FastMath dist4

Percentage Accurate: 87.3% → 98.3%

Time: 4.2s

Alternatives: 12

Speedup: 1.6×

• 33.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * 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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Initial Program: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * 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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(\left(d4 - d3\right) - d1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)) INFINITY)
   (fma (- d2 d3) d1 (* d1 (- d4 d1)))
   (* d1 (- (- d4 d3) d1))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (((((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)) <= ((double) INFINITY)) {
		tmp = fma((d2 - d3), d1, (d1 * (d4 - d1)));
	} else {
		tmp = d1 * ((d4 - d3) - d1);
	}
	return tmp;
}

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1)) <= Inf)
		tmp = fma(Float64(d2 - d3), d1, Float64(d1 * Float64(d4 - d1)));
	else
		tmp = Float64(d1 * Float64(Float64(d4 - d3) - d1));
	end
	return tmp
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(d2 - d3), $MachinePrecision] * d1 + N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1)) < +inf.0

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{{d1}^{2}} \]

      9. associate--l+ N/A

         \[\leadsto \color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right) + \left(d4 \cdot d1 - {d1}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) + \left(\color{blue}{d1 \cdot d4} - {d1}^{2}\right) \]

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

          \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(d2 - d3\right) \cdot d1} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - {d1}^{2}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{d2 - d3}, d1, d1 \cdot d4 - {d1}^{2}\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - \color{blue}{d1 \cdot d1}\right) \]

      16. distribute-lft-out-- N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 - d1\right)}\right) \]

      17. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 + \left(\mathsf{neg}\left(d1\right)\right)\right)}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 + -1 \cdot d1\right)}\right) \]

      20. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right)\right) \]

      21. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

      22. lower--.f64 97.0

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)} \]

   if +inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1))

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - d1 \cdot \color{blue}{d1} \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - \color{blue}{d1}\right) \]

      7. lower--.f64 76.7

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - d1\right) \]

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d4 - d3\right) - d1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d4 \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;d1 \cdot \left(\left(d2 - d3\right) - d1\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(\left(d4 - d3\right) - d1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 4.2e-19) (* d1 (- (- d2 d3) d1)) (* d1 (- (- d4 d3) d1))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 4.2e-19) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = d1 * ((d4 - d3) - d1);
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 4.2d-19) then
        tmp = d1 * ((d2 - d3) - d1)
    else
        tmp = d1 * ((d4 - d3) - d1)
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 4.2e-19) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = d1 * ((d4 - d3) - d1);
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 4.2e-19:
		tmp = d1 * ((d2 - d3) - d1)
	else:
		tmp = d1 * ((d4 - d3) - d1)
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 4.2e-19)
		tmp = Float64(d1 * Float64(Float64(d2 - d3) - d1));
	else
		tmp = Float64(d1 * Float64(Float64(d4 - d3) - d1));
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 4.2e-19)
		tmp = d1 * ((d2 - d3) - d1);
	else
		tmp = d1 * ((d4 - d3) - d1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 4.2e-19], N[(d1 * N[(N[(d2 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 4.1999999999999998e-19

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   if 4.1999999999999998e-19 < d4

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - d1 \cdot \color{blue}{d1} \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - \color{blue}{d1}\right) \]

      7. lower--.f64 76.7

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - d1\right) \]

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d4 - d3\right) - d1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d4 \leq 1.65 \cdot 10^{+105}:\\ \;\;\;\;d1 \cdot \left(\left(d2 - d3\right) - d1\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d2 - \left(d1 - d4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 1.65e+105) (* d1 (- (- d2 d3) d1)) (* d1 (- d2 (- d1 d4)))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1.65e+105) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = d1 * (d2 - (d1 - d4));
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 1.65d+105) then
        tmp = d1 * ((d2 - d3) - d1)
    else
        tmp = d1 * (d2 - (d1 - d4))
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1.65e+105) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = d1 * (d2 - (d1 - d4));
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 1.65e+105:
		tmp = d1 * ((d2 - d3) - d1)
	else:
		tmp = d1 * (d2 - (d1 - d4))
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 1.65e+105)
		tmp = Float64(d1 * Float64(Float64(d2 - d3) - d1));
	else
		tmp = Float64(d1 * Float64(d2 - Float64(d1 - d4)));
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 1.65e+105)
		tmp = d1 * ((d2 - d3) - d1);
	else
		tmp = d1 * (d2 - (d1 - d4));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 1.65e+105], N[(d1 * N[(N[(d2 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d2 - N[(d1 - d4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 1.64999999999999999e105

