FastMath test3

Percentage Accurate: 97.3% → 100.0%

Time: 3.6s

Alternatives: 7

Speedup: 1.8×

• 21.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = ((d1 * 3.0d0) + (d1 * d2)) + (d1 * d3)
end function

Java

public static double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

Python

def code(d1, d2, d3):
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3)

Julia

function code(d1, d2, d3)
	return Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
end

Wolfram

code[d1_, d2_, d3_] := N[(N[(N[(d1 * 3.0), $MachinePrecision] + N[(d1 * d2), $MachinePrecision]), $MachinePrecision] + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = ((d1 * 3.0d0) + (d1 * d2)) + (d1 * d3)
end function

Java

public static double code(double d1, double d2, double d3) {
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
}

Python

def code(d1, d2, d3):
	return ((d1 * 3.0) + (d1 * d2)) + (d1 * d3)

Julia

function code(d1, d2, d3)
	return Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
end

Wolfram

code[d1_, d2_, d3_] := N[(N[(N[(d1 * 3.0), $MachinePrecision] + N[(d1 * d2), $MachinePrecision]), $MachinePrecision] + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(d1, 3, \left(d3 + d2\right) \cdot d1\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (fma d1 3.0 (* (+ d3 d2) d1)))

C

double code(double d1, double d2, double d3) {
	return fma(d1, 3.0, ((d3 + d2) * d1));
}

Julia

function code(d1, d2, d3)
	return fma(d1, 3.0, Float64(Float64(d3 + d2) * d1))
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * 3.0 + N[(N[(d3 + d2), $MachinePrecision] * d1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right)} + d1 \cdot d3 \]

   3. lift-*.f64 N/A

      \[\leadsto \left(d1 \cdot 3 + \color{blue}{d1 \cdot d2}\right) + d1 \cdot d3 \]

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

      \[\leadsto \color{blue}{\left(d1 \cdot 3 - \left(\mathsf{neg}\left(d1\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

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

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{d1 \cdot 3 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \color{blue}{d1 \cdot 3} + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right) \]

   8. remove-double-neg N/A

      \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1} \cdot d2 + d1 \cdot d3\right) \]

   9. lift-*.f64 N/A

      \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, d1 \cdot d2 + d1 \cdot d3\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, d1 \cdot d2 + \color{blue}{d1 \cdot d3}\right) \]

   13. distribute-lft-out N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot \left(d2 + d3\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

   17. lower-+.f64 100.0

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, \left(d3 + d2\right) \cdot d1\right)} \]
5. Add Preprocessing

Alternative 2: 40.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-135} \lor \neg \left(t\_0 \leq 10^{+228} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+280}\right)\right):\\ \;\;\;\;3 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d3 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (let* ((t_0 (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3))))
   (if (<= t_0 -2e-293)
     (* d2 d1)
     (if (or (<= t_0 2e-135) (not (or (<= t_0 1e+228) (not (<= t_0 2e+280)))))
       (* 3.0 d1)
       (* d3 d1)))))

C

double code(double d1, double d2, double d3) {
	double t_0 = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
	double tmp;
	if (t_0 <= -2e-293) {
		tmp = d2 * d1;
	} else if ((t_0 <= 2e-135) || !((t_0 <= 1e+228) || !(t_0 <= 2e+280))) {
		tmp = 3.0 * d1;
	} else {
		tmp = d3 * d1;
	}
	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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d1 * 3.0d0) + (d1 * d2)) + (d1 * d3)
    if (t_0 <= (-2d-293)) then
        tmp = d2 * d1
    else if ((t_0 <= 2d-135) .or. (.not. (t_0 <= 1d+228) .or. (.not. (t_0 <= 2d+280)))) then
        tmp = 3.0d0 * d1
    else
        tmp = d3 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double t_0 = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
	double tmp;
	if (t_0 <= -2e-293) {
		tmp = d2 * d1;
	} else if ((t_0 <= 2e-135) || !((t_0 <= 1e+228) || !(t_0 <= 2e+280))) {
		tmp = 3.0 * d1;
	} else {
		tmp = d3 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	t_0 = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3)
	tmp = 0
	if t_0 <= -2e-293:
		tmp = d2 * d1
	elif (t_0 <= 2e-135) or not ((t_0 <= 1e+228) or not (t_0 <= 2e+280)):
		tmp = 3.0 * d1
	else:
		tmp = d3 * d1
	return tmp

