Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A

Percentage Accurate: 90.1% → 96.2%

Time: 6.0s

Alternatives: 12

Speedup: 0.5×

• 47.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot t\\ \mathbf{if}\;2 \cdot \left(t\_1 - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \leq \infty:\\ \;\;\;\;2 \cdot \left(t\_1 - \mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z t))))
   (if (<= (* 2.0 (- t_1 (* (* (+ a (* b c)) c) i))) INFINITY)
     (* 2.0 (- t_1 (* (fma c b a) (* i c))))
     (* (* c (* (* i c) b)) -2.0))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x * y) + (z * t);
	double tmp;
	if ((2.0 * (t_1 - (((a + (b * c)) * c) * i))) <= ((double) INFINITY)) {
		tmp = 2.0 * (t_1 - (fma(c, b, a) * (i * c)));
	} else {
		tmp = (c * ((i * c) * b)) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (Float64(2.0 * Float64(t_1 - Float64(Float64(Float64(a + Float64(b * c)) * c) * i))) <= Inf)
		tmp = Float64(2.0 * Float64(t_1 - Float64(fma(c, b, a) * Float64(i * c))));
	else
		tmp = Float64(Float64(c * Float64(Float64(i * c) * b)) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(t$95$1 - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 * N[(t$95$1 - N[(N[(c * b + a), $MachinePrecision] * N[(i * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right) \]

      3. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{\left(a + b \cdot c\right)} \cdot c\right) \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + \color{blue}{b \cdot c}\right) \cdot c\right) \cdot i\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(b \cdot c + a\right)} \cdot \left(c \cdot i\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{c \cdot b} + a\right) \cdot \left(c \cdot i\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c, b, a\right)} \cdot \left(c \cdot i\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{\left(i \cdot c\right)}\right) \]

      11. lower-*.f64 99.5

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{\left(i \cdot c\right)}\right) \]

   4. Applied rewrites 99.5%

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)}\right) \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 82.0

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      6. lift-*.f64 82.0

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

   7. Applied rewrites 82.0%

      \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      3. associate-*l* N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot i\right) \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot i\right) \cdot b\right)\right) \cdot -2 \]

      9. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      11. lift-*.f64 82.0

          \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

   9. Applied rewrites 82.0%

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

Alternative 2: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+113}:\\ \;\;\;\;2 \cdot \left(y \cdot x - a \cdot \left(i \cdot c\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+237}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* (* (fma c b a) i) c))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -1e+276)
     t_1
     (if (<= t_2 -2e+113)
       (* 2.0 (- (* y x) (* a (* i c))))
       (if (<= t_2 2e+237) (* 2.0 (fma t z (* y x))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -2.0 * ((fma(c, b, a) * i) * c);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -1e+276) {
		tmp = t_1;
	} else if (t_2 <= -2e+113) {
		tmp = 2.0 * ((y * x) - (a * (i * c)));
	} else if (t_2 <= 2e+237) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+276)
		tmp = t_1;
	elseif (t_2 <= -2e+113)
		tmp = Float64(2.0 * Float64(Float64(y * x) - Float64(a * Float64(i * c))));
	elseif (t_2 <= 2e+237)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+276], t$95$1, If[LessEqual[t$95$2, -2e+113], N[(2.0 * N[(N[(y * x), $MachinePrecision] - N[(a * N[(i * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+237], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.0000000000000001e276 or 1.99999999999999988e237 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.3%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 92.9

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -1.0000000000000001e276 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e113

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right) \]

      3. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{\left(a + b \cdot c\right)} \cdot c\right) \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + \color{blue}{b \cdot c}\right) \cdot c\right) \cdot i\right) \]

      5. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(b \cdot c + a\right)} \cdot \left(c \cdot i\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{c \cdot b} + a\right) \cdot \left(c \cdot i\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c, b, a\right)} \cdot \left(c \cdot i\right)\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{\left(i \cdot c\right)}\right) \]

      11. lower-*.f64 99.7

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c, b, a\right) \cdot \color{blue}{\left(i \cdot c\right)}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)}\right) \]

   5. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \left(\color{blue}{x \cdot y} - \mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot \color{blue}{x} - \mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)\right) \]

