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

Percentage Accurate: 79.8% → 89.1%

Time: 5.7s

Alternatives: 14

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% 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: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot x, 9, b\right)\\ t_2 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ t_3 := \frac{\mathsf{fma}\left(-4, t, \frac{t\_1}{a \cdot z}\right)}{c} \cdot a\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{a \cdot t}{c}, \frac{t\_1}{c \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* y x) 9.0 b))
        (t_2 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))
        (t_3 (* (/ (fma -4.0 t (/ t_1 (* a z))) c) a)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 2e+295) (fma -4.0 (/ (* a t) c) (/ t_1 (* c z))) t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((y * x), 9.0, b);
	double t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double t_3 = (fma(-4.0, t, (t_1 / (a * z))) / c) * a;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 2e+295) {
		tmp = fma(-4.0, ((a * t) / c), (t_1 / (c * z)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(y * x), 9.0, b)
	t_2 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
	t_3 = Float64(Float64(fma(-4.0, t, Float64(t_1 / Float64(a * z))) / c) * a)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 2e+295)
		tmp = fma(-4.0, Float64(Float64(a * t) / c), Float64(t_1 / Float64(c * z)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[(N[(-4.0 * t + N[(t$95$1 / N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 2e+295], N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] + N[(t$95$1 / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < -inf.0 or 2e295 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c))

   1. Initial program 70.7%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 86.8

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

   7. Applied rewrites 86.8%

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

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x #s(literal 9 binary64)) y) (*.f64 (*.f64 (*.f64 z #s(literal 4 binary64)) t) a)) b) (*.f64 z c)) < 2e295

   1. Initial program 90.7%

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

   2. Taylor expanded in 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 91.3

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

   4. Applied rewrites 91.3%

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

Alternative 2: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot x, 9, b\right)\\ \mathbf{if}\;c \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{t\_1}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{c}, -4, \frac{t\_1}{\left(c \cdot z\right) \cdot a}\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* y x) 9.0 b)))
   (if (<= c 2.6e+48)
     (/ (fma (* a t) -4.0 (/ t_1 z)) c)
     (* (fma (/ t c) -4.0 (/ t_1 (* (* c z) a))) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((y * x), 9.0, b);
	double tmp;
	if (c <= 2.6e+48) {
		tmp = fma((a * t), -4.0, (t_1 / z)) / c;
	} else {
		tmp = fma((t / c), -4.0, (t_1 / ((c * z) * a))) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(y * x), 9.0, b)
	tmp = 0.0
	if (c <= 2.6e+48)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(t_1 / z)) / c);
	else
		tmp = Float64(fma(Float64(t / c), -4.0, Float64(t_1 / Float64(Float64(c * z) * a))) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]}, If[LessEqual[c, 2.6e+48], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(t / c), $MachinePrecision] * -4.0 + N[(t$95$1 / N[(N[(c * z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 2.59999999999999995e48

   1. Initial program 83.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 35.9%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   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 34.3

