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

Percentage Accurate: 96.1% → 97.8%

Time: 3.6s

Alternatives: 11

Speedup: 0.5×

• 41.4% 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: 96.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   4. Step-by-step derivation

      1. lower-*.f64 58.8

         \[\leadsto t \cdot \color{blue}{z} + c \cdot i \]

   5. Applied rewrites 58.8%

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{t \cdot z + c \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto t \cdot z + \color{blue}{c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + t \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + t \cdot z \]

      5. lower-fma.f64 58.8

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

      6. +-commutative 58.8

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

      7. *-commutative 58.8

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

      8. +-commutative 58.8

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

      9. *-commutative 58.8

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

      10. *-commutative 58.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 58.8

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

   7. Applied rewrites 58.8%

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

Alternative 2: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(i, c, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* x y) -2e+177)
   (fma i c (* y x))
   (if (<= (* x y) -2e+92)
     (fma b a (* z t))
     (if (<= (* x y) 1e+151) (fma i c (* z t)) (fma b a (* x y))))))

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+177) {
		tmp = fma(i, c, (y * x));
	} else if ((x * y) <= -2e+92) {
		tmp = fma(b, a, (z * t));
	} else if ((x * y) <= 1e+151) {
		tmp = fma(i, c, (z * t));
	} else {
		tmp = fma(b, a, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(x * y) <= -2e+177)
		tmp = fma(i, c, Float64(y * x));
	elseif (Float64(x * y) <= -2e+92)
		tmp = fma(b, a, Float64(z * t));
	elseif (Float64(x * y) <= 1e+151)
		tmp = fma(i, c, Float64(z * t));
	else
		tmp = fma(b, a, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+177], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2e+92], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+151], N[(i * c + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2e177

   1. Initial program 91.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 94.4

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

   7. Applied rewrites 94.4%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 94.4

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

   10. Applied rewrites 94.4%

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

   if -2e177 < (*.f64 x y) < -2.0000000000000001e92

   1. Initial program 84.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.9

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

   8. Applied rewrites 67.9%

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

   if -2.0000000000000001e92 < (*.f64 x y) < 1.00000000000000002e151

   1. Initial program 97.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   4. Step-by-step derivation

      1. lower-*.f64 70.8

         \[\leadsto t \cdot \color{blue}{z} + c \cdot i \]

   5. Applied rewrites 70.8%

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{t \cdot z + c \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto t \cdot z + \color{blue}{c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + t \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + t \cdot z \]

      5. lower-fma.f64 70.8

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

      6. +-commutative 70.8

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

      7. *-commutative 70.8

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

      8. +-commutative 70.8

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

      9. *-commutative 70.8

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

      10. *-commutative 70.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 70.8

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

   7. Applied rewrites 70.8%

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

   if 1.00000000000000002e151 < (*.f64 x y)

   1. Initial program 90.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 85.8

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

   8. Applied rewrites 85.8%

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

Alternative 3: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+217} \lor \neg \left(c \cdot i \leq 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -5e+217) (not (<= (* c i) 1e+95)))
   (fma i c (* z t))
   (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 tmp;
	if (((c * i) <= -5e+217) || !((c * i) <= 1e+95)) {
		tmp = fma(i, c, (z * t));
	} else {
		tmp = fma(b, a, fma(t, z, (y * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -5e+217) || !(Float64(c * i) <= 1e+95))
		tmp = fma(i, c, Float64(z * t));
	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_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -5e+217], N[Not[LessEqual[N[(c * i), $MachinePrecision], 1e+95]], $MachinePrecision]], N[(i * c + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -5.00000000000000041e217 or 1.00000000000000002e95 < (*.f64 c i)

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   4. Step-by-step derivation

      1. lower-*.f64 86.1

         \[\leadsto t \cdot \color{blue}{z} + c \cdot i \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{t \cdot z + c \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto t \cdot z + \color{blue}{c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + t \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + t \cdot z \]

      5. lower-fma.f64 86.1

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

      6. +-commutative 86.1

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

      7. *-commutative 86.1

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

      8. +-commutative 86.1

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

      9. *-commutative 86.1

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

      10. *-commutative 86.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.1

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

   7. Applied rewrites 86.1%

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

   if -5.00000000000000041e217 < (*.f64 c i) < 1.00000000000000002e95

   1. Initial program 97.2%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 91.2

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

   5. Applied rewrites 91.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+217} \lor \neg \left(c \cdot i \leq 10^{+95}\right):\\ \;\;\;\;\mathsf{fma}\left(i, c, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(i, c, z \cdot t\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma t z (* y x))))
   (if (<= (* c i) -5e+217)
     (fma i c (* z t))
     (if (<= (* c i) 2e+86) (fma b a t_1) (fma i c t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(t, z, (y * x));
	double tmp;
	if ((c * i) <= -5e+217) {
		tmp = fma(i, c, (z * t));
	} else if ((c * i) <= 2e+86) {
		tmp = fma(b, a, t_1);
	} else {
		tmp = fma(i, c, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(t, z, Float64(y * x))
	tmp = 0.0
	if (Float64(c * i) <= -5e+217)
		tmp = fma(i, c, Float64(z * t));
	elseif (Float64(c * i) <= 2e+86)
		tmp = fma(b, a, t_1);
	else
		tmp = fma(i, c, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(c * i), $MachinePrecision], -5e+217], N[(i * c + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+86], N[(b * a + t$95$1), $MachinePrecision], N[(i * c + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5.00000000000000041e217