      2. lower-*.f64 95.6

         \[\leadsto 2 \cdot \left(y \cdot \color{blue}{x} - \mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)\right) \]

   7. Applied rewrites 95.6%

      \[\leadsto 2 \cdot \left(\color{blue}{y \cdot x} - \mathsf{fma}\left(c, b, a\right) \cdot \left(i \cdot c\right)\right) \]

   8. Taylor expanded in a around inf

      \[\leadsto 2 \cdot \left(y \cdot x - \color{blue}{a} \cdot \left(i \cdot c\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 81.9%

       \[\leadsto 2 \cdot \left(y \cdot x - \color{blue}{a} \cdot \left(i \cdot c\right)\right) \]

   if -2e113 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999988e237

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

      3. lower-*.f64 80.7

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

   5. Applied rewrites 80.7%

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

Alternative 3: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+276}:\\ \;\;\;\;\left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 10^{+238}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i \cdot \left(b \cdot c\right)\right) \cdot c\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -1e+276)
     (* (* c (* (* i c) b)) -2.0)
     (if (<= t_1 -2e+113)
       (* (* (* a c) i) -2.0)
       (if (<= t_1 1e+238)
         (* 2.0 (fma t z (* y x)))
         (* (* (* i (* b c)) c) -2.0))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -1e+276) {
		tmp = (c * ((i * c) * b)) * -2.0;
	} else if (t_1 <= -2e+113) {
		tmp = ((a * c) * i) * -2.0;
	} else if (t_1 <= 1e+238) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = ((i * (b * c)) * c) * -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -1e+276)
		tmp = Float64(Float64(c * Float64(Float64(i * c) * b)) * -2.0);
	elseif (t_1 <= -2e+113)
		tmp = Float64(Float64(Float64(a * c) * i) * -2.0);
	elseif (t_1 <= 1e+238)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = Float64(Float64(Float64(i * Float64(b * c)) * c) * -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+276], N[(N[(c * N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -2e+113], N[(N[(N[(a * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+238], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.0000000000000001e276

   1. Initial program 80.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 78.9

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      6. lift-*.f64 78.9

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

   7. Applied rewrites 78.9%

      \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      3. associate-*l* N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot i\right) \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot i\right) \cdot b\right)\right) \cdot -2 \]

      9. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      11. lift-*.f64 78.9

          \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

   9. Applied rewrites 78.9%

      \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

   if -1.0000000000000001e276 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e113

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      6. lower-*.f64 58.9

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\left(i \cdot c\right) \cdot a\right) \cdot -2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot \left(i \cdot c\right)\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a \cdot \left(i \cdot c\right)\right) \cdot -2 \]

      4. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot -2 \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

      7. lower-*.f64 59.0

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

   7. Applied rewrites 59.0%

      \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

   if -2e113 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e238

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

      3. lower-*.f64 80.1

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

   5. Applied rewrites 80.1%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(t, z, y \cdot x\right)} \]

   if 1e238 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 79.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto 2 \cdot \color{blue}{\left(\left(t \cdot z + x \cdot y\right) - b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left({c}^{2} \cdot i\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) + \left(-1 \cdot b\right) \cdot \left(\color{blue}{{c}^{2}} \cdot i\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) + -1 \cdot \color{blue}{\left(b \cdot \left({c}^{2} \cdot i\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \left(-1 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right) + \color{blue}{\left(t \cdot z + x \cdot y\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(-1 \cdot b\right) \cdot \left({c}^{2} \cdot i\right) + \left(\color{blue}{t \cdot z} + x \cdot y\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(b\right)\right) \cdot \left({c}^{2} \cdot i\right) + \left(\color{blue}{t} \cdot z + x \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{neg}\left(b\right), \color{blue}{{c}^{2} \cdot i}, t \cdot z + x \cdot y\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(-b, \color{blue}{{c}^{2}} \cdot i, t \cdot z + x \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(-b, {c}^{2} \cdot \color{blue}{i}, t \cdot z + x \cdot y\right) \]

      10. unpow2 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(-b, \left(c \cdot c\right) \cdot i, t \cdot z + x \cdot y\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(-b, \left(c \cdot c\right) \cdot i, t \cdot z + x \cdot y\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(-b, \left(c \cdot c\right) \cdot i, \mathsf{fma}\left(t, z, x \cdot y\right)\right) \]