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

      6. associate-+l- 34.3

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

      7. associate-*l* 34.3

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

      8. *-commutative 34.3

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

      9. *-commutative 34.3

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

      10. associate-+l- 34.3

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

   6. Applied rewrites 34.3%

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

   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 89.9

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

   9. Applied rewrites 89.9%

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

   if 2.59999999999999995e48 < c

   1. Initial program 67.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 74.5%

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

Alternative 3: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot x, 9, b\right)\\ t_2 := \frac{\mathsf{fma}\left(-4, t, \frac{t\_1}{a \cdot z}\right)}{c} \cdot a\\ \mathbf{if}\;a \leq -20000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{t\_1}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* y x) 9.0 b))
        (t_2 (* (/ (fma -4.0 t (/ t_1 (* a z))) c) a)))
   (if (<= a -20000000.0)
     t_2
     (if (<= a 3.5e+23) (/ (fma (* a t) -4.0 (/ t_1 z)) c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((y * x), 9.0, b);
	double t_2 = (fma(-4.0, t, (t_1 / (a * z))) / c) * a;
	double tmp;
	if (a <= -20000000.0) {
		tmp = t_2;
	} else if (a <= 3.5e+23) {
		tmp = fma((a * t), -4.0, (t_1 / z)) / c;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(y * x), 9.0, b)
	t_2 = Float64(Float64(fma(-4.0, t, Float64(t_1 / Float64(a * z))) / c) * a)
	tmp = 0.0
	if (a <= -20000000.0)
		tmp = t_2;
	elseif (a <= 3.5e+23)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(t_1 / z)) / c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-4.0 * t + N[(t$95$1 / N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -20000000.0], t$95$2, If[LessEqual[a, 3.5e+23], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if a < -2e7 or 3.5000000000000002e23 < a

   1. Initial program 78.6%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if -2e7 < a < 3.5000000000000002e23

   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 b around inf

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

   4. Applied rewrites 42.3%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   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 41.4

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

      6. associate-+l- 41.4

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

      7. associate-*l* 41.4

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

      8. *-commutative 41.4

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

      9. *-commutative 41.4

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

      10. associate-+l- 41.4

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

   6. Applied rewrites 41.4%

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

   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 89.8

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

   9. Applied rewrites 89.8%

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

Alternative 4: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, t, \frac{b}{a \cdot z}\right)}{c} \cdot a\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+219}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{\mathsf{fma}\left(y \cdot x, 9, b\right)}{z}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (/ (fma -4.0 t (/ b (* a z))) c) a)))
   (if (<= t -5.6e+219)
     t_1
     (if (<= t 1.08e-19)
       (/ (fma (* a t) -4.0 (/ (fma (* y x) 9.0 b) z)) c)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (fma(-4.0, t, (b / (a * z))) / c) * a;
	double tmp;
	if (t <= -5.6e+219) {
		tmp = t_1;
	} else if (t <= 1.08e-19) {
		tmp = fma((a * t), -4.0, (fma((y * x), 9.0, b) / z)) / c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(fma(-4.0, t, Float64(b / Float64(a * z))) / c) * a)
	tmp = 0.0
	if (t <= -5.6e+219)
		tmp = t_1;
	elseif (t <= 1.08e-19)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(fma(Float64(y * x), 9.0, b) / z)) / c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(-4.0 * t + N[(b / N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -5.6e+219], t$95$1, If[LessEqual[t, 1.08e-19], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(N[(N[(y * x), $MachinePrecision] * 9.0 + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.60000000000000031e219 or 1.08e-19 < t

   1. Initial program 73.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.1%

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

   if -5.60000000000000031e219 < t < 1.08e-19

   1. Initial program 83.1%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 40.0%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   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 38.7

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

      6. associate-+l- 38.7

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

      7. associate-*l* 38.7

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

      8. *-commutative 38.7

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

      9. *-commutative 38.7

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

      10. associate-+l- 38.7

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

   6. Applied rewrites 38.7%

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

   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 88.6

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

   9. Applied rewrites 88.6%

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

Alternative 5: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a \cdot t}{c}, -4, \frac{b}{c \cdot z}\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{y \cdot x}{z} \cdot 9\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, t, \frac{b}{a \cdot z}\right)}{c} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= b -1.5e-48)
   (fma (/ (* a t) c) -4.0 (/ b (* c z)))
   (if (<= b 5.5e-13)
     (/ (fma (* a t) -4.0 (* (/ (* y x) z) 9.0)) c)
     (* (/ (fma -4.0 t (/ b (* a z))) c) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (b <= -1.5e-48) {
		tmp = fma(((a * t) / c), -4.0, (b / (c * z)));
	} else if (b <= 5.5e-13) {
		tmp = fma((a * t), -4.0, (((y * x) / z) * 9.0)) / c;
	} else {
		tmp = (fma(-4.0, t, (b / (a * z))) / c) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (b <= -1.5e-48)
		tmp = fma(Float64(Float64(a * t) / c), -4.0, Float64(b / Float64(c * z)));
	elseif (b <= 5.5e-13)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(Float64(Float64(y * x) / z) * 9.0)) / c);
	else
		tmp = Float64(Float64(fma(-4.0, t, Float64(b / Float64(a * z))) / c) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[b, -1.5e-48], N[(N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision] * -4.0 + N[(b / N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e-13], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(-4.0 * t + N[(b / N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e-48