Julia

function code(d1, d2, d3)
	t_0 = Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3))
	tmp = 0.0
	if (t_0 <= -2e-293)
		tmp = Float64(d2 * d1);
	elseif ((t_0 <= 2e-135) || !((t_0 <= 1e+228) || !(t_0 <= 2e+280)))
		tmp = Float64(3.0 * d1);
	else
		tmp = Float64(d3 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	t_0 = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3);
	tmp = 0.0;
	if (t_0 <= -2e-293)
		tmp = d2 * d1;
	elseif ((t_0 <= 2e-135) || ~(((t_0 <= 1e+228) || ~((t_0 <= 2e+280)))))
		tmp = 3.0 * d1;
	else
		tmp = d3 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := Block[{t$95$0 = N[(N[(N[(d1 * 3.0), $MachinePrecision] + N[(d1 * d2), $MachinePrecision]), $MachinePrecision] + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-293], N[(d2 * d1), $MachinePrecision], If[Or[LessEqual[t$95$0, 2e-135], N[Not[Or[LessEqual[t$95$0, 1e+228], N[Not[LessEqual[t$95$0, 2e+280]], $MachinePrecision]]], $MachinePrecision]], N[(3.0 * d1), $MachinePrecision], N[(d3 * d1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < -2.0000000000000001e-293

   1. Initial program 99.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right)} + d1 \cdot d3 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(d1 \cdot 3 + \color{blue}{d1 \cdot d2}\right) + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 - \left(\mathsf{neg}\left(d1\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{d1 \cdot 3 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{d1 \cdot 3} + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right) \]

      8. remove-double-neg N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1} \cdot d2 + d1 \cdot d3\right) \]

      9. lift-*.f64 N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, d1 \cdot d2 + d1 \cdot d3\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, d1 \cdot d2 + \color{blue}{d1 \cdot d3}\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot \left(d2 + d3\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

      17. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, \left(d3 + d2\right) \cdot d1\right)} \]

   5. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{d2 \cdot d1} \]

      2. lower-*.f64 36.9

         \[\leadsto \color{blue}{d2 \cdot d1} \]

   7. Applied rewrites 36.9%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -2.0000000000000001e-293 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < 2.0000000000000001e-135 or 9.9999999999999992e227 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < 2.0000000000000001e280

   1. Initial program 99.7%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{3 \cdot d1 + d1 \cdot d3} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{d1 \cdot d3 + 3 \cdot d1} \]

      2. *-commutative N/A

         \[\leadsto d1 \cdot d3 + \color{blue}{d1 \cdot 3} \]

      3. distribute-lft-out N/A

         \[\leadsto \color{blue}{d1 \cdot \left(d3 + 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(d3 + 3\right) \cdot d1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(d3 + 3\right) \cdot d1} \]

      6. metadata-eval N/A

         \[\leadsto \left(d3 + \color{blue}{-3 \cdot -1}\right) \cdot d1 \]

      7. metadata-eval N/A

         \[\leadsto \left(d3 + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot -1\right) \cdot d1 \]

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

         \[\leadsto \color{blue}{\left(d3 - 3 \cdot -1\right)} \cdot d1 \]

      9. metadata-eval N/A

         \[\leadsto \left(d3 - \color{blue}{-3}\right) \cdot d1 \]

      10. lower--.f64 64.1

          \[\leadsto \color{blue}{\left(d3 - -3\right)} \cdot d1 \]

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\left(d3 - -3\right) \cdot d1} \]

   6. Taylor expanded in d3 around 0

      \[\leadsto 3 \cdot \color{blue}{d1} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.9%

      \[\leadsto 3 \cdot \color{blue}{d1} \]

   if 2.0000000000000001e-135 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < 9.9999999999999992e227 or 2.0000000000000001e280 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 95.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing

   3. Taylor expanded in d3 around inf

      \[\leadsto \color{blue}{d1 \cdot d3} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{d3 \cdot d1} \]

