Linear.V4:$cdot from linear-1.19.1.3, C

Percentage Accurate: 96.0% → 97.8%

Time: 5.0s

Alternatives: 10

Speedup: 0.5×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i))))
   (if (<= t_1 INFINITY) t_1 (fma b a (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 = (((x * y) + (z * t)) + (a * b)) + (c * i);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(b, a, fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) + Float64(a * b)) + Float64(c * i))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(b, a, 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[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

Alternative 2: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;c \cdot i \leq 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+79)
   (fma i c (* y x))
   (if (<= (* c i) 1e+264) (fma b a (fma t z (* y x))) (fma i c (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+79) {
		tmp = fma(i, c, (y * x));
	} else if ((c * i) <= 1e+264) {
		tmp = fma(b, a, fma(t, z, (y * x)));
	} else {
		tmp = fma(i, c, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+79)
		tmp = fma(i, c, Float64(y * x));
	elseif (Float64(c * i) <= 1e+264)
		tmp = fma(b, a, fma(t, z, Float64(y * x)));
	else
		tmp = fma(i, c, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+79], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 1e+264], N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5e79

   1. Initial program 93.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

      2. lower-*.f64 72.6

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{y \cdot x + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + y \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + y \cdot x \]

      5. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if -5e79 < (*.f64 c i) < 1.00000000000000004e264

   1. Initial program 98.1%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.2

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

   4. Applied rewrites 88.2%

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

   if 1.00000000000000004e264 < (*.f64 c i)

   1. Initial program 84.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

      2. lower-*.f64 86.4

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

   4. Applied rewrites 86.4%

      \[\leadsto \color{blue}{b \cdot a} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot a + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot a + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + b \cdot a} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + b \cdot a \]

      5. lower-fma.f64 88.8

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

   6. Applied rewrites 88.8%

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

Alternative 3: 67.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, b \cdot a\right)\\ t_2 := \mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* b a))) (t_2 (fma i c (* y x))))
   (if (<= (* x y) -2.5e+100)
     t_2
     (if (<= (* x y) -1e-141)
       t_1
       (if (<= (* x y) 2e-312)
         (fma i c (* t z))
         (if (<= (* x y) 5e-13) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, c, (b * a));
	double t_2 = fma(i, c, (y * x));
	double tmp;
	if ((x * y) <= -2.5e+100) {
		tmp = t_2;
	} else if ((x * y) <= -1e-141) {
		tmp = t_1;
	} else if ((x * y) <= 2e-312) {
		tmp = fma(i, c, (t * z));
	} else if ((x * y) <= 5e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(b * a))
	t_2 = fma(i, c, Float64(y * x))
	tmp = 0.0
	if (Float64(x * y) <= -2.5e+100)
		tmp = t_2;
	elseif (Float64(x * y) <= -1e-141)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-312)
		tmp = fma(i, c, Float64(t * z));
	elseif (Float64(x * y) <= 5e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.5e+100], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], -1e-141], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-312], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-13], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.4999999999999999e100 or 4.9999999999999999e-13 < (*.f64 x y)

   1. Initial program 93.8%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

      2. lower-*.f64 68.7

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

   4. Applied rewrites 68.7%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{y \cdot x + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + y \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + y \cdot x \]

      5. lower-fma.f64 69.4

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

   6. Applied rewrites 69.4%

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

   if -2.4999999999999999e100 < (*.f64 x y) < -1e-141 or 2.0000000000019e-312 < (*.f64 x y) < 4.9999999999999999e-13

   1. Initial program 97.8%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

      2. lower-*.f64 61.8

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{b \cdot a} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot a + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot a + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + b \cdot a} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + b \cdot a \]

      5. lower-fma.f64 62.2

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

   6. Applied rewrites 62.2%

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

   if -1e-141 < (*.f64 x y) < 2.0000000000019e-312

   1. Initial program 97.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

      2. lower-*.f64 35.9

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{y \cdot x + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + y \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + y \cdot x \]