   1. Initial program 79.8%

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

   2. 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 70.7%

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

   5. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 69.5

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

   7. Applied rewrites 69.5%

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

   if -1.5e-48 < b < 5.49999999999999979e-13

   1. Initial program 79.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 17.4%

      \[\leadsto \frac{\color{blue}{b}}{z \cdot c} \]
   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 15.4

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

      6. associate-+l- 15.4

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

      7. associate-*l* 15.4

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

      8. *-commutative 15.4

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

      9. *-commutative 15.4

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

      10. associate-+l- 15.4

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

   6. Applied rewrites 15.4%

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

   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 87.9

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

   9. Applied rewrites 87.9%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 80.7

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

   12. Applied rewrites 80.7%

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

   if 5.49999999999999979e-13 < b

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 67.9%

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

Alternative 6: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(-4, t, \frac{b}{a \cdot z}\right)}{c} \cdot a\\ \mathbf{if}\;t \leq -11000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y \cdot 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (/ (fma -4.0 t (/ b (* a z))) c) a)))
   (if (<= t -11000000000.0)
     t_1
     (if (<= t 6.1e-89) (/ (fma x (* y 9.0) b) (* z c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (fma(-4.0, t, (b / (a * z))) / c) * a;
	double tmp;
	if (t <= -11000000000.0) {
		tmp = t_1;
	} else if (t <= 6.1e-89) {
		tmp = fma(x, (y * 9.0), b) / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(fma(-4.0, t, Float64(b / Float64(a * z))) / c) * a)
	tmp = 0.0
	if (t <= -11000000000.0)
		tmp = t_1;
	elseif (t <= 6.1e-89)
		tmp = Float64(fma(x, Float64(y * 9.0), b) / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(-4.0 * t + N[(b / N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -11000000000.0], t$95$1, If[LessEqual[t, 6.1e-89], N[(N[(x * N[(y * 9.0), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1e10 or 6.1000000000000003e-89 < t

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 70.8%

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

   if -1.1e10 < t < 6.1000000000000003e-89

   1. Initial program 85.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.1

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

   6. Applied rewrites 75.1%

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

Alternative 7: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{t}{c} \cdot -4\right) \cdot a\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -11000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot t, -4, \frac{b}{z}\right)}{c}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y \cdot 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* (/ t c) -4.0) a)))
   (if (<= t -2.75e+210)
     t_1
     (if (<= t -11000000000.0)
       (/ (fma (* a t) -4.0 (/ b z)) c)
       (if (<= t 3.6e-20) (/ (fma x (* y 9.0) b) (* z c)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -2.75e+210) {
		tmp = t_1;
	} else if (t <= -11000000000.0) {
		tmp = fma((a * t), -4.0, (b / z)) / c;
	} else if (t <= 3.6e-20) {
		tmp = fma(x, (y * 9.0), b) / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t / c) * -4.0) * a)
	tmp = 0.0
	if (t <= -2.75e+210)
		tmp = t_1;
	elseif (t <= -11000000000.0)
		tmp = Float64(fma(Float64(a * t), -4.0, Float64(b / z)) / c);
	elseif (t <= 3.6e-20)
		tmp = Float64(fma(x, Float64(y * 9.0), b) / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t / c), $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -2.75e+210], t$95$1, If[LessEqual[t, -11000000000.0], N[(N[(N[(a * t), $MachinePrecision] * -4.0 + N[(b / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t, 3.6e-20], N[(N[(x * N[(y * 9.0), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.75000000000000006e210 or 3.59999999999999974e-20 < t

