1;
	elseif (b <= -1.2e-79)
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	elseif (b <= 1.86e+46)
		tmp = Float64(Float64(Float64(y * x) * 9.0) / Float64(z * c_m));
	elseif (b <= 5.8e+196)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = b / (z * c_m);
	tmp = 0.0;
	if (b <= -1.25e+45)
		tmp = t_1;
	elseif (b <= -1.2e-79)
		tmp = ((a / c_m) * -4.0) * t;
	elseif (b <= 1.86e+46)
		tmp = ((y * x) * 9.0) / (z * c_m);
	elseif (b <= 5.8e+196)
		tmp = -4.0 * ((a * t) / c_m);
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[b, -1.25e+45], t$95$1, If[LessEqual[b, -1.2e-79], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 1.86e+46], N[(N[(N[(y * x), $MachinePrecision] * 9.0), $MachinePrecision] / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e+196], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if b < -1.25e45 or 5.8e196 < b

   1. Initial program 79.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 57.5%

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

   if -1.25e45 < b < -1.20000000000000003e-79

   1. Initial program 80.4%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 42.8

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

   7. Applied rewrites 42.8%

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

   if -1.20000000000000003e-79 < b < 1.8600000000000001e46

   1. Initial program 81.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 45.9

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

   4. Applied rewrites 45.9%

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

   if 1.8600000000000001e46 < b < 5.8e196

   1. Initial program 79.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 33.6

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

   4. Applied rewrites 33.6%

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

Alternative 12: 47.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-83}:\\ \;\;\;\;\left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ \mathbf{elif}\;b \leq 1.86 \cdot 10^{+46}:\\ \;\;\;\;\left(\frac{y}{c\_m \cdot z} \cdot x\right) \cdot 9\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+196}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ b (* z c_m))))
   (*
    c_s
    (if (<= b -1.25e+45)
      t_1
      (if (<= b -5.4e-83)
        (* (* (/ a c_m) -4.0) t)
        (if (<= b 1.86e+46)
          (* (* (/ y (* c_m z)) x) 9.0)
          (if (<= b 5.8e+196) (* -4.0 (/ (* a t) c_m)) t_1)))))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = b / (z * c_m);
	double tmp;
	if (b <= -1.25e+45) {
		tmp = t_1;
	} else if (b <= -5.4e-83) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (b <= 1.86e+46) {
		tmp = ((y / (c_m * z)) * x) * 9.0;
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
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) :: tmp
    t_1 = b / (z * c_m)
    if (b <= (-1.25d+45)) then
        tmp = t_1
    else if (b <= (-5.4d-83)) then
        tmp = ((a / c_m) * (-4.0d0)) * t
    else if (b <= 1.86d+46) then
        tmp = ((y / (c_m * z)) * x) * 9.0d0
    else if (b <= 5.8d+196) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    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);
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 = b / (z * c_m);
	double tmp;
	if (b <= -1.25e+45) {
		tmp = t_1;
	} else if (b <= -5.4e-83) {
		tmp = ((a / c_m) * -4.0) * t;
	} else if (b <= 1.86e+46) {
		tmp = ((y / (c_m * z)) * x) * 9.0;
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = b / (z * c_m)
	tmp = 0
	if b <= -1.25e+45:
		tmp = t_1
	elif b <= -5.4e-83:
		tmp = ((a / c_m) * -4.0) * t
	elif b <= 1.86e+46:
		tmp = ((y / (c_m * z)) * x) * 9.0
	elif b <= 5.8e+196:
		tmp = -4.0 * ((a * t) / c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(b / Float64(z * c_m))
	tmp = 0.0
	if (b <= -1.25e+45)
		tmp = t_1;
	elseif (b <= -5.4e-83)
		tmp = Float64(Float64(Float64(a / c_m) * -4.0) * t);
	elseif (b <= 1.86e+46)
		tmp = Float64(Float64(Float64(y / Float64(c_m * z)) * x) * 9.0);
	elseif (b <= 5.8e+196)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = b / (z * c_m);
	tmp = 0.0;
	if (b <= -1.25e+45)
		tmp = t_1;
	elseif (b <= -5.4e-83)
		tmp = ((a / c_m) * -4.0) * t;
	elseif (b <= 1.86e+46)
		tmp = ((y / (c_m * z)) * x) * 9.0;
	elseif (b <= 5.8e+196)
		tmp = -4.0 * ((a * t) / c_m);
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[b, -1.25e+45], t$95$1, If[LessEqual[b, -5.4e-83], N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 1.86e+46], N[(N[(N[(y / N[(c$95$m * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[b, 5.8e+196], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if b < -1.25e45 or 5.8e196 < b

   1. Initial program 79.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 57.5%

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

   if -1.25e45 < b < -5.39999999999999982e-83

   1. Initial program 80.6%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 42.7

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

   7. Applied rewrites 42.7%

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

   if -5.39999999999999982e-83 < b < 1.8600000000000001e46

   1. Initial program 81.1%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 19.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 17.7

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

   6. Applied rewrites 17.7%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. div-add N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