      5. lower-fma.f64 35.9

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

   6. Applied rewrites 35.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{t \cdot z}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 68.2

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

   9. Applied rewrites 68.2%

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

Alternative 4: 66.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, b \cdot a\right)\\ t_2 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+119}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* b a))) (t_2 (fma b a (* y x))))
   (if (<= (* x y) -5e+119)
     t_2
     (if (<= (* x y) -1e-141)
       t_1
       (if (<= (* x y) 2e-312)
         (fma i c (* t z))
         (if (<= (* x y) 2e+133) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, c, (b * a));
	double t_2 = fma(b, a, (y * x));
	double tmp;
	if ((x * y) <= -5e+119) {
		tmp = t_2;
	} else if ((x * y) <= -1e-141) {
		tmp = t_1;
	} else if ((x * y) <= 2e-312) {
		tmp = fma(i, c, (t * z));
	} else if ((x * y) <= 2e+133) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(b * a))
	t_2 = fma(b, a, Float64(y * x))
	tmp = 0.0
	if (Float64(x * y) <= -5e+119)
		tmp = t_2;
	elseif (Float64(x * y) <= -1e-141)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-312)
		tmp = fma(i, c, Float64(t * z));
	elseif (Float64(x * y) <= 2e+133)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e+119], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], -1e-141], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-312], N[(i * c + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+133], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.9999999999999999e119 or 2e133 < (*.f64 x y)

   1. Initial program 92.5%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

   if -4.9999999999999999e119 < (*.f64 x y) < -1e-141 or 2.0000000000019e-312 < (*.f64 x y) < 2e133

   1. Initial program 97.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

      2. lower-*.f64 60.1

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

   4. Applied rewrites 60.1%

      \[\leadsto \color{blue}{b \cdot a} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot a + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot a + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + b \cdot a} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + b \cdot a \]

      5. lower-fma.f64 60.5

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

   6. Applied rewrites 60.5%

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

   if -1e-141 < (*.f64 x y) < 2.0000000000019e-312

   1. Initial program 97.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

      2. lower-*.f64 35.9

         \[\leadsto y \cdot \color{blue}{x} + c \cdot i \]

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot x + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{y \cdot x + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + y \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + y \cdot x \]

      5. lower-fma.f64 35.9

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

   6. Applied rewrites 35.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(i, c, \color{blue}{t \cdot z}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 68.2

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

   9. Applied rewrites 68.2%

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

Alternative 5: 66.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, c, b \cdot a\right)\\ t_2 := \mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+119}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma i c (* b a))) (t_2 (fma b a (* y x))))
   (if (<= (* x y) -5e+119)
     t_2
     (if (<= (* x y) -2e-208)
       t_1
       (if (<= (* x y) 2e-312)
         (fma b a (* t z))
         (if (<= (* x y) 2e+133) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(i, c, (b * a));
	double t_2 = fma(b, a, (y * x));
	double tmp;
	if ((x * y) <= -5e+119) {
		tmp = t_2;
	} else if ((x * y) <= -2e-208) {
		tmp = t_1;
	} else if ((x * y) <= 2e-312) {
		tmp = fma(b, a, (t * z));
	} else if ((x * y) <= 2e+133) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(i, c, Float64(b * a))
	t_2 = fma(b, a, Float64(y * x))
	tmp = 0.0
	if (Float64(x * y) <= -5e+119)
		tmp = t_2;
	elseif (Float64(x * y) <= -2e-208)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-312)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(x * y) <= 2e+133)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(i * c + N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e+119], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], -2e-208], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-312], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+133], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.9999999999999999e119 or 2e133 < (*.f64 x y)

   1. Initial program 92.5%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

   if -4.9999999999999999e119 < (*.f64 x y) < -2.0000000000000002e-208 or 2.0000000000019e-312 < (*.f64 x y) < 2e133