   1. Initial program 86.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   4. Step-by-step derivation

      1. lower-*.f64 93.8

         \[\leadsto t \cdot \color{blue}{z} + c \cdot i \]

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{t \cdot z + c \cdot i} \]

      2. lift-*.f64 N/A

         \[\leadsto t \cdot z + \color{blue}{c \cdot i} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{c \cdot i + t \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{i \cdot c} + t \cdot z \]

      5. lower-fma.f64 93.8

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

      6. +-commutative 93.8

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

      7. *-commutative 93.8

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

      8. +-commutative 93.8

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

      9. *-commutative 93.8

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

      10. *-commutative 93.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if -5.00000000000000041e217 < (*.f64 c i) < 2e86

   1. Initial program 97.1%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 91.1

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

   5. Applied rewrites 91.1%

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

   if 2e86 < (*.f64 c i)

   1. Initial program 95.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

Alternative 5: 43.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;z \cdot t \leq 10^{-105}:\\ \;\;\;\;i \cdot c\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+123}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+93)
   (* t z)
   (if (<= (* z t) 1e-105) (* i c) (if (<= (* z t) 2e+123) (* y x) (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -2e+93) {
		tmp = t * z;
	} else if ((z * t) <= 1e-105) {
		tmp = i * c;
	} else if ((z * t) <= 2e+123) {
		tmp = y * x;
	} else {
		tmp = 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 ((z * t) <= (-2d+93)) then
        tmp = t * z
    else if ((z * t) <= 1d-105) then
        tmp = i * c
    else if ((z * t) <= 2d+123) then
        tmp = y * x
    else
        tmp = 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 ((z * t) <= -2e+93) {
		tmp = t * z;
	} else if ((z * t) <= 1e-105) {
		tmp = i * c;
	} else if ((z * t) <= 2e+123) {
		tmp = y * x;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -2e+93:
		tmp = t * z
	elif (z * t) <= 1e-105:
		tmp = i * c
	elif (z * t) <= 2e+123:
		tmp = y * x
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -2e+93)
		tmp = Float64(t * z);
	elseif (Float64(z * t) <= 1e-105)
		tmp = Float64(i * c);
	elseif (Float64(z * t) <= 2e+123)
		tmp = Float64(y * x);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -2e+93)
		tmp = t * z;
	elseif ((z * t) <= 1e-105)
		tmp = i * c;
	elseif ((z * t) <= 2e+123)
		tmp = y * x;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+93], N[(t * z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e-105], N[(i * c), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+123], N[(y * x), $MachinePrecision], N[(t * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000009e93 or 1.99999999999999996e123 < (*.f64 z t)

   1. Initial program 90.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if -2.00000000000000009e93 < (*.f64 z t) < 9.99999999999999965e-106

   1. Initial program 96.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 45.3

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

   5. Applied rewrites 45.3%

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

   if 9.99999999999999965e-106 < (*.f64 z t) < 1.99999999999999996e123

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 37.1

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

   5. Applied rewrites 37.1%

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

Alternative 6: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93} \lor \neg \left(z \cdot t \leq 3 \cdot 10^{+190}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -2e+93) (not (<= (* z t) 3e+190)))
   (fma b a (* z t))
   (fma 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 (((z * t) <= -2e+93) || !((z * t) <= 3e+190)) {
		tmp = fma(b, a, (z * t));
	} else {
		tmp = fma(i, c, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+93) || !(Float64(z * t) <= 3e+190))
		tmp = fma(b, a, Float64(z * t));
	else
		tmp = fma(i, c, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+93], N[Not[LessEqual[N[(z * t), $MachinePrecision], 3e+190]], $MachinePrecision]], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000009e93 or 2.99999999999999982e190 < (*.f64 z t)