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   if 1.64999999999999999e105 < d4

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - {\color{blue}{d1}}^{2} \]

      2. pow2 N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - d1 \cdot \color{blue}{d1} \]

      3. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 + d4\right) - d1\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d1}\right) \]

      6. associate-+r+ N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 + -1 \cdot d1\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 + -1 \cdot d1\right)\right)} \]

      8. add-flip N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

      9. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

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

          \[\leadsto d1 \cdot \left(d2 - \left(\mathsf{neg}\left(\left(d4 - \left(\mathsf{neg}\left(-1\right)\right) \cdot d1\right)\right)\right)\right) \]

      11. sub-negate N/A

          \[\leadsto d1 \cdot \left(d2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot d1 - \color{blue}{d4}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto d1 \cdot \left(d2 - \left(1 \cdot d1 - d4\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - d4\right)\right) \]

      14. lower--.f64 76.8

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - \color{blue}{d4}\right)\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 - \left(d1 - d4\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d3 \leq -1.2 \cdot 10^{+201}:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{elif}\;d3 \leq 4.5 \cdot 10^{+88}:\\ \;\;\;\;d1 \cdot \left(d2 - \left(d1 - d4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d3 -1.2e+201)
   (* d1 (- d2 d3))
   (if (<= d3 4.5e+88) (* d1 (- d2 (- d1 d4))) (* d1 (- d4 d3)))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d3 <= -1.2e+201) {
		tmp = d1 * (d2 - d3);
	} else if (d3 <= 4.5e+88) {
		tmp = d1 * (d2 - (d1 - d4));
	} else {
		tmp = d1 * (d4 - d3);
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d3 <= (-1.2d+201)) then
        tmp = d1 * (d2 - d3)
    else if (d3 <= 4.5d+88) then
        tmp = d1 * (d2 - (d1 - d4))
    else
        tmp = d1 * (d4 - d3)
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d3 <= -1.2e+201) {
		tmp = d1 * (d2 - d3);
	} else if (d3 <= 4.5e+88) {
		tmp = d1 * (d2 - (d1 - d4));
	} else {
		tmp = d1 * (d4 - d3);
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d3 <= -1.2e+201:
		tmp = d1 * (d2 - d3)
	elif d3 <= 4.5e+88:
		tmp = d1 * (d2 - (d1 - d4))
	else:
		tmp = d1 * (d4 - d3)
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d3 <= -1.2e+201)
		tmp = Float64(d1 * Float64(d2 - d3));
	elseif (d3 <= 4.5e+88)
		tmp = Float64(d1 * Float64(d2 - Float64(d1 - d4)));
	else
		tmp = Float64(d1 * Float64(d4 - d3));
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d3 <= -1.2e+201)
		tmp = d1 * (d2 - d3);
	elseif (d3 <= 4.5e+88)
		tmp = d1 * (d2 - (d1 - d4));
	else
		tmp = d1 * (d4 - d3);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d3, -1.2e+201], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], If[LessEqual[d3, 4.5e+88], N[(d1 * N[(d2 - N[(d1 - d4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - d3), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d3 < -1.19999999999999996e201

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in d1 around 0

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

      1. lift--.f64 57.0

         \[\leadsto d1 \cdot \left(d2 - d3\right) \]

   7. Applied rewrites 57.0%

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

   if -1.19999999999999996e201 < d3 < 4.5e88

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - {\color{blue}{d1}}^{2} \]

      2. pow2 N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - d1 \cdot \color{blue}{d1} \]

      3. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 + d4\right) - d1\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d1}\right) \]