      13. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(-b, \left(c \cdot c\right) \cdot i, \mathsf{fma}\left(t, z, y \cdot x\right)\right) \]

      14. lower-*.f64 63.3

          \[\leadsto 2 \cdot \mathsf{fma}\left(-b, \left(c \cdot c\right) \cdot i, \mathsf{fma}\left(t, z, y \cdot x\right)\right) \]

   5. Applied rewrites 63.3%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-b, \left(c \cdot c\right) \cdot i, \mathsf{fma}\left(t, z, y \cdot x\right)\right)} \]

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot -2 \]

      4. *-commutative N/A

         \[\leadsto \left(\left(i \cdot \left(a + c \cdot b\right)\right) \cdot c\right) \cdot -2 \]

      5. +-commutative N/A

         \[\leadsto \left(\left(i \cdot \left(c \cdot b + a\right)\right) \cdot c\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(\left(i \cdot \left(c \cdot b + a\right)\right) \cdot c\right) \cdot -2 \]

      9. +-commutative N/A

         \[\leadsto \left(\left(i \cdot \left(a + c \cdot b\right)\right) \cdot c\right) \cdot -2 \]

      10. *-commutative N/A

          \[\leadsto \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot -2 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot c\right) \cdot -2 \]

      12. *-commutative N/A

          \[\leadsto \left(\left(i \cdot \left(a + c \cdot b\right)\right) \cdot c\right) \cdot -2 \]

      13. +-commutative N/A

          \[\leadsto \left(\left(i \cdot \left(c \cdot b + a\right)\right) \cdot c\right) \cdot -2 \]

      14. lower-fma.f64 93.1

          \[\leadsto \left(\left(i \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot c\right) \cdot -2 \]

   8. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(\left(i \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot c\right) \cdot -2} \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(\left(i \cdot \left(b \cdot c\right)\right) \cdot c\right) \cdot -2 \]
   10. Step-by-step derivation

       1. lower-*.f64 73.9

          \[\leadsto \left(\left(i \cdot \left(b \cdot c\right)\right) \cdot c\right) \cdot -2 \]

   11. Applied rewrites 73.9%

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

Alternative 4: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 10^{+238}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (* (* i c) b)) -2.0)) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -1e+276)
     t_1
     (if (<= t_2 -2e+113)
       (* (* (* a c) i) -2.0)
       (if (<= t_2 1e+238) (* 2.0 (fma t z (* y x))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * ((i * c) * b)) * -2.0;
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -1e+276) {
		tmp = t_1;
	} else if (t_2 <= -2e+113) {
		tmp = ((a * c) * i) * -2.0;
	} else if (t_2 <= 1e+238) {
		tmp = 2.0 * fma(t, z, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(Float64(i * c) * b)) * -2.0)
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+276)
		tmp = t_1;
	elseif (t_2 <= -2e+113)
		tmp = Float64(Float64(Float64(a * c) * i) * -2.0);
	elseif (t_2 <= 1e+238)
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(N[(i * c), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+276], t$95$1, If[LessEqual[t$95$2, -2e+113], N[(N[(N[(a * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 1e+238], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.0000000000000001e276 or 1e238 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 80.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-2 \cdot \left(b \cdot \left({c}^{2} \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \left({c}^{2} \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left({c}^{2} \cdot i\right) \cdot b\right) \cdot -2 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      7. lower-*.f64 73.7

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(c \cdot c\right) \cdot i\right) \cdot b\right) \cdot -2 \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(c \cdot i\right)\right) \cdot b\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      6. lift-*.f64 73.6

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

   7. Applied rewrites 73.6%

      \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(c \cdot \left(i \cdot c\right)\right) \cdot b\right) \cdot -2 \]

      3. associate-*l* N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot i\right) \cdot b\right)\right) \cdot -2 \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \cdot -2 \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(c \cdot i\right) \cdot b\right)\right) \cdot -2 \]

      9. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

      11. lift-*.f64 75.4

          \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

   9. Applied rewrites 75.4%

      \[\leadsto \left(c \cdot \left(\left(i \cdot c\right) \cdot b\right)\right) \cdot -2 \]

   if -1.0000000000000001e276 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e113