   10. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 45.9

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

   12. Applied rewrites 45.9%

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

   if 1.8600000000000001e46 < b < 5.8e196

   1. Initial program 79.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 33.6

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

   4. Applied rewrites 33.6%

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

Alternative 13: 47.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \left(\frac{a}{c\_m} \cdot -4\right) \cdot t\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{b}{z \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (* (* (/ a c_m) -4.0) t)))
   (* c_s (if (<= a -5.8e-111) t_1 (if (<= a 1.8e+73) (/ b (* z c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = ((a / c_m) * -4.0) * t;
	double tmp;
	if (a <= -5.8e-111) {
		tmp = t_1;
	} else if (a <= 1.8e+73) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
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) :: tmp
    t_1 = ((a / c_m) * (-4.0d0)) * t
    if (a <= (-5.8d-111)) then
        tmp = t_1
    else if (a <= 1.8d+73) then
        tmp = b / (z * c_m)
    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);
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 = ((a / c_m) * -4.0) * t;
	double tmp;
	if (a <= -5.8e-111) {
		tmp = t_1;
	} else if (a <= 1.8e+73) {
		tmp = b / (z * c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = ((a / c_m) * -4.0) * t
	tmp = 0
	if a <= -5.8e-111:
		tmp = t_1
	elif a <= 1.8e+73:
		tmp = b / (z * c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(Float64(Float64(a / c_m) * -4.0) * t)
	tmp = 0.0
	if (a <= -5.8e-111)
		tmp = t_1;
	elseif (a <= 1.8e+73)
		tmp = Float64(b / Float64(z * c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = ((a / c_m) * -4.0) * t;
	tmp = 0.0;
	if (a <= -5.8e-111)
		tmp = t_1;
	elseif (a <= 1.8e+73)
		tmp = b / (z * c_m);
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(N[(N[(a / c$95$m), $MachinePrecision] * -4.0), $MachinePrecision] * t), $MachinePrecision]}, N[(c$95$s * If[LessEqual[a, -5.8e-111], t$95$1, If[LessEqual[a, 1.8e+73], N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.80000000000000003e-111 or 1.7999999999999999e73 < a

   1. Initial program 79.3%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 55.7

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

   7. Applied rewrites 55.7%

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

   if -5.80000000000000003e-111 < a < 1.7999999999999999e73

   1. Initial program 81.4%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 44.4%

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

Alternative 14: 47.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} c\_m = \left|c\right| \\ c\_s = \mathsf{copysign}\left(1, c\right) \\ \begin{array}{l} t_1 := \frac{b}{z \cdot c\_m}\\ c\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+196}:\\ \;\;\;\;-4 \cdot \frac{a \cdot t}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m)
 :precision binary64
 (let* ((t_1 (/ b (* z c_m))))
   (*
    c_s
    (if (<= b -3.1e+42)
      t_1
      (if (<= b 5.8e+196) (* -4.0 (/ (* a t) c_m)) t_1)))))

C

c\_m = fabs(c);
c\_s = copysign(1.0, c);
double code(double c_s, double x, double y, double z, double t, double a, double b, double c_m) {
	double t_1 = b / (z * c_m);
	double tmp;
	if (b <= -3.1e+42) {
		tmp = t_1;
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Fortran

c\_m =     private
c\_s =     private
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) :: tmp
    t_1 = b / (z * c_m)
    if (b <= (-3.1d+42)) then
        tmp = t_1
    else if (b <= 5.8d+196) then
        tmp = (-4.0d0) * ((a * t) / c_m)
    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);
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 = b / (z * c_m);
	double tmp;
	if (b <= -3.1e+42) {
		tmp = t_1;
	} else if (b <= 5.8e+196) {
		tmp = -4.0 * ((a * t) / c_m);
	} else {
		tmp = t_1;
	}
	return c_s * tmp;
}

Python

c\_m = math.fabs(c)
c\_s = math.copysign(1.0, c)
def code(c_s, x, y, z, t, a, b, c_m):
	t_1 = b / (z * c_m)
	tmp = 0
	if b <= -3.1e+42:
		tmp = t_1
	elif b <= 5.8e+196:
		tmp = -4.0 * ((a * t) / c_m)
	else:
		tmp = t_1
	return c_s * tmp

Julia

c\_m = abs(c)
c\_s = copysign(1.0, c)
function code(c_s, x, y, z, t, a, b, c_m)
	t_1 = Float64(b / Float64(z * c_m))
	tmp = 0.0
	if (b <= -3.1e+42)
		tmp = t_1;
	elseif (b <= 5.8e+196)
		tmp = Float64(-4.0 * Float64(Float64(a * t) / c_m));
	else
		tmp = t_1;
	end
	return Float64(c_s * tmp)
end

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp_2 = code(c_s, x, y, z, t, a, b, c_m)
	t_1 = b / (z * c_m);
	tmp = 0.0;
	if (b <= -3.1e+42)
		tmp = t_1;
	elseif (b <= 5.8e+196)
		tmp = -4.0 * ((a * t) / c_m);
	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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := Block[{t$95$1 = N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(c$95$s * If[LessEqual[b, -3.1e+42], t$95$1, If[LessEqual[b, 5.8e+196], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.1000000000000002e42 or 5.8e196 < b

   1. Initial program 79.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 57.3%

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

   if -3.1000000000000002e42 < b < 5.8e196

   1. Initial program 80.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 43.1

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

   4. Applied rewrites 43.1%

      \[\leadsto \color{blue}{-4 \cdot \frac{a \cdot t}{c}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 36.1% accurate, 3.8× speedup

Math

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

FPCore

c\_m = (fabs.f64 c)
c\_s = (copysign.f64 #s(literal 1 binary64) c)
(FPCore (c_s x y z t a b c_m) :precision binary64 (* c_s (/ b (* z c_m))))

C

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

Fortran

c\_m =     private
c\_s =     private
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 / (z * c_m))
end function

Java

c\_m = Math.abs(c);
c\_s = Math.copySign(1.0, c);
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 / (z * c_m));
}

Python

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

Julia

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

MATLAB

c\_m = abs(c);
c\_s = sign(c) * abs(1.0);
function tmp = code(c_s, x, y, z, t, a, b, c_m)
	tmp = c_s * (b / (z * c_m));
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]
code[c$95$s_, x_, y_, z_, t_, a_, b_, c$95$m_] := N[(c$95$s * N[(b / N[(z * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2025117 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64
  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))