      6. associate-+r+ N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 + -1 \cdot d1\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 + -1 \cdot d1\right)\right)} \]

      8. add-flip N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

      9. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

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

          \[\leadsto d1 \cdot \left(d2 - \left(\mathsf{neg}\left(\left(d4 - \left(\mathsf{neg}\left(-1\right)\right) \cdot d1\right)\right)\right)\right) \]

      11. sub-negate N/A

          \[\leadsto d1 \cdot \left(d2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot d1 - \color{blue}{d4}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto d1 \cdot \left(d2 - \left(1 \cdot d1 - d4\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - d4\right)\right) \]

      14. lower--.f64 76.8

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - \color{blue}{d4}\right)\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 - \left(d1 - d4\right)\right)} \]

   if 4.5e88 < d3

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - d1 \cdot \color{blue}{d1} \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - \color{blue}{d1}\right) \]

      7. lower--.f64 76.7

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - d1\right) \]

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d4 - d3\right) - d1\right)} \]

   5. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d3}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 55.9

         \[\leadsto d1 \cdot \left(d4 - d3\right) \]

   7. Applied rewrites 55.9%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d3}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} t_0 := d1 \cdot \left(d4 - d3\right)\\ t_1 := d1 \cdot \left(d2 - d1\right)\\ \mathbf{if}\;d4 \leq 2.1 \cdot 10^{-182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d4 \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{elif}\;d4 \leq 1.46 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d4 \leq 8.2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (* d1 (- d4 d3))) (t_1 (* d1 (- d2 d1))))
   (if (<= d4 2.1e-182)
     t_1
     (if (<= d4 4.2e-19)
       (* d1 (- d2 d3))
       (if (<= d4 1.46e+106) t_0 (if (<= d4 8.2e+148) t_1 t_0))))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d4 - d3);
	double t_1 = d1 * (d2 - d1);
	double tmp;
	if (d4 <= 2.1e-182) {
		tmp = t_1;
	} else if (d4 <= 4.2e-19) {
		tmp = d1 * (d2 - d3);
	} else if (d4 <= 1.46e+106) {
		tmp = t_0;
	} else if (d4 <= 8.2e+148) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d1 * (d4 - d3)
    t_1 = d1 * (d2 - d1)
    if (d4 <= 2.1d-182) then
        tmp = t_1
    else if (d4 <= 4.2d-19) then
        tmp = d1 * (d2 - d3)
    else if (d4 <= 1.46d+106) then
        tmp = t_0
    else if (d4 <= 8.2d+148) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d4 - d3);
	double t_1 = d1 * (d2 - d1);
	double tmp;
	if (d4 <= 2.1e-182) {
		tmp = t_1;
	} else if (d4 <= 4.2e-19) {
		tmp = d1 * (d2 - d3);
	} else if (d4 <= 1.46e+106) {
		tmp = t_0;
	} else if (d4 <= 8.2e+148) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	t_0 = d1 * (d4 - d3)
	t_1 = d1 * (d2 - d1)
	tmp = 0
	if d4 <= 2.1e-182:
		tmp = t_1
	elif d4 <= 4.2e-19:
		tmp = d1 * (d2 - d3)
	elif d4 <= 1.46e+106:
		tmp = t_0
	elif d4 <= 8.2e+148:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	t_0 = Float64(d1 * Float64(d4 - d3))
	t_1 = Float64(d1 * Float64(d2 - d1))
	tmp = 0.0
	if (d4 <= 2.1e-182)
		tmp = t_1;
	elseif (d4 <= 4.2e-19)
		tmp = Float64(d1 * Float64(d2 - d3));
	elseif (d4 <= 1.46e+106)
		tmp = t_0;
	elseif (d4 <= 8.2e+148)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	t_0 = d1 * (d4 - d3);
	t_1 = d1 * (d2 - d1);
	tmp = 0.0;
	if (d4 <= 2.1e-182)
		tmp = t_1;
	elseif (d4 <= 4.2e-19)
		tmp = d1 * (d2 - d3);
	elseif (d4 <= 1.46e+106)
		tmp = t_0;
	elseif (d4 <= 8.2e+148)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(d1 * N[(d4 - d3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d1 * N[(d2 - d1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d4, 2.1e-182], t$95$1, If[LessEqual[d4, 4.2e-19], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], If[LessEqual[d4, 1.46e+106], t$95$0, If[LessEqual[d4, 8.2e+148], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d4 < 2.1e-182 or 1.46000000000000005e106 < d4 < 8.1999999999999996e148