      2. lower-*.f64 38.2

         \[\leadsto \color{blue}{d3 \cdot d1} \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{d3 \cdot d1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq 2 \cdot 10^{-135} \lor \neg \left(\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq 10^{+228} \lor \neg \left(\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq 2 \cdot 10^{+280}\right)\right):\\ \;\;\;\;3 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d3 \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d3 - -3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)) -2e-293)
   (* (- d2 -3.0) d1)
   (* (- d3 -3.0) d1)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if ((((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -2e-293) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = (d3 - -3.0) * d1;
	}
	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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if ((((d1 * 3.0d0) + (d1 * d2)) + (d1 * d3)) <= (-2d-293)) then
        tmp = (d2 - (-3.0d0)) * d1
    else
        tmp = (d3 - (-3.0d0)) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if ((((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -2e-293) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = (d3 - -3.0) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if (((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -2e-293:
		tmp = (d2 - -3.0) * d1
	else:
		tmp = (d3 - -3.0) * d1
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3)) <= -2e-293)
		tmp = Float64(Float64(d2 - -3.0) * d1);
	else
		tmp = Float64(Float64(d3 - -3.0) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if ((((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -2e-293)
		tmp = (d2 - -3.0) * d1;
	else
		tmp = (d3 - -3.0) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[N[(N[(N[(d1 * 3.0), $MachinePrecision] + N[(d1 * d2), $MachinePrecision]), $MachinePrecision] + N[(d1 * d3), $MachinePrecision]), $MachinePrecision], -2e-293], N[(N[(d2 - -3.0), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d3 - -3.0), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < -2.0000000000000001e-293

   1. Initial program 99.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right)} + d1 \cdot d3 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(d1 \cdot 3 + \color{blue}{d1 \cdot d2}\right) + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 - \left(\mathsf{neg}\left(d1\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{d1 \cdot 3 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{d1 \cdot 3} + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right) \]

      8. remove-double-neg N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1} \cdot d2 + d1 \cdot d3\right) \]

      9. lift-*.f64 N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, d1 \cdot d2 + d1 \cdot d3\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, d1 \cdot d2 + \color{blue}{d1 \cdot d3}\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot \left(d2 + d3\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

      17. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, \left(d3 + d2\right) \cdot d1\right)} \]

   5. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{3 \cdot d1 + d1 \cdot d2} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 3 \cdot d1 + \color{blue}{d2 \cdot d1} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{d1 \cdot \left(3 + d2\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(3 + d2\right) \cdot d1} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 + d2\right) \cdot d1} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(d2 + 3\right)} \cdot d1 \]

      6. metadata-eval N/A

         \[\leadsto \left(d2 + \color{blue}{3 \cdot 1}\right) \cdot d1 \]

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

         \[\leadsto \color{blue}{\left(d2 - \left(\mathsf{neg}\left(3\right)\right) \cdot 1\right)} \cdot d1 \]

      8. metadata-eval N/A

         \[\leadsto \left(d2 - \color{blue}{-3} \cdot 1\right) \cdot d1 \]

      9. metadata-eval N/A

         \[\leadsto \left(d2 - \color{blue}{-3}\right) \cdot d1 \]

      10. lower--.f64 59.9

          \[\leadsto \color{blue}{\left(d2 - -3\right)} \cdot d1 \]

   7. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(d2 - -3\right) \cdot d1} \]

   if -2.0000000000000001e-293 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 97.0%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{3 \cdot d1 + d1 \cdot d3} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{d1 \cdot d3 + 3 \cdot d1} \]

      2. *-commutative N/A

         \[\leadsto d1 \cdot d3 + \color{blue}{d1 \cdot 3} \]

      3. distribute-lft-out N/A

         \[\leadsto \color{blue}{d1 \cdot \left(d3 + 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(d3 + 3\right) \cdot d1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(d3 + 3\right) \cdot d1} \]

      6. metadata-eval N/A

         \[\leadsto \left(d3 + \color{blue}{-3 \cdot -1}\right) \cdot d1 \]

      7. metadata-eval N/A

         \[\leadsto \left(d3 + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot -1\right) \cdot d1 \]

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

         \[\leadsto \color{blue}{\left(d3 - 3 \cdot -1\right)} \cdot d1 \]