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      6. lower-*.f64 58.9

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\left(\left(i \cdot c\right) \cdot a\right) \cdot -2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot \left(i \cdot c\right)\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a \cdot \left(i \cdot c\right)\right) \cdot -2 \]

      4. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot -2 \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

      7. lower-*.f64 59.0

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

   7. Applied rewrites 59.0%

      \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

   if -2e113 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1e238

   1. Initial program 99.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

      3. lower-*.f64 80.1

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

   5. Applied rewrites 80.1%

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

Alternative 5: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-79} \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;2 \cdot \left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 -1e-79) (not (<= t_1 50000.0)))
     (* 2.0 (- (* t z) (* (* (fma c b a) i) c)))
     (* 2.0 (fma t z (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -1e-79) || !(t_1 <= 50000.0)) {
		tmp = 2.0 * ((t * z) - ((fma(c, b, a) * i) * c));
	} else {
		tmp = 2.0 * fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= -1e-79) || !(t_1 <= 50000.0))
		tmp = Float64(2.0 * Float64(Float64(t * z) - Float64(Float64(fma(c, b, a) * i) * c)));
	else
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-79], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(2.0 * N[(N[(t * z), $MachinePrecision] - N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e-79 or 5e4 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      9. lower-fma.f64 82.5

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   5. Applied rewrites 82.5%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -1e-79 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5e4

   1. Initial program 98.7%

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

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

      3. lower-*.f64 94.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

   5. Applied rewrites 94.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{-79} \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 50000\right):\\ \;\;\;\;2 \cdot \left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+105} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+237}\right):\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 -1e+105) (not (<= t_1 2e+237)))
     (* -2.0 (* (* (fma c b a) i) c))
     (* 2.0 (fma t z (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -1e+105) || !(t_1 <= 2e+237)) {
		tmp = -2.0 * ((fma(c, b, a) * i) * c);
	} else {
		tmp = 2.0 * fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= -1e+105) || !(t_1 <= 2e+237))
		tmp = Float64(-2.0 * Float64(Float64(fma(c, b, a) * i) * c));
	else
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+105], N[Not[LessEqual[t$95$1, 2e+237]], $MachinePrecision]], N[(-2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -9.9999999999999994e104 or 1.99999999999999988e237 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 83.8%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. +-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto -2 \cdot \left(\left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      8. lower-fma.f64 85.5

         \[\leadsto -2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   5. Applied rewrites 85.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -9.9999999999999994e104 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999988e237

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

      3. lower-*.f64 81.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

   5. Applied rewrites 81.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -1 \cdot 10^{+105} \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+237}\right):\\ \;\;\;\;-2 \cdot \left(\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+123} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+180}\right):\\ \;\;\;\;2 \cdot \left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ a (* b c)) c)))
   (if (or (<= t_1 -4e+123) (not (<= t_1 2e+180)))
     (* 2.0 (- (* t z) (* (* (fma c b a) i) c)))
     (* 2.0 (- (fma t z (* y x)) (* (* i c) a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a + (b * c)) * c;
	double tmp;
	if ((t_1 <= -4e+123) || !(t_1 <= 2e+180)) {
		tmp = 2.0 * ((t * z) - ((fma(c, b, a) * i) * c));
	} else {
		tmp = 2.0 * (fma(t, z, (y * x)) - ((i * c) * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if ((t_1 <= -4e+123) || !(t_1 <= 2e+180))
		tmp = Float64(2.0 * Float64(Float64(t * z) - Float64(Float64(fma(c, b, a) * i) * c)));
	else
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(y * x)) - Float64(Float64(i * c) * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+123], N[Not[LessEqual[t$95$1, 2e+180]], $MachinePrecision]], N[(2.0 * N[(N[(t * z), $MachinePrecision] - N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -3.99999999999999991e123 or 2e180 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 83.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{c}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(a + b \cdot c\right) \cdot i\right) \cdot c\right) \]

      7. +-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(b \cdot c + a\right) \cdot i\right) \cdot c\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\left(c \cdot b + a\right) \cdot i\right) \cdot c\right) \]

      9. lower-fma.f64 91.0

         \[\leadsto 2 \cdot \left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right) \]