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in d3 around 0

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

      1. lower--.f64 54.6

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

   7. Applied rewrites 54.6%

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

   if 2.1e-182 < d4 < 4.1999999999999998e-19

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in d1 around 0

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

      1. lift--.f64 57.0

         \[\leadsto d1 \cdot \left(d2 - d3\right) \]

   7. Applied rewrites 57.0%

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

   if 4.1999999999999998e-19 < d4 < 1.46000000000000005e106 or 8.1999999999999996e148 < d4

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - d1 \cdot \color{blue}{d1} \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - \color{blue}{d1}\right) \]

      7. lower--.f64 76.7

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - d1\right) \]

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d4 - d3\right) - d1\right)} \]

   5. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d3}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 55.9

         \[\leadsto d1 \cdot \left(d4 - d3\right) \]

   7. Applied rewrites 55.9%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d3}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d2 \leq -2.6 \cdot 10^{+17}:\\ \;\;\;\;d1 \cdot \left(d4 + d2\right)\\ \mathbf{elif}\;d2 \leq -7.7 \cdot 10^{-271}:\\ \;\;\;\;d1 \cdot \left(\left(-d3\right) - d1\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -2.6e+17)
   (* d1 (+ d4 d2))
   (if (<= d2 -7.7e-271) (* d1 (- (- d3) d1)) (* d1 (- d4 d3)))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -2.6e+17) {
		tmp = d1 * (d4 + d2);
	} else if (d2 <= -7.7e-271) {
		tmp = d1 * (-d3 - d1);
	} else {
		tmp = d1 * (d4 - d3);
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-2.6d+17)) then
        tmp = d1 * (d4 + d2)
    else if (d2 <= (-7.7d-271)) then
        tmp = d1 * (-d3 - d1)
    else
        tmp = d1 * (d4 - d3)
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -2.6e+17) {
		tmp = d1 * (d4 + d2);
	} else if (d2 <= -7.7e-271) {
		tmp = d1 * (-d3 - d1);
	} else {
		tmp = d1 * (d4 - d3);
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -2.6e+17:
		tmp = d1 * (d4 + d2)
	elif d2 <= -7.7e-271:
		tmp = d1 * (-d3 - d1)
	else:
		tmp = d1 * (d4 - d3)
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -2.6e+17)
		tmp = Float64(d1 * Float64(d4 + d2));
	elseif (d2 <= -7.7e-271)
		tmp = Float64(d1 * Float64(Float64(-d3) - d1));
	else
		tmp = Float64(d1 * Float64(d4 - d3));
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -2.6e+17)
		tmp = d1 * (d4 + d2);
	elseif (d2 <= -7.7e-271)
		tmp = d1 * (-d3 - d1);
	else
		tmp = d1 * (d4 - d3);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -2.6e+17], N[(d1 * N[(d4 + d2), $MachinePrecision]), $MachinePrecision], If[LessEqual[d2, -7.7e-271], N[(d1 * N[((-d3) - d1), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - d3), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -2.6e17

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - {\color{blue}{d1}}^{2} \]

      2. pow2 N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - d1 \cdot \color{blue}{d1} \]

      3. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 + d4\right) - d1\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d1}\right) \]

      6. associate-+r+ N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 + -1 \cdot d1\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 + -1 \cdot d1\right)\right)} \]

      8. add-flip N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

      9. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

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

          \[\leadsto d1 \cdot \left(d2 - \left(\mathsf{neg}\left(\left(d4 - \left(\mathsf{neg}\left(-1\right)\right) \cdot d1\right)\right)\right)\right) \]

