FastMath test3

Percentage Accurate: 97.7% → 99.9%

Time: 2.5s

Alternatives: 7

Speedup: 1.6×

• 21.0% 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: 97.7% 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: 99.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double d1, double d2, double d3) {
	return ((d2 + d3) - -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 = ((d2 + d3) - (-3.0d0)) * d1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f64 N/A

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

   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. lift-+.f64 N/A

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

   6. associate-+l+ N/A

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

   7. distribute-lft-out N/A

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

   8. distribute-lft-in N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.9

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\left(d3 + d2\right) + 3\right) \cdot d1} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(\left(d3 + d2\right) + \color{blue}{3 \cdot 1}\right) \cdot d1 \]

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

      \[\leadsto \color{blue}{\left(\left(d3 + d2\right) - \left(\mathsf{neg}\left(3\right)\right) \cdot 1\right)} \cdot d1 \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(d3 + d2\right) - \color{blue}{-3} \cdot 1\right) \cdot d1 \]

   6. metadata-eval N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d2 + d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 2.5e-7) (* (- d2 -3.0) d1) (* d1 (+ d2 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 2.5e-7) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = d1 * (d2 + d3);
	}
	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 <= 2.5d-7) then
        tmp = (d2 - (-3.0d0)) * d1
    else
        tmp = d1 * (d2 + d3)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 2.5e-7) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = d1 * (d2 + d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 2.5e-7:
		tmp = (d2 - -3.0) * d1
	else:
		tmp = d1 * (d2 + d3)
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 2.5e-7)
		tmp = Float64(Float64(d2 - -3.0) * d1);
	else
		tmp = Float64(d1 * Float64(d2 + d3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 2.5e-7)
		tmp = (d2 - -3.0) * d1;
	else
		tmp = d1 * (d2 + d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 2.5e-7], N[(N[(d2 - -3.0), $MachinePrecision] * d1), $MachinePrecision], N[(d1 * N[(d2 + d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 2.49999999999999989e-7

   1. Initial program 98.3%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(3 + d2\right)} \cdot d1 \]
   5. Step-by-step derivation

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(d2 - -3 \cdot 1\right) \cdot d1 \]

      5. metadata-eval N/A

         \[\leadsto \left(d2 - -3\right) \cdot d1 \]

      6. lower--.f64 74.8

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

   6. Applied rewrites 74.8%

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

   if 2.49999999999999989e-7 < d3

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 97.8

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

   3. Applied rewrites 97.8%

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

   4. Taylor expanded in d2 around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{d2}, d1, d3 \cdot d1\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 95.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{d2}, d1, d3 \cdot d1\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \color{blue}{d2 \cdot d1 + d3 \cdot d1} \]

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 98.0

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

   8. Applied rewrites 98.0%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + d3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d3 - -3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)) 2e-227)
   (* (- 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-227) {
		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-227) 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-227) {
		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-227:
		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-227)
		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-227)
		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-227], 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)) < 1.99999999999999989e-227

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(3 + d2\right)} \cdot d1 \]
   5. Step-by-step derivation

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(d2 - -3 \cdot 1\right) \cdot d1 \]

      5. metadata-eval N/A

         \[\leadsto \left(d2 - -3\right) \cdot d1 \]

      6. lower--.f64 64.3

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

   6. Applied rewrites 64.3%

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

   if 1.99999999999999989e-227 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 95.5%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{\left(3 + d3\right)} \cdot d1 \]
   5. Step-by-step derivation

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(d3 - -3 \cdot 1\right) \cdot d1 \]

      5. metadata-eval N/A

         \[\leadsto \left(d3 - -3\right) \cdot d1 \]

      6. lower--.f64 63.9

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

   6. Applied rewrites 63.9%

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

Alternative 4: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 2000:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d3 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 2000.0) (* (- d2 -3.0) d1) (* d3 d1)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 2000.0) {
		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 <= 2000.0d0) 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 <= 2000.0) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = d3 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 2000.0:
		tmp = (d2 - -3.0) * d1
	else:
		tmp = d3 * d1
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 2000.0)
		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 <= 2000.0)
		tmp = (d2 - -3.0) * d1;
	else
		tmp = d3 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 2000.0], N[(N[(d2 - -3.0), $MachinePrecision] * d1), $MachinePrecision], N[(d3 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 2e3

