FastMath test3

Percentage Accurate: 97.8% → 99.9%

Time: 3.2s

Alternatives: 8

Speedup: 1.7×

• 20.8% 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: 97.8% 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, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
2. 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. add-flip N/A

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

   12. lower--.f64 N/A

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

   13. distribute-neg-in N/A

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

   14. sub-flip-reverse N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq -1 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(d2, d1, 3 \cdot d1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, d1, d1 \cdot d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)) -1e-222)
   (fma d2 d1 (* 3.0 d1))
   (fma 3.0 d1 (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if ((((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -1e-222) {
		tmp = fma(d2, d1, (3.0 * d1));
	} else {
		tmp = fma(3.0, d1, (d1 * d3));
	}
	return tmp;
}

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3)) <= -1e-222)
		tmp = fma(d2, d1, Float64(3.0 * d1));
	else
		tmp = fma(3.0, d1, Float64(d1 * d3));
	end
	return 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], -1e-222], N[(d2 * d1 + N[(3.0 * d1), $MachinePrecision]), $MachinePrecision], N[(3.0 * d1 + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < -1.00000000000000005e-222

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. distribute-lft-out N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. distribute-lft-out N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-lft-out N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. add-flip N/A

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

      23. lower--.f64 N/A

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

      24. metadata-eval 98.8

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in d3 around 0

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

   6. Applied rewrites 63.9%

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

   if -1.00000000000000005e-222 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 97.8%

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

   2. Taylor expanded in d2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 63.0

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

   4. Applied rewrites 63.0%

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

Alternative 3: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq -1 \cdot 10^{-222}:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3, d1, d1 \cdot d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)) -1e-222)
   (* (- d2 -3.0) d1)
   (fma 3.0 d1 (* d1 d3))))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if ((((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -1e-222) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = fma(3.0, d1, (d1 * d3));
	}
	return tmp;
}

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3)) <= -1e-222)
		tmp = Float64(Float64(d2 - -3.0) * d1);
	else
		tmp = fma(3.0, d1, Float64(d1 * d3));
	end
	return 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], -1e-222], N[(N[(d2 - -3.0), $MachinePrecision] * d1), $MachinePrecision], N[(3.0 * d1 + N[(d1 * d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < -1.00000000000000005e-222

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. sub-flip-reverse N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

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

      1. lower-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 63.9

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

   8. Applied rewrites 63.9%

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

   if -1.00000000000000005e-222 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 97.8%

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

   2. Taylor expanded in d2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 63.0

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

   4. Applied rewrites 63.0%

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

Alternative 4: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq -1 \cdot 10^{-222}:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d3 - -3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)) -1e-222)
   (* (- 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)) <= -1e-222) {
		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)) <= (-1d-222)) 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)) <= -1e-222) {
		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)) <= -1e-222:
		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)) <= -1e-222)
		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)) <= -1e-222)
		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], -1e-222], 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)) < -1.00000000000000005e-222

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. sub-flip-reverse N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

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

      1. lower-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 63.9

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

   8. Applied rewrites 63.9%

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

   if -1.00000000000000005e-222 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. sub-flip-reverse N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d2 around 0

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

   6. Applied rewrites 63.0%

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

Alternative 5: 63.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq 380:\\ \;\;\;\;\left(d2 - -3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= d3 380.0) (* (- d2 -3.0) d1) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 380.0) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = d1 * 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 <= 380.0d0) then
        tmp = (d2 - (-3.0d0)) * d1
    else
        tmp = d1 * d3
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3) {
	double tmp;
	if (d3 <= 380.0) {
		tmp = (d2 - -3.0) * d1;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if d3 <= 380.0:
		tmp = (d2 - -3.0) * d1
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (d3 <= 380.0)
		tmp = Float64(Float64(d2 - -3.0) * d1);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if (d3 <= 380.0)
		tmp = (d2 - -3.0) * d1;
	else
		tmp = d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_] := If[LessEqual[d3, 380.0], N[(N[(d2 - -3.0), $MachinePrecision] * d1), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < 380