   1. Initial program 97.6%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

      2. lower-*.f64 60.3

         \[\leadsto b \cdot \color{blue}{a} + c \cdot i \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{b \cdot a} + c \cdot i \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto b \cdot a + \color{blue}{c \cdot i} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot a + c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + b \cdot a} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + b \cdot a \]

      5. lower-fma.f64 60.7

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

   6. Applied rewrites 60.7%

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

   if -2.0000000000000002e-208 < (*.f64 x y) < 2.0000000000019e-312

   1. Initial program 97.4%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. lift-*.f64 66.9

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

   7. Applied rewrites 66.9%

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

Alternative 6: 63.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+79}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq -5 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+79)
   (* i c)
   (if (<= (* c i) -5e-161)
     (fma b a (* t z))
     (if (<= (* c i) 5e+151) (fma b a (* y x)) (* i c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+79) {
		tmp = i * c;
	} else if ((c * i) <= -5e-161) {
		tmp = fma(b, a, (t * z));
	} else if ((c * i) <= 5e+151) {
		tmp = fma(b, a, (y * x));
	} else {
		tmp = i * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+79)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= -5e-161)
		tmp = fma(b, a, Float64(t * z));
	elseif (Float64(c * i) <= 5e+151)
		tmp = fma(b, a, Float64(y * x));
	else
		tmp = Float64(i * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+79], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], -5e-161], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+151], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(i * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5e79 or 5.0000000000000002e151 < (*.f64 c i)

   1. Initial program 91.5%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{c} \]

      2. lower-*.f64 65.2

         \[\leadsto i \cdot \color{blue}{c} \]

   4. Applied rewrites 65.2%

      \[\leadsto \color{blue}{i \cdot c} \]

   if -5e79 < (*.f64 c i) < -4.9999999999999999e-161

   1. Initial program 98.2%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in x around 0

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

      1. lift-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

   if -4.9999999999999999e-161 < (*.f64 c i) < 5.0000000000000002e151

   1. Initial program 98.1%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

Alternative 7: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+168}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -2e+168)
   (* y x)
   (if (<= (* x y) 4e+118) (fma b a (* t z)) (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -2e+168) {
		tmp = y * x;
	} else if ((x * y) <= 4e+118) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -2e+168)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 4e+118)
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+168], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+118], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.9999999999999999e168 or 3.99999999999999987e118 < (*.f64 x y)

   1. Initial program 92.1%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 66.3

         \[\leadsto y \cdot \color{blue}{x} \]

   4. Applied rewrites 66.3%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -1.9999999999999999e168 < (*.f64 x y) < 3.99999999999999987e118

   1. Initial program 97.6%

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

   2. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in x around 0

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

      1. lift-*.f64 61.3

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

   7. Applied rewrites 61.3%

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

Alternative 8: 43.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq -4 \cdot 10^{-130}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-312}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+142}:\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -2.5e+100)
   (* y x)
   (if (<= (* x y) -4e-130)
     (* b a)
     (if (<= (* x y) 2e-312)
       (* t z)
       (if (<= (* x y) 2e+142) (* i c) (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x * y) <= -2.5e+100) {
		tmp = y * x;
	} else if ((x * y) <= -4e-130) {
		tmp = b * a;
	} else if ((x * y) <= 2e-312) {
		tmp = t * z;
	} else if ((x * y) <= 2e+142) {
		tmp = i * c;
	} else {
		tmp = y * x;
	}
	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) <= (-2.5d+100)) then
        tmp = y * x
    else if ((x * y) <= (-4d-130)) then
        tmp = b * a
    else if ((x * y) <= 2d-312) then
        tmp = t * z
    else if ((x * y) <= 2d+142) then
        tmp = i * c
    else
        tmp = y * x
    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) <= -2.5e+100) {
		tmp = y * x;
	} else if ((x * y) <= -4e-130) {
		tmp = b * a;
	} else if ((x * y) <= 2e-312) {
		tmp = t * z;
	} else if ((x * y) <= 2e+142) {
		tmp = i * c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (x * y) <= -2.5e+100:
		tmp = y * x
	elif (x * y) <= -4e-130:
		tmp = b * a
	elif (x * y) <= 2e-312:
		tmp = t * z
	elif (x * y) <= 2e+142:
		tmp = i * c
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -2.5e+100)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= -4e-130)
		tmp = Float64(b * a);
	elseif (Float64(x * y) <= 2e-312)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= 2e+142)
		tmp = Float64(i * c);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((x * y) <= -2.5e+100)
		tmp = y * x;
	elseif ((x * y) <= -4e-130)
		tmp = b * a;
	elseif ((x * y) <= 2e-312)
		tmp = t * z;
	elseif ((x * y) <= 2e+142)
		tmp = i * c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -2.5e+100], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -4e-130], N[(b * a), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-312], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+142], N[(i * c), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2.4999999999999999e100 or 2.0000000000000001e142 < (*.f64 x y)