   5. Applied rewrites 91.0%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)} \]

   if -3.99999999999999991e123 < (*.f64 (+.f64 a (*.f64 b c)) c) < 2e180

   1. Initial program 99.2%

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

   3. Taylor expanded in b around 0

      \[\leadsto 2 \cdot \color{blue}{\left(\left(t \cdot z + x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) - \color{blue}{a \cdot \left(c \cdot i\right)}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - \color{blue}{a} \cdot \left(c \cdot i\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - a \cdot \left(c \cdot i\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - a \cdot \left(c \cdot i\right)\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(c \cdot i\right) \cdot \color{blue}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(c \cdot i\right) \cdot \color{blue}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right) \]

      8. lower-*.f64 94.0

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right) \]

   5. Applied rewrites 94.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a + b \cdot c\right) \cdot c \leq -4 \cdot 10^{+123} \lor \neg \left(\left(a + b \cdot c\right) \cdot c \leq 2 \cdot 10^{+180}\right):\\ \;\;\;\;2 \cdot \left(t \cdot z - \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, y \cdot x\right) - \left(i \cdot c\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+113} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+237}\right):\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 -2e+113) (not (<= t_1 2e+237)))
     (* (* (* i c) a) -2.0)
     (* 2.0 (fma t z (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -2e+113) || !(t_1 <= 2e+237)) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = 2.0 * fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= -2e+113) || !(t_1 <= 2e+237))
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	else
		tmp = Float64(2.0 * fma(t, z, Float64(y * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+113], N[Not[LessEqual[t$95$1, 2e+237]], $MachinePrecision]], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e113 or 1.99999999999999988e237 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 83.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      6. lower-*.f64 40.3

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\left(\left(i \cdot c\right) \cdot a\right) \cdot -2} \]

   if -2e113 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.99999999999999988e237

   1. Initial program 99.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

      3. lower-*.f64 80.7

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right) \]

   5. Applied rewrites 80.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+113} \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 2 \cdot 10^{+237}\right):\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+113} \lor \neg \left(t\_1 \leq 10^{+68}\right):\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 -2e+113) (not (<= t_1 1e+68)))
     (* (* (* i c) a) -2.0)
     (* 2.0 (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -2e+113) || !(t_1 <= 1e+68)) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = 2.0 * (t * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((a + (b * c)) * c) * i
    if ((t_1 <= (-2d+113)) .or. (.not. (t_1 <= 1d+68))) then
        tmp = ((i * c) * a) * (-2.0d0)
    else
        tmp = 2.0d0 * (t * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -2e+113) || !(t_1 <= 1e+68)) {
		tmp = ((i * c) * a) * -2.0;
	} else {
		tmp = 2.0 * (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((a + (b * c)) * c) * i
	tmp = 0
	if (t_1 <= -2e+113) or not (t_1 <= 1e+68):
		tmp = ((i * c) * a) * -2.0
	else:
		tmp = 2.0 * (t * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= -2e+113) || !(t_1 <= 1e+68))
		tmp = Float64(Float64(Float64(i * c) * a) * -2.0);
	else
		tmp = Float64(2.0 * Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((a + (b * c)) * c) * i;
	tmp = 0.0;
	if ((t_1 <= -2e+113) || ~((t_1 <= 1e+68)))
		tmp = ((i * c) * a) * -2.0;
	else
		tmp = 2.0 * (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+113], N[Not[LessEqual[t$95$1, 1e+68]], $MachinePrecision]], N[(N[(N[(i * c), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e113 or 9.99999999999999953e67 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      6. lower-*.f64 40.4

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

   5. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\left(\left(i \cdot c\right) \cdot a\right) \cdot -2} \]