      11. sub-negate N/A

          \[\leadsto d1 \cdot \left(d2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot d1 - \color{blue}{d4}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto d1 \cdot \left(d2 - \left(1 \cdot d1 - d4\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - d4\right)\right) \]

      14. lower--.f64 76.8

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - \color{blue}{d4}\right)\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 - \left(d1 - d4\right)\right)} \]

   5. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \left(d2 + \color{blue}{d4}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto d1 \cdot \left(d4 + d2\right) \]

      2. lower-+.f64 55.8

         \[\leadsto d1 \cdot \left(d4 + d2\right) \]

   7. Applied rewrites 55.8%

      \[\leadsto d1 \cdot \left(d4 + \color{blue}{d2}\right) \]

   if -2.6e17 < d2 < -7.70000000000000007e-271

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in d2 around 0

      \[\leadsto d1 \cdot \left(-1 \cdot d3 - d1\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if -7.70000000000000007e-271 < d2

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around 0

      \[\leadsto \color{blue}{d1 \cdot d4 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d4 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

         \[\leadsto d1 \cdot \left(d4 - d3\right) - d1 \cdot \color{blue}{d1} \]

      4. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d4 - d3\right) - d1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - \color{blue}{d1}\right) \]

      7. lower--.f64 76.7

         \[\leadsto d1 \cdot \left(\left(d4 - d3\right) - d1\right) \]

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d4 - d3\right) - d1\right)} \]

   5. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d3}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 55.9

         \[\leadsto d1 \cdot \left(d4 - d3\right) \]

   7. Applied rewrites 55.9%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d3}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d4 \leq 2.1 \cdot 10^{-182}:\\ \;\;\;\;d1 \cdot \left(d2 - d1\right)\\ \mathbf{elif}\;d4 \leq 1.85 \cdot 10^{+78}:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 2.1e-182)
   (* d1 (- d2 d1))
   (if (<= d4 1.85e+78) (* d1 (- d2 d3)) (* d1 (- d4 d1)))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 2.1e-182) {
		tmp = d1 * (d2 - d1);
	} else if (d4 <= 1.85e+78) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 2.1d-182) then
        tmp = d1 * (d2 - d1)
    else if (d4 <= 1.85d+78) then
        tmp = d1 * (d2 - d3)
    else
        tmp = d1 * (d4 - d1)
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 2.1e-182) {
		tmp = d1 * (d2 - d1);
	} else if (d4 <= 1.85e+78) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 2.1e-182:
		tmp = d1 * (d2 - d1)
	elif d4 <= 1.85e+78:
		tmp = d1 * (d2 - d3)
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 2.1e-182)
		tmp = Float64(d1 * Float64(d2 - d1));
	elseif (d4 <= 1.85e+78)
		tmp = Float64(d1 * Float64(d2 - d3));
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 2.1e-182)
		tmp = d1 * (d2 - d1);
	elseif (d4 <= 1.85e+78)
		tmp = d1 * (d2 - d3);
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 2.1e-182], N[(d1 * N[(d2 - d1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d4, 1.85e+78], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d4 < 2.1e-182

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in d3 around 0

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

      1. lower--.f64 54.6

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

   7. Applied rewrites 54.6%

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

   if 2.1e-182 < d4 < 1.84999999999999992e78

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in d1 around 0

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

      1. lift--.f64 57.0

         \[\leadsto d1 \cdot \left(d2 - d3\right) \]

   7. Applied rewrites 57.0%

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

   if 1.84999999999999992e78 < d4

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - {\color{blue}{d1}}^{2} \]

      2. pow2 N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - d1 \cdot \color{blue}{d1} \]

      3. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 + d4\right) - d1\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d1}\right) \]

      6. associate-+r+ N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 + -1 \cdot d1\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 + -1 \cdot d1\right)\right)} \]

      8. add-flip N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

      9. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

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

          \[\leadsto d1 \cdot \left(d2 - \left(\mathsf{neg}\left(\left(d4 - \left(\mathsf{neg}\left(-1\right)\right) \cdot d1\right)\right)\right)\right) \]