   1. Initial program 98.3%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. distribute-lft-out N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(3 + d2\right)} \cdot d1 \]
   5. Step-by-step derivation

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(d2 - -3 \cdot 1\right) \cdot d1 \]

      5. metadata-eval N/A

         \[\leadsto \left(d2 - -3\right) \cdot d1 \]

      6. lower--.f64 74.7

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

   6. Applied rewrites 74.7%

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

   if 2e3 < d3

   1. Initial program 96.0%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]

   2. Taylor expanded in d3 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

Alternative 5: 44.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-211}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;t\_0 \leq 50000:\\ \;\;\;\;d1 \cdot 3\\ \mathbf{else}:\\ \;\;\;\;d3 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (let* ((t_0 (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3))))
   (if (<= t_0 -1e-211) (* d2 d1) (if (<= t_0 50000.0) (* d1 3.0) (* 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 <= -1e-211) {
		tmp = d2 * d1;
	} else if (t_0 <= 50000.0) {
		tmp = d1 * 3.0;
	} 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 <= (-1d-211)) then
        tmp = d2 * d1
    else if (t_0 <= 50000.0d0) then
        tmp = d1 * 3.0d0
    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 <= -1e-211) {
		tmp = d2 * d1;
	} else if (t_0 <= 50000.0) {
		tmp = d1 * 3.0;
	} 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 <= -1e-211:
		tmp = d2 * d1
	elif t_0 <= 50000.0:
		tmp = d1 * 3.0
	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 <= -1e-211)
		tmp = Float64(d2 * d1);
	elseif (t_0 <= 50000.0)
		tmp = Float64(d1 * 3.0);
	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 <= -1e-211)
		tmp = d2 * d1;
	elseif (t_0 <= 50000.0)
		tmp = d1 * 3.0;
	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, -1e-211], N[(d2 * d1), $MachinePrecision], If[LessEqual[t$95$0, 50000.0], N[(d1 * 3.0), $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)) < -1.00000000000000009e-211

   1. Initial program 99.9%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]

   2. Taylor expanded in d2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.1

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

   4. Applied rewrites 40.1%

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

   if -1.00000000000000009e-211 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < 5e4

   1. Initial program 99.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]

   2. Taylor expanded in d2 around 0

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

      1. *-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in d3 around 0

      \[\leadsto d1 \cdot 3 \]
   6. Step-by-step derivation

   7. Applied rewrites 48.5%

      \[\leadsto d1 \cdot 3 \]

   if 5e4 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 94.0%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]

   2. Taylor expanded in d3 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 44.9

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

   4. Applied rewrites 44.9%

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

Alternative 6: 43.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -170:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d2 -170.0) (* d2 d1) (* d1 3.0)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -170.0) {
		tmp = d2 * d1;
	} else {
		tmp = d1 * 3.0;
	}
	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 <= (-170.0d0)) then
        tmp = d2 * d1
    else
        tmp = d1 * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d2 <= -170.0) {
		tmp = d2 * d1;
	} else {
		tmp = d1 * 3.0;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d2 <= -170.0:
		tmp = d2 * d1
	else:
		tmp = d1 * 3.0
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d2 <= -170.0)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(d1 * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d2 <= -170.0)
		tmp = d2 * d1;
	else
		tmp = d1 * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d2, -170.0], N[(d2 * d1), $MachinePrecision], N[(d1 * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -170

   1. Initial program 95.6%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]

   2. Taylor expanded in d2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

   if -170 < d2

   1. Initial program 98.4%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]

   2. Taylor expanded in d2 around 0

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

      1. *-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in d3 around 0

      \[\leadsto d1 \cdot 3 \]
   6. Step-by-step derivation

   7. Applied rewrites 34.7%

      \[\leadsto d1 \cdot 3 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 26.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ d1 \cdot 3 \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]

2. Taylor expanded in d2 around 0

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

   1. *-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 64.8

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in d3 around 0

   \[\leadsto d1 \cdot 3 \]
6. Step-by-step derivation

7. Applied rewrites 26.9%

   \[\leadsto d1 \cdot 3 \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.6× 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 2025112 
(FPCore (d1 d2 d3)
  :name "FastMath test3"
  :precision binary64

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

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