   if -2e113 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999953e67

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 53.3

         \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

   5. Applied rewrites 53.3%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+113} \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 10^{+68}\right):\\ \;\;\;\;\left(\left(i \cdot c\right) \cdot a\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+113} \lor \neg \left(t\_1 \leq 0.01\right):\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (or (<= t_1 -2e+113) (not (<= t_1 0.01)))
     (* (* (* a c) i) -2.0)
     (* 2.0 (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -2e+113) || !(t_1 <= 0.01)) {
		tmp = ((a * c) * i) * -2.0;
	} else {
		tmp = 2.0 * (t * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((a + (b * c)) * c) * i
    if ((t_1 <= (-2d+113)) .or. (.not. (t_1 <= 0.01d0))) then
        tmp = ((a * c) * i) * (-2.0d0)
    else
        tmp = 2.0d0 * (t * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if ((t_1 <= -2e+113) || !(t_1 <= 0.01)) {
		tmp = ((a * c) * i) * -2.0;
	} else {
		tmp = 2.0 * (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((a + (b * c)) * c) * i
	tmp = 0
	if (t_1 <= -2e+113) or not (t_1 <= 0.01):
		tmp = ((a * c) * i) * -2.0
	else:
		tmp = 2.0 * (t * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if ((t_1 <= -2e+113) || !(t_1 <= 0.01))
		tmp = Float64(Float64(Float64(a * c) * i) * -2.0);
	else
		tmp = Float64(2.0 * Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((a + (b * c)) * c) * i;
	tmp = 0.0;
	if ((t_1 <= -2e+113) || ~((t_1 <= 0.01)))
		tmp = ((a * c) * i) * -2.0;
	else
		tmp = 2.0 * (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+113], N[Not[LessEqual[t$95$1, 0.01]], $MachinePrecision]], N[(N[(N[(a * c), $MachinePrecision] * i), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2e113 or 0.0100000000000000002 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot \color{blue}{-2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(c \cdot i\right) \cdot a\right) \cdot -2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      6. lower-*.f64 40.0

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\left(\left(i \cdot c\right) \cdot a\right) \cdot -2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(i \cdot c\right) \cdot a\right) \cdot -2 \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot \left(i \cdot c\right)\right) \cdot -2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a \cdot \left(i \cdot c\right)\right) \cdot -2 \]

      4. *-commutative N/A

         \[\leadsto \left(a \cdot \left(c \cdot i\right)\right) \cdot -2 \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

      7. lower-*.f64 38.2

         \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

   7. Applied rewrites 38.2%

      \[\leadsto \left(\left(a \cdot c\right) \cdot i\right) \cdot -2 \]

   if -2e113 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 0.0100000000000000002

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 54.3

         \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

   5. Applied rewrites 54.3%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq -2 \cdot 10^{+113} \lor \neg \left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i \leq 0.01\right):\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot i\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+169} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+201}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* x y) -1e+169) (not (<= (* x y) 5e+201)))
   (* 2.0 (* y x))
   (* 2.0 (* t z))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -1e+169) || !((x * y) <= 5e+201)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = 2.0 * (t * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((x * y) <= (-1d+169)) .or. (.not. ((x * y) <= 5d+201))) then
        tmp = 2.0d0 * (y * x)
    else
        tmp = 2.0d0 * (t * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((x * y) <= -1e+169) || !((x * y) <= 5e+201)) {
		tmp = 2.0 * (y * x);
	} else {
		tmp = 2.0 * (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((x * y) <= -1e+169) or not ((x * y) <= 5e+201):
		tmp = 2.0 * (y * x)
	else:
		tmp = 2.0 * (t * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+169) || !(Float64(x * y) <= 5e+201))
		tmp = Float64(2.0 * Float64(y * x));
	else
		tmp = Float64(2.0 * Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((x * y) <= -1e+169) || ~(((x * y) <= 5e+201)))
		tmp = 2.0 * (y * x);
	else
		tmp = 2.0 * (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+169], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+201]], $MachinePrecision]], N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.99999999999999934e168 or 4.9999999999999995e201 < (*.f64 x y)

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot \color{blue}{x}\right) \]

      2. lower-*.f64 67.5

         \[\leadsto 2 \cdot \left(y \cdot \color{blue}{x}\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot x\right)} \]

   if -9.99999999999999934e168 < (*.f64 x y) < 4.9999999999999995e201

   1. Initial program 92.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 36.4

         \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

   5. Applied rewrites 36.4%

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+169} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+201}\right):\\ \;\;\;\;2 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(t \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* 2.0 (* t z)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (t * z);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (t * z)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (t * z);
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (t * z)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(t * z))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (t * z);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(t * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.2%

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

3. Taylor expanded in z around inf

   \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 30.7

      \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

5. Applied rewrites 30.7%

   \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* 2 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i)))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))