      11. sub-negate N/A

          \[\leadsto d1 \cdot \left(d2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot d1 - \color{blue}{d4}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto d1 \cdot \left(d2 - \left(1 \cdot d1 - d4\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - d4\right)\right) \]

      14. lower--.f64 76.8

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - \color{blue}{d4}\right)\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 - \left(d1 - d4\right)\right)} \]

   5. Taylor expanded in d2 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 53.6

         \[\leadsto d1 \cdot \left(d4 - d1\right) \]

   7. Applied rewrites 53.6%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d4 \leq 1.12 \cdot 10^{-18}:\\ \;\;\;\;d1 \cdot \left(d2 - d1\right)\\ \mathbf{elif}\;d4 \leq 1.85 \cdot 10^{+78}:\\ \;\;\;\;\left(-d1\right) \cdot d3\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 1.12e-18)
   (* d1 (- d2 d1))
   (if (<= d4 1.85e+78) (* (- d1) d3) (* d1 (- d4 d1)))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1.12e-18) {
		tmp = d1 * (d2 - d1);
	} else if (d4 <= 1.85e+78) {
		tmp = -d1 * d3;
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 1.12d-18) then
        tmp = d1 * (d2 - d1)
    else if (d4 <= 1.85d+78) then
        tmp = -d1 * d3
    else
        tmp = d1 * (d4 - d1)
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1.12e-18) {
		tmp = d1 * (d2 - d1);
	} else if (d4 <= 1.85e+78) {
		tmp = -d1 * d3;
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 1.12e-18:
		tmp = d1 * (d2 - d1)
	elif d4 <= 1.85e+78:
		tmp = -d1 * d3
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 1.12e-18)
		tmp = Float64(d1 * Float64(d2 - d1));
	elseif (d4 <= 1.85e+78)
		tmp = Float64(Float64(-d1) * d3);
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 1.12e-18)
		tmp = d1 * (d2 - d1);
	elseif (d4 <= 1.85e+78)
		tmp = -d1 * d3;
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 1.12e-18], N[(d1 * N[(d2 - d1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d4, 1.85e+78], N[((-d1) * d3), $MachinePrecision], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d4 < 1.12000000000000001e-18

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. pow2 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 77.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in d3 around 0

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

      1. lower--.f64 54.6

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

   7. Applied rewrites 54.6%

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

   if 1.12000000000000001e-18 < d4 < 1.84999999999999992e78

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{{d1}^{2}} \]

      9. associate--l+ N/A

         \[\leadsto \color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right) + \left(d4 \cdot d1 - {d1}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) + \left(\color{blue}{d1 \cdot d4} - {d1}^{2}\right) \]

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

          \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(d2 - d3\right) \cdot d1} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - {d1}^{2}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{d2 - d3}, d1, d1 \cdot d4 - {d1}^{2}\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - \color{blue}{d1 \cdot d1}\right) \]

      16. distribute-lft-out-- N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 - d1\right)}\right) \]

      17. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 + \left(\mathsf{neg}\left(d1\right)\right)\right)}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 + -1 \cdot d1\right)}\right) \]

      20. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right)\right) \]

      21. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

      22. lower--.f64 97.0

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)} \]

   4. Taylor expanded in d3 around inf

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 31.3%

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

   if 1.84999999999999992e78 < d4

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - {\color{blue}{d1}}^{2} \]

      2. pow2 N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - d1 \cdot \color{blue}{d1} \]

      3. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 + d4\right) - d1\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d1}\right) \]

      6. associate-+r+ N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 + -1 \cdot d1\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 + -1 \cdot d1\right)\right)} \]

      8. add-flip N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

      9. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

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

          \[\leadsto d1 \cdot \left(d2 - \left(\mathsf{neg}\left(\left(d4 - \left(\mathsf{neg}\left(-1\right)\right) \cdot d1\right)\right)\right)\right) \]

      11. sub-negate N/A

          \[\leadsto d1 \cdot \left(d2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot d1 - \color{blue}{d4}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto d1 \cdot \left(d2 - \left(1 \cdot d1 - d4\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - d4\right)\right) \]