   1. Initial program 92.5%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 64.1

         \[\leadsto y \cdot \color{blue}{x} \]

   4. Applied rewrites 64.1%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -2.4999999999999999e100 < (*.f64 x y) < -4.0000000000000003e-130

   1. Initial program 98.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{a} \]

      2. lower-*.f64 30.9

         \[\leadsto b \cdot \color{blue}{a} \]

   4. Applied rewrites 30.9%

      \[\leadsto \color{blue}{b \cdot a} \]

   if -4.0000000000000003e-130 < (*.f64 x y) < 2.0000000000019e-312

   1. Initial program 97.3%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 36.1

         \[\leadsto t \cdot \color{blue}{z} \]

   4. Applied rewrites 36.1%

      \[\leadsto \color{blue}{t \cdot z} \]

   if 2.0000000000019e-312 < (*.f64 x y) < 2.0000000000000001e142

   1. Initial program 97.6%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{c} \]

      2. lower-*.f64 32.6

         \[\leadsto i \cdot \color{blue}{c} \]

   4. Applied rewrites 32.6%

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

Alternative 9: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+79}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;c \cdot i \leq 5 \cdot 10^{+55}:\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+79) (* i c) (if (<= (* c i) 5e+55) (* b a) (* i c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+79) {
		tmp = i * c;
	} else if ((c * i) <= 5e+55) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	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 ((c * i) <= (-5d+79)) then
        tmp = i * c
    else if ((c * i) <= 5d+55) then
        tmp = b * a
    else
        tmp = i * c
    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 ((c * i) <= -5e+79) {
		tmp = i * c;
	} else if ((c * i) <= 5e+55) {
		tmp = b * a;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c * i) <= -5e+79:
		tmp = i * c
	elif (c * i) <= 5e+55:
		tmp = b * a
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+79)
		tmp = Float64(i * c);
	elseif (Float64(c * i) <= 5e+55)
		tmp = Float64(b * a);
	else
		tmp = Float64(i * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c * i) <= -5e+79)
		tmp = i * c;
	elseif ((c * i) <= 5e+55)
		tmp = b * a;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+79], N[(i * c), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 5e+55], N[(b * a), $MachinePrecision], N[(i * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5e79 or 5.00000000000000046e55 < (*.f64 c i)

   1. Initial program 92.6%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{c} \]

      2. lower-*.f64 59.9

         \[\leadsto i \cdot \color{blue}{c} \]

   4. Applied rewrites 59.9%

      \[\leadsto \color{blue}{i \cdot c} \]

   if -5e79 < (*.f64 c i) < 5.00000000000000046e55

   1. Initial program 98.1%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{a} \]

      2. lower-*.f64 33.0

         \[\leadsto b \cdot \color{blue}{a} \]

   4. Applied rewrites 33.0%

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

Alternative 10: 27.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ b \cdot a \end{array} \]

FPCore

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

C

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

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 = b * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

      \[\leadsto b \cdot \color{blue}{a} \]

   2. lower-*.f64 27.1

      \[\leadsto b \cdot \color{blue}{a} \]

4. Applied rewrites 27.1%

   \[\leadsto \color{blue}{b \cdot a} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))