      9. metadata-eval N/A

         \[\leadsto \left(d3 - \color{blue}{-3}\right) \cdot d1 \]

      10. lower--.f64 57.1

          \[\leadsto \color{blue}{\left(d3 - -3\right)} \cdot d1 \]

   5. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\left(d3 - -3\right) \cdot d1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d3 - -3\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 6500000000000:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d3 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 6500000000000.0) (* (- d2 -3.0) d1) (* d3 d1)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6500000000000.0) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = d3 * d1;
	}
	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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d3 <= 6500000000000.0d0) then
        tmp = (d2 - (-3.0d0)) * d1
    else
        tmp = d3 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 6500000000000.0) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = d3 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 6500000000000.0:
		tmp = (d2 - -3.0) * d1
	else:
		tmp = d3 * d1
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 6500000000000.0)
		tmp = Float64(Float64(d2 - -3.0) * d1);
	else
		tmp = Float64(d3 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 6500000000000.0)
		tmp = (d2 - -3.0) * d1;
	else
		tmp = d3 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 6500000000000.0], N[(N[(d2 - -3.0), $MachinePrecision] * d1), $MachinePrecision], N[(d3 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 6.5e12

   1. Initial program 99.3%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right)} + d1 \cdot d3 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(d1 \cdot 3 + \color{blue}{d1 \cdot d2}\right) + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 - \left(\mathsf{neg}\left(d1\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{d1 \cdot 3 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{d1 \cdot 3} + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right) \]

      8. remove-double-neg N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1} \cdot d2 + d1 \cdot d3\right) \]

      9. lift-*.f64 N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, d1 \cdot d2 + d1 \cdot d3\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, d1 \cdot d2 + \color{blue}{d1 \cdot d3}\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot \left(d2 + d3\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

      17. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, \left(d3 + d2\right) \cdot d1\right)} \]

   5. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{3 \cdot d1 + d1 \cdot d2} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 3 \cdot d1 + \color{blue}{d2 \cdot d1} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{d1 \cdot \left(3 + d2\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(3 + d2\right) \cdot d1} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(3 + d2\right) \cdot d1} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(d2 + 3\right)} \cdot d1 \]

      6. metadata-eval N/A

         \[\leadsto \left(d2 + \color{blue}{3 \cdot 1}\right) \cdot d1 \]

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

         \[\leadsto \color{blue}{\left(d2 - \left(\mathsf{neg}\left(3\right)\right) \cdot 1\right)} \cdot d1 \]

      8. metadata-eval N/A

         \[\leadsto \left(d2 - \color{blue}{-3} \cdot 1\right) \cdot d1 \]

      9. metadata-eval N/A

         \[\leadsto \left(d2 - \color{blue}{-3}\right) \cdot d1 \]

      10. lower--.f64 79.6

          \[\leadsto \color{blue}{\left(d2 - -3\right)} \cdot d1 \]

   7. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(d2 - -3\right) \cdot d1} \]

   if 6.5e12 < d3

   1. Initial program 95.1%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing

   3. Taylor expanded in d3 around inf

      \[\leadsto \color{blue}{d1 \cdot d3} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{d3 \cdot d1} \]

      2. lower-*.f64 82.2

         \[\leadsto \color{blue}{d3 \cdot d1} \]

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{d3 \cdot d1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq 6500000000000:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d3 \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -3:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;3 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (if (<= d2 -3.0) (* d2 d1) (* 3.0 d1)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -3.0) {
		tmp = d2 * d1;
	} else {
		tmp = 3.0 * d1;
	}
	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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8) :: tmp
    if (d2 <= (-3.0d0)) then
        tmp = d2 * d1
    else
        tmp = 3.0d0 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -3.0) {
		tmp = d2 * d1;
	} else {
		tmp = 3.0 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -3.0:
		tmp = d2 * d1
	else:
		tmp = 3.0 * d1
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -3.0)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(3.0 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -3.0)
		tmp = d2 * d1;
	else
		tmp = 3.0 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -3.0], N[(d2 * d1), $MachinePrecision], N[(3.0 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -3

   1. Initial program 95.6%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right)} + d1 \cdot d3 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(d1 \cdot 3 + \color{blue}{d1 \cdot d2}\right) + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 - \left(\mathsf{neg}\left(d1\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