   1. Initial program 72.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -2.75000000000000006e210 < t < -1.1e10

   1. Initial program 76.3%

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

   2. Taylor expanded in 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 70.4%

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

   5. Taylor expanded in c around -inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 70.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-/.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 68.0

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

   10. Applied rewrites 68.0%

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

   if -1.1e10 < t < 3.59999999999999974e-20

   1. Initial program 85.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.7

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

   6. Applied rewrites 73.7%

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

Alternative 8: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{t}{c} \cdot -4\right) \cdot a\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y \cdot 9, b\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* (/ t c) -4.0) a)))
   (if (<= t -3.2e+106)
     t_1
     (if (<= t 3.6e-20) (/ (fma x (* y 9.0) b) (* z c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -3.2e+106) {
		tmp = t_1;
	} else if (t <= 3.6e-20) {
		tmp = fma(x, (y * 9.0), b) / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t / c) * -4.0) * a)
	tmp = 0.0
	if (t <= -3.2e+106)
		tmp = t_1;
	elseif (t <= 3.6e-20)
		tmp = Float64(fma(x, Float64(y * 9.0), b) / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t / c), $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -3.2e+106], t$95$1, If[LessEqual[t, 3.6e-20], N[(N[(x * N[(y * 9.0), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1999999999999998e106 or 3.59999999999999974e-20 < t

   1. Initial program 73.0%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 60.9

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

   7. Applied rewrites 60.9%

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

   if -3.1999999999999998e106 < t < 3.59999999999999974e-20

   1. Initial program 84.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 71.2

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

   6. Applied rewrites 71.2%

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

Alternative 9: 51.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{t}{c} \cdot -4\right) \cdot a\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} \cdot 9\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* (/ t c) -4.0) a)))
   (if (<= t -9.2e+61)
     t_1
     (if (<= t -6.5e-12)
       (/ b (* z c))
       (if (<= t 1.4e-22) (/ (* y (* (/ x z) 9.0)) c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= -6.5e-12) {
		tmp = b / (z * c);
	} else if (t <= 1.4e-22) {
		tmp = (y * ((x / z) * 9.0)) / c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t / c) * (-4.0d0)) * a
    if (t <= (-9.2d+61)) then
        tmp = t_1
    else if (t <= (-6.5d-12)) then
        tmp = b / (z * c)
    else if (t <= 1.4d-22) then
        tmp = (y * ((x / z) * 9.0d0)) / c
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= -6.5e-12) {
		tmp = b / (z * c);
	} else if (t <= 1.4e-22) {
		tmp = (y * ((x / z) * 9.0)) / c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((t / c) * -4.0) * a
	tmp = 0
	if t <= -9.2e+61:
		tmp = t_1
	elif t <= -6.5e-12:
		tmp = b / (z * c)
	elif t <= 1.4e-22:
		tmp = (y * ((x / z) * 9.0)) / c
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t / c) * -4.0) * a)
	tmp = 0.0
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= -6.5e-12)
		tmp = Float64(b / Float64(z * c));
	elseif (t <= 1.4e-22)
		tmp = Float64(Float64(y * Float64(Float64(x / z) * 9.0)) / c);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((t / c) * -4.0) * a;
	tmp = 0.0;
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= -6.5e-12)
		tmp = b / (z * c);
	elseif (t <= 1.4e-22)
		tmp = (y * ((x / z) * 9.0)) / c;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t / c), $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -9.2e+61], t$95$1, If[LessEqual[t, -6.5e-12], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-22], N[(N[(y * N[(N[(x / z), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.1999999999999998e61 or 1.39999999999999997e-22 < t

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if -9.1999999999999998e61 < t < -6.5000000000000002e-12