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. sub-flip-reverse N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

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

      1. lower-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 63.9

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

   8. Applied rewrites 63.9%

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

   if 380 < d3

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. distribute-lft-neg-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-commutative N/A

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

      14. add-flip N/A

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

      15. associate--r- N/A

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

      16. distribute-rgt-in N/A

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

      17. *-commutative N/A

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

      18. sub-flip N/A

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

      19. distribute-neg-in N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. distribute-lft-neg-out N/A

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

      22. remove-double-neg N/A

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

      23. *-commutative N/A

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

      24. distribute-rgt-neg-in N/A

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

      25. distribute-lft-neg-in N/A

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

      26. *-commutative N/A

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

      27. lift-*.f64 N/A

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in d3 around 0

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

   6. Applied rewrites 64.0%

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

   7. Taylor expanded in d3 around inf

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

      1. lower-*.f64 39.7

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

   9. Applied rewrites 39.7%

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

Alternative 6: 43.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-222}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;t\_0 \leq 10^{-181}:\\ \;\;\;\;3 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (let* ((t_0 (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3))))
   (if (<= t_0 -1e-222) (* d1 d2) (if (<= t_0 1e-181) (* 3.0 d1) (* d1 d3)))))

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-222) {
		tmp = d1 * d2;
	} else if (t_0 <= 1e-181) {
		tmp = 3.0 * d1;
	} else {
		tmp = d1 * 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) :: t_0
    real(8) :: tmp
    t_0 = ((d1 * 3.0d0) + (d1 * d2)) + (d1 * d3)
    if (t_0 <= (-1d-222)) then
        tmp = d1 * d2
    else if (t_0 <= 1d-181) then
        tmp = 3.0d0 * d1
    else
        tmp = d1 * d3
    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-222) {
		tmp = d1 * d2;
	} else if (t_0 <= 1e-181) {
		tmp = 3.0 * d1;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	t_0 = ((d1 * 3.0) + (d1 * d2)) + (d1 * d3)
	tmp = 0
	if t_0 <= -1e-222:
		tmp = d1 * d2
	elif t_0 <= 1e-181:
		tmp = 3.0 * d1
	else:
		tmp = d1 * d3
	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-222)
		tmp = Float64(d1 * d2);
	elseif (t_0 <= 1e-181)
		tmp = Float64(3.0 * d1);
	else
		tmp = Float64(d1 * d3);
	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-222)
		tmp = d1 * d2;
	elseif (t_0 <= 1e-181)
		tmp = 3.0 * d1;
	else
		tmp = d1 * d3;
	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-222], N[(d1 * d2), $MachinePrecision], If[LessEqual[t$95$0, 1e-181], N[(3.0 * d1), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < -1.00000000000000005e-222

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. sub-flip-reverse N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

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

      1. lower-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 63.9

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

   8. Applied rewrites 63.9%

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

   9. Taylor expanded in d2 around inf

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

       1. lower-*.f64 40.6

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

   11. Applied rewrites 40.6%

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

   if -1.00000000000000005e-222 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < 1.00000000000000005e-181

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. sub-flip-reverse N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

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

      1. lower-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

   7. Taylor expanded in d2 around 0

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

   9. Applied rewrites 25.9%

      \[\leadsto 3 \cdot d1 \]

   if 1.00000000000000005e-181 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. distribute-lft-neg-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-commutative N/A

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

      14. add-flip N/A

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

      15. associate--r- N/A

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

      16. distribute-rgt-in N/A

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

      17. *-commutative N/A

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

      18. sub-flip N/A

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

      19. distribute-neg-in N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. distribute-lft-neg-out N/A