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

         \[\leadsto \color{blue}{\left(d1 \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{d1 \cdot 3 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{d1 \cdot 3} + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right) \]

      8. remove-double-neg N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1} \cdot d2 + d1 \cdot d3\right) \]

      9. lift-*.f64 N/A

         \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, d1 \cdot d2 + d1 \cdot d3\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, d1 \cdot d2 + \color{blue}{d1 \cdot d3}\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot \left(d2 + d3\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

      17. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, \left(d3 + d2\right) \cdot d1\right)} \]

   5. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{d2 \cdot d1} \]

      2. lower-*.f64 69.2

         \[\leadsto \color{blue}{d2 \cdot d1} \]

   7. Applied rewrites 69.2%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -3 < d2

   1. Initial program 99.3%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{3 \cdot d1 + d1 \cdot d3} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{d1 \cdot d3 + 3 \cdot d1} \]

      2. *-commutative N/A

         \[\leadsto d1 \cdot d3 + \color{blue}{d1 \cdot 3} \]

      3. distribute-lft-out N/A

         \[\leadsto \color{blue}{d1 \cdot \left(d3 + 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(d3 + 3\right) \cdot d1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(d3 + 3\right) \cdot d1} \]

      6. metadata-eval N/A

         \[\leadsto \left(d3 + \color{blue}{-3 \cdot -1}\right) \cdot d1 \]

      7. metadata-eval N/A

         \[\leadsto \left(d3 + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot -1\right) \cdot d1 \]

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

         \[\leadsto \color{blue}{\left(d3 - 3 \cdot -1\right)} \cdot d1 \]

      9. metadata-eval N/A

         \[\leadsto \left(d3 - \color{blue}{-3}\right) \cdot d1 \]

      10. lower--.f64 72.5

          \[\leadsto \color{blue}{\left(d3 - -3\right)} \cdot d1 \]

   5. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\left(d3 - -3\right) \cdot d1} \]

   6. Taylor expanded in d3 around 0

      \[\leadsto 3 \cdot \color{blue}{d1} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.7%

      \[\leadsto 3 \cdot \color{blue}{d1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -3:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;3 \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(d3 + d2\right) + 3\right) \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* (+ (+ d3 d2) 3.0) d1))

C

double code(double d1, double d2, double d3) {
	return ((d3 + d2) + 3.0) * d1;
}

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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = ((d3 + d2) + 3.0d0) * d1
end function

Java

public static double code(double d1, double d2, double d3) {
	return ((d3 + d2) + 3.0) * d1;
}

Python

def code(d1, d2, d3):
	return ((d3 + d2) + 3.0) * d1

Julia

function code(d1, d2, d3)
	return Float64(Float64(Float64(d3 + d2) + 3.0) * d1)
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = ((d3 + d2) + 3.0) * d1;
end

Wolfram

code[d1_, d2_, d3_] := N[(N[(N[(d3 + d2), $MachinePrecision] + 3.0), $MachinePrecision] * d1), $MachinePrecision]

Derivation

1. Initial program 98.3%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right)} + d1 \cdot d3 \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{d1 \cdot 3} + d1 \cdot d2\right) + d1 \cdot d3 \]

   4. lift-*.f64 N/A

      \[\leadsto \left(d1 \cdot 3 + \color{blue}{d1 \cdot d2}\right) + d1 \cdot d3 \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{d1 \cdot \left(3 + d2\right)} + d1 \cdot d3 \]

   6. lift-*.f64 N/A

      \[\leadsto d1 \cdot \left(3 + d2\right) + \color{blue}{d1 \cdot d3} \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{d1 \cdot \left(\left(3 + d2\right) + d3\right)} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 + d2\right) + d3\right) \cdot d1} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(3 + d2\right) + d3\right) \cdot d1} \]

   10. +-commutative N/A

       \[\leadsto \color{blue}{\left(d3 + \left(3 + d2\right)\right)} \cdot d1 \]

   11. +-commutative N/A

       \[\leadsto \left(d3 + \color{blue}{\left(d2 + 3\right)}\right) \cdot d1 \]