   1. Initial program 81.7%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 34.3%

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

   if -6.5000000000000002e-12 < t < 1.39999999999999997e-22

   1. Initial program 85.6%

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

   2. Taylor expanded in 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 75.5%

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

   5. Taylor expanded in c around -inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 44.9

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

   10. Applied rewrites 44.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 43.7

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

   12. Applied rewrites 43.7%

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

Alternative 10: 50.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{t}{c} \cdot -4\right) \cdot a\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-23}:\\ \;\;\;\;y \cdot \left(\frac{x}{c \cdot z} \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* (/ t c) -4.0) a)))
   (if (<= t -9.2e+61)
     t_1
     (if (<= t -2.2e-133)
       (/ b (* z c))
       (if (<= t 4e-23) (* y (* (/ x (* c z)) 9.0)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= -2.2e-133) {
		tmp = b / (z * c);
	} else if (t <= 4e-23) {
		tmp = y * ((x / (c * z)) * 9.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t / c) * (-4.0d0)) * a
    if (t <= (-9.2d+61)) then
        tmp = t_1
    else if (t <= (-2.2d-133)) then
        tmp = b / (z * c)
    else if (t <= 4d-23) then
        tmp = y * ((x / (c * z)) * 9.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= -2.2e-133) {
		tmp = b / (z * c);
	} else if (t <= 4e-23) {
		tmp = y * ((x / (c * z)) * 9.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((t / c) * -4.0) * a
	tmp = 0
	if t <= -9.2e+61:
		tmp = t_1
	elif t <= -2.2e-133:
		tmp = b / (z * c)
	elif t <= 4e-23:
		tmp = y * ((x / (c * z)) * 9.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t / c) * -4.0) * a)
	tmp = 0.0
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= -2.2e-133)
		tmp = Float64(b / Float64(z * c));
	elseif (t <= 4e-23)
		tmp = Float64(y * Float64(Float64(x / Float64(c * z)) * 9.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((t / c) * -4.0) * a;
	tmp = 0.0;
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= -2.2e-133)
		tmp = b / (z * c);
	elseif (t <= 4e-23)
		tmp = y * ((x / (c * z)) * 9.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t / c), $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -9.2e+61], t$95$1, If[LessEqual[t, -2.2e-133], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-23], N[(y * N[(N[(x / N[(c * z), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.1999999999999998e61 or 3.99999999999999984e-23 < t

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 60.2

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

   7. Applied rewrites 60.2%

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

   if -9.1999999999999998e61 < t < -2.2000000000000001e-133

   1. Initial program 83.3%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 37.3%

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

   if -2.2000000000000001e-133 < t < 3.99999999999999984e-23

   1. Initial program 86.0%

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

   2. Taylor expanded in 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 75.7%

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

   5. Taylor expanded in c around -inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 47.1

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

   10. Applied rewrites 47.1%

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

Alternative 11: 50.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+61}:\\ \;\;\;\;\left(\frac{t}{c} \cdot -4\right) \cdot a\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-87}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot t}{c} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -9.2e+61)
   (* (* (/ t c) -4.0) a)
   (if (<= t 1.7e-87) (/ b (* z c)) (* (/ (* -4.0 t) c) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -9.2e+61) {
		tmp = ((t / c) * -4.0) * a;
	} else if (t <= 1.7e-87) {
		tmp = b / (z * c);
	} else {
		tmp = ((-4.0 * t) / c) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (t <= (-9.2d+61)) then
        tmp = ((t / c) * (-4.0d0)) * a
    else if (t <= 1.7d-87) then
        tmp = b / (z * c)
    else
        tmp = (((-4.0d0) * t) / c) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -9.2e+61) {
		tmp = ((t / c) * -4.0) * a;
	} else if (t <= 1.7e-87) {
		tmp = b / (z * c);
	} else {
		tmp = ((-4.0 * t) / c) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if t <= -9.2e+61:
		tmp = ((t / c) * -4.0) * a
	elif t <= 1.7e-87:
		tmp = b / (z * c)
	else:
		tmp = ((-4.0 * t) / c) * a
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -9.2e+61)
		tmp = Float64(Float64(Float64(t / c) * -4.0) * a);
	elseif (t <= 1.7e-87)
		tmp = Float64(b / Float64(z * c));
	else
		tmp = Float64(Float64(Float64(-4.0 * t) / c) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (t <= -9.2e+61)
		tmp = ((t / c) * -4.0) * a;
	elseif (t <= 1.7e-87)
		tmp = b / (z * c);
	else
		tmp = ((-4.0 * t) / c) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -9.2e+61], N[(N[(N[(t / c), $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t, 1.7e-87], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * t), $MachinePrecision] / c), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.1999999999999998e61