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

      22. remove-double-neg N/A

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

      23. *-commutative N/A

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

      24. distribute-rgt-neg-in N/A

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

      25. distribute-lft-neg-in N/A

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

      26. *-commutative N/A

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

      27. lift-*.f64 N/A

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in d3 around 0

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

   6. Applied rewrites 64.0%

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

   7. Taylor expanded in d3 around inf

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

      1. lower-*.f64 39.7

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

   9. Applied rewrites 39.7%

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

Alternative 7: 40.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \leq -1 \cdot 10^{-222}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d3\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3)
 :precision binary64
 (if (<= (+ (+ (* d1 3.0) (* d1 d2)) (* d1 d3)) -1e-222) (* d1 d2) (* d1 d3)))

C

double code(double d1, double d2, double d3) {
	double tmp;
	if ((((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -1e-222) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * 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 ((((d1 * 3.0d0) + (d1 * d2)) + (d1 * d3)) <= (-1d-222)) then
        tmp = d1 * d2
    else
        tmp = d1 * d3
    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)) <= -1e-222) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * d3;
	}
	return tmp;
}

Python

def code(d1, d2, d3):
	tmp = 0
	if (((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -1e-222:
		tmp = d1 * d2
	else:
		tmp = d1 * d3
	return tmp

Julia

function code(d1, d2, d3)
	tmp = 0.0
	if (Float64(Float64(Float64(d1 * 3.0) + Float64(d1 * d2)) + Float64(d1 * d3)) <= -1e-222)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * d3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3)
	tmp = 0.0;
	if ((((d1 * 3.0) + (d1 * d2)) + (d1 * d3)) <= -1e-222)
		tmp = d1 * d2;
	else
		tmp = d1 * d3;
	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], -1e-222], N[(d1 * d2), $MachinePrecision], N[(d1 * d3), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3)) < -1.00000000000000005e-222

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. sub-flip-reverse N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in d3 around 0

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

      1. lower-+.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 63.9

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

   8. Applied rewrites 63.9%

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

   9. Taylor expanded in d2 around inf

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

       1. lower-*.f64 40.6

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

   11. Applied rewrites 40.6%

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

   if -1.00000000000000005e-222 < (+.f64 (+.f64 (*.f64 d1 #s(literal 3 binary64)) (*.f64 d1 d2)) (*.f64 d1 d3))

   1. Initial program 97.8%

      \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
   2. 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. add-flip N/A

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

      10. sub-negate-rev N/A

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

      11. distribute-lft-neg-out N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-commutative N/A

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

      14. add-flip N/A

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

      15. associate--r- N/A

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

      16. distribute-rgt-in N/A

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

      17. *-commutative N/A

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

      18. sub-flip N/A

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

      19. distribute-neg-in N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. distribute-lft-neg-out N/A

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

      22. remove-double-neg N/A

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

      23. *-commutative N/A

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

      24. distribute-rgt-neg-in N/A

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

      25. distribute-lft-neg-in N/A

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

      26. *-commutative N/A

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

      27. lift-*.f64 N/A

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in d3 around 0

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

   6. Applied rewrites 64.0%

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

   7. Taylor expanded in d3 around inf

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

      1. lower-*.f64 39.7

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

   9. Applied rewrites 39.7%

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

Alternative 8: 40.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

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 * d2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

   \[\left(d1 \cdot 3 + d1 \cdot d2\right) + d1 \cdot d3 \]
2. 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. add-flip N/A

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

   12. lower--.f64 N/A

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

   13. distribute-neg-in N/A

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

   14. sub-flip-reverse N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval 99.9

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in d3 around 0

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

   1. lower-+.f64 63.9

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

6. Applied rewrites 63.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip-rev N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 63.9

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

8. Applied rewrites 63.9%

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

9. Taylor expanded in d2 around inf

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

    1. lower-*.f64 40.6

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

11. Applied rewrites 40.6%

    \[\leadsto \color{blue}{d1 \cdot d2} \]
12. 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 2025143 
(FPCore (d1 d2 d3)
  :name "FastMath test3"
  :precision binary64

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

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