   12. associate-+r+ N/A

       \[\leadsto \color{blue}{\left(\left(d3 + d2\right) + 3\right)} \cdot d1 \]

   13. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(d2 + d3\right)} + 3\right) \cdot d1 \]

   14. lower-+.f64 N/A

       \[\leadsto \color{blue}{\left(\left(d2 + d3\right) + 3\right)} \cdot d1 \]

   15. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(d3 + d2\right)} + 3\right) \cdot d1 \]

   16. lower-+.f64 99.9

       \[\leadsto \left(\color{blue}{\left(d3 + d2\right)} + 3\right) \cdot d1 \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\left(d3 + d2\right) + 3\right) \cdot d1} \]
5. Add Preprocessing

Alternative 7: 39.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ d2 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d2 d1))

C

double code(double d1, double d2, double d3) {
	return d2 * d1;
}

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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = d2 * d1
end function

Java

public static double code(double d1, double d2, double d3) {
	return d2 * d1;
}

Python

def code(d1, d2, d3):
	return d2 * d1

Julia

function code(d1, d2, d3)
	return Float64(d2 * d1)
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d2 * d1;
end

Wolfram

code[d1_, d2_, d3_] := N[(d2 * d1), $MachinePrecision]

Derivation

1. Initial program 98.3%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + d1 \cdot d2\right)} + d1 \cdot d3 \]

   3. lift-*.f64 N/A

      \[\leadsto \left(d1 \cdot 3 + \color{blue}{d1 \cdot d2}\right) + d1 \cdot d3 \]

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

      \[\leadsto \color{blue}{\left(d1 \cdot 3 - \left(\mathsf{neg}\left(d1\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

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

      \[\leadsto \color{blue}{\left(d1 \cdot 3 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2\right)} + d1 \cdot d3 \]

   6. associate-+l+ N/A

      \[\leadsto \color{blue}{d1 \cdot 3 + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \color{blue}{d1 \cdot 3} + \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(d1\right)\right)\right)\right) \cdot d2 + d1 \cdot d3\right) \]

   8. remove-double-neg N/A

      \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1} \cdot d2 + d1 \cdot d3\right) \]

   9. lift-*.f64 N/A

      \[\leadsto d1 \cdot 3 + \left(\color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, d1 \cdot d2 + d1 \cdot d3\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot d2} + d1 \cdot d3\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, d1 \cdot d2 + \color{blue}{d1 \cdot d3}\right) \]

   13. distribute-lft-out N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{d1 \cdot \left(d2 + d3\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d2 + d3\right) \cdot d1}\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

   17. lower-+.f64 100.0

       \[\leadsto \mathsf{fma}\left(d1, 3, \color{blue}{\left(d3 + d2\right)} \cdot d1\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 3, \left(d3 + d2\right) \cdot d1\right)} \]

5. Taylor expanded in d2 around inf

   \[\leadsto \color{blue}{d1 \cdot d2} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   2. lower-*.f64 40.8

      \[\leadsto \color{blue}{d2 \cdot d1} \]

7. Applied rewrites 40.8%

   \[\leadsto \color{blue}{d2 \cdot d1} \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(\left(3 + d2\right) + d3\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3) :precision binary64 (* d1 (+ (+ 3.0 d2) d3)))

C

double code(double d1, double d2, double d3) {
	return d1 * ((3.0 + d2) + d3);
}

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(d1, d2, d3)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    code = d1 * ((3.0d0 + d2) + d3)
end function

Java

public static double code(double d1, double d2, double d3) {
	return d1 * ((3.0 + d2) + d3);
}

Python

def code(d1, d2, d3):
	return d1 * ((3.0 + d2) + d3)

Julia

function code(d1, d2, d3)
	return Float64(d1 * Float64(Float64(3.0 + d2) + d3))
end

MATLAB

function tmp = code(d1, d2, d3)
	tmp = d1 * ((3.0 + d2) + d3);
end

Wolfram

code[d1_, d2_, d3_] := N[(d1 * N[(N[(3.0 + d2), $MachinePrecision] + d3), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025015 
(FPCore (d1 d2 d3)
  :name "FastMath test3"
  :precision binary64

  :alt
  (! :herbie-platform default (* d1 (+ 3 d2 d3)))

  (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)))