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 61.2

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

   7. Applied rewrites 61.2%

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

   if -9.1999999999999998e61 < t < 1.6999999999999999e-87

   1. Initial program 85.4%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 42.9%

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

   if 1.6999999999999999e-87 < t

   1. Initial program 75.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 81.7

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

   7. Applied rewrites 81.7%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 54.3

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

   10. Applied rewrites 54.3%

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

Alternative 12: 50.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{t}{c} \cdot -4\right) \cdot a\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-87}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* (/ t c) -4.0) a)))
   (if (<= t -9.2e+61) t_1 (if (<= t 1.7e-87) (/ b (* z c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= 1.7e-87) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t / c) * (-4.0d0)) * a
    if (t <= (-9.2d+61)) then
        tmp = t_1
    else if (t <= 1.7d-87) then
        tmp = b / (z * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t / c) * -4.0) * a;
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= 1.7e-87) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((t / c) * -4.0) * a
	tmp = 0
	if t <= -9.2e+61:
		tmp = t_1
	elif t <= 1.7e-87:
		tmp = b / (z * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t / c) * -4.0) * a)
	tmp = 0.0
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= 1.7e-87)
		tmp = Float64(b / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((t / c) * -4.0) * a;
	tmp = 0.0;
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= 1.7e-87)
		tmp = b / (z * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t / c), $MachinePrecision] * -4.0), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -9.2e+61], t$95$1, If[LessEqual[t, 1.7e-87], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.1999999999999998e61 or 1.6999999999999999e-87 < t

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if -9.1999999999999998e61 < t < 1.6999999999999999e-87

   1. Initial program 85.4%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 42.9%

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

Alternative 13: 47.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -4 \cdot \frac{a \cdot t}{c}\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-87}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -4.0 (/ (* a t) c))))
   (if (<= t -9.2e+61) t_1 (if (<= t 1.7e-87) (/ b (* z c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -4.0 * ((a * t) / c);
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= 1.7e-87) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-4.0d0) * ((a * t) / c)
    if (t <= (-9.2d+61)) then
        tmp = t_1
    else if (t <= 1.7d-87) then
        tmp = b / (z * c)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -4.0 * ((a * t) / c);
	double tmp;
	if (t <= -9.2e+61) {
		tmp = t_1;
	} else if (t <= 1.7e-87) {
		tmp = b / (z * c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = -4.0 * ((a * t) / c)
	tmp = 0
	if t <= -9.2e+61:
		tmp = t_1
	elif t <= 1.7e-87:
		tmp = b / (z * c)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(-4.0 * Float64(Float64(a * t) / c))
	tmp = 0.0
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= 1.7e-87)
		tmp = Float64(b / Float64(z * c));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = -4.0 * ((a * t) / c);
	tmp = 0.0;
	if (t <= -9.2e+61)
		tmp = t_1;
	elseif (t <= 1.7e-87)
		tmp = b / (z * c);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-4.0 * N[(N[(a * t), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.2e+61], t$95$1, If[LessEqual[t, 1.7e-87], N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.1999999999999998e61 or 1.6999999999999999e-87 < t

   1. Initial program 74.5%

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

   2. Taylor expanded in 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 51.2

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

   4. Applied rewrites 51.2%

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

   if -9.1999999999999998e61 < t < 1.6999999999999999e-87

   1. Initial program 85.4%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 42.9%

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

Alternative 14: 35.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{b}{z \cdot c} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 35.0%

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

Reproduce

herbie shell --seed 2025114 
(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)))