   1. Initial program 90.1%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 89.7

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.3

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

   8. Applied rewrites 80.3%

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

   if -2.00000000000000009e93 < (*.f64 z t) < 2.99999999999999982e190

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.8

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

   10. Applied rewrites 65.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93} \lor \neg \left(z \cdot t \leq 3 \cdot 10^{+190}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93} \lor \neg \left(z \cdot t \leq 3 \cdot 10^{+190}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -2e+93) (not (<= (* z t) 3e+190)))
   (* t z)
   (fma 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 (((z * t) <= -2e+93) || !((z * t) <= 3e+190)) {
		tmp = t * z;
	} else {
		tmp = fma(i, c, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+93) || !(Float64(z * t) <= 3e+190))
		tmp = Float64(t * z);
	else
		tmp = fma(i, c, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+93], N[Not[LessEqual[N[(z * t), $MachinePrecision], 3e+190]], $MachinePrecision]], N[(t * z), $MachinePrecision], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000009e93 or 2.99999999999999982e190 < (*.f64 z t)

   1. Initial program 90.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if -2.00000000000000009e93 < (*.f64 z t) < 2.99999999999999982e190

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.8

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

   10. Applied rewrites 65.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93} \lor \neg \left(z \cdot t \leq 3 \cdot 10^{+190}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(i, c, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+93)
   (fma b a (* z t))
   (if (<= (* z t) 2e+122) (fma i c (* y x)) (fma x y (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -2e+93) {
		tmp = fma(b, a, (z * t));
	} else if ((z * t) <= 2e+122) {
		tmp = fma(i, c, (y * x));
	} else {
		tmp = fma(x, y, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -2e+93)
		tmp = fma(b, a, Float64(z * t));
	elseif (Float64(z * t) <= 2e+122)
		tmp = fma(i, c, Float64(y * x));
	else
		tmp = fma(x, y, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+93], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+122], N[(i * c + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000009e93

   1. Initial program 89.4%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 86.5

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.4

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

   8. Applied rewrites 78.4%

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

   if -2.00000000000000009e93 < (*.f64 z t) < 2.00000000000000003e122

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 73.1

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto i \cdot c + x \cdot y \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 65.9

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

   10. Applied rewrites 65.9%

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

   if 2.00000000000000003e122 < (*.f64 z t)

   1. Initial program 92.9%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      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

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

      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 90.5

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

   5. Applied rewrites 90.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 81.5

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

   10. Applied rewrites 81.5%

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

Alternative 9: 42.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+122}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* z t) -2e+93) (not (<= (* z t) 2e+122))) (* t z) (* i c)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((z * t) <= -2e+93) || !((z * t) <= 2e+122)) {
		tmp = t * z;
	} 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 (((z * t) <= (-2d+93)) .or. (.not. ((z * t) <= 2d+122))) then
        tmp = t * z
    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 (((z * t) <= -2e+93) || !((z * t) <= 2e+122)) {
		tmp = t * z;
	} else {
		tmp = i * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((z * t) <= -2e+93) or not ((z * t) <= 2e+122):
		tmp = t * z
	else:
		tmp = i * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+93) || !(Float64(z * t) <= 2e+122))
		tmp = Float64(t * z);
	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 (((z * t) <= -2e+93) || ~(((z * t) <= 2e+122)))
		tmp = t * z;
	else
		tmp = i * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+93], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+122]], $MachinePrecision]], N[(t * z), $MachinePrecision], N[(i * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000009e93 or 2.00000000000000003e122 < (*.f64 z t)

   1. Initial program 91.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -2.00000000000000009e93 < (*.f64 z t) < 2.00000000000000003e122

   1. Initial program 97.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.1

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

   5. Applied rewrites 40.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+93} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+122}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;i \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -4 \cdot 10^{+129} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+90}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= (* c i) -4e+129) (not (<= (* c i) 2e+90))) (* 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) <= -4e+129) || !((c * i) <= 2e+90)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, 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) <= (-4d+129)) .or. (.not. ((c * i) <= 2d+90))) then
        tmp = i * c
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((c * i) <= -4e+129) || !((c * i) <= 2e+90)) {
		tmp = i * c;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((c * i) <= -4e+129) or not ((c * i) <= 2e+90):
		tmp = i * c
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((Float64(c * i) <= -4e+129) || !(Float64(c * i) <= 2e+90))
		tmp = Float64(i * c);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((c * i) <= -4e+129) || ~(((c * i) <= 2e+90)))
		tmp = i * c;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[N[(c * i), $MachinePrecision], -4e+129], N[Not[LessEqual[N[(c * i), $MachinePrecision], 2e+90]], $MachinePrecision]], N[(i * c), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c i) < -4e129 or 1.99999999999999993e90 < (*.f64 c i)

   1. Initial program 92.1%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -4e129 < (*.f64 c i) < 1.99999999999999993e90

   1. Initial program 97.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.1

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

   5. Applied rewrites 31.1%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -4 \cdot 10^{+129} \lor \neg \left(c \cdot i \leq 2 \cdot 10^{+90}\right):\\ \;\;\;\;i \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.7% accurate, 5.0× 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 95.3%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 23.6

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

5. Applied rewrites 23.6%

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

Reproduce

herbie shell --seed 2025037 
(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)))