      14. lower--.f64 76.8

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - \color{blue}{d4}\right)\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 - \left(d1 - d4\right)\right)} \]

   5. Taylor expanded in d2 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 53.6

         \[\leadsto d1 \cdot \left(d4 - d1\right) \]

   7. Applied rewrites 53.6%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d2 \leq -4 \cdot 10^{+134}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -4e+134) (* d1 d2) (* d1 (- d4 d1))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -4e+134) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-4d+134)) then
        tmp = d1 * d2
    else
        tmp = d1 * (d4 - d1)
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -4e+134) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -4e+134:
		tmp = d1 * d2
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -4e+134)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -4e+134)
		tmp = d1 * d2;
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -4e+134], N[(d1 * d2), $MachinePrecision], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -3.99999999999999969e134

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{{d1}^{2}} \]

      9. associate--l+ N/A

         \[\leadsto \color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right) + \left(d4 \cdot d1 - {d1}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) + \left(\color{blue}{d1 \cdot d4} - {d1}^{2}\right) \]

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

          \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(d2 - d3\right) \cdot d1} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - {d1}^{2}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{d2 - d3}, d1, d1 \cdot d4 - {d1}^{2}\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - \color{blue}{d1 \cdot d1}\right) \]

      16. distribute-lft-out-- N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 - d1\right)}\right) \]

      17. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 + \left(\mathsf{neg}\left(d1\right)\right)\right)}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 + -1 \cdot d1\right)}\right) \]

      20. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right)\right) \]

      21. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

      22. lower--.f64 97.0

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)} \]

   4. Taylor expanded in d2 around inf

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

   6. Applied rewrites 31.5%

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

   if -3.99999999999999969e134 < d2

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - {\color{blue}{d1}}^{2} \]

      2. pow2 N/A

         \[\leadsto d1 \cdot \left(d2 + d4\right) - d1 \cdot \color{blue}{d1} \]

      3. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 + d4\right) - d1\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto d1 \cdot \left(\left(d2 + d4\right) + -1 \cdot \color{blue}{d1}\right) \]

      6. associate-+r+ N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 + -1 \cdot d1\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 + -1 \cdot d1\right)\right)} \]

      8. add-flip N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

      9. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(d2 - \color{blue}{\left(\mathsf{neg}\left(\left(d4 + -1 \cdot d1\right)\right)\right)}\right) \]

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

          \[\leadsto d1 \cdot \left(d2 - \left(\mathsf{neg}\left(\left(d4 - \left(\mathsf{neg}\left(-1\right)\right) \cdot d1\right)\right)\right)\right) \]

      11. sub-negate N/A

          \[\leadsto d1 \cdot \left(d2 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot d1 - \color{blue}{d4}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto d1 \cdot \left(d2 - \left(1 \cdot d1 - d4\right)\right) \]

      13. *-lft-identity N/A

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - d4\right)\right) \]

      14. lower--.f64 76.8

          \[\leadsto d1 \cdot \left(d2 - \left(d1 - \color{blue}{d4}\right)\right) \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 - \left(d1 - d4\right)\right)} \]

   5. Taylor expanded in d2 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 53.6

         \[\leadsto d1 \cdot \left(d4 - d1\right) \]

   7. Applied rewrites 53.6%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d2 \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq 6.2 \cdot 10^{-306}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(-d1\right) \cdot d3\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -3.3e-21)
   (* d1 d2)
   (if (<= d2 6.2e-306) (* (- d1) d1) (* (- d1) d3))))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -3.3e-21) {
		tmp = d1 * d2;
	} else if (d2 <= 6.2e-306) {
		tmp = -d1 * d1;
	} else {
		tmp = -d1 * d3;
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-3.3d-21)) then
        tmp = d1 * d2
    else if (d2 <= 6.2d-306) then
        tmp = -d1 * d1
    else
        tmp = -d1 * d3
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -3.3e-21) {
		tmp = d1 * d2;
	} else if (d2 <= 6.2e-306) {
		tmp = -d1 * d1;
	} else {
		tmp = -d1 * d3;
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -3.3e-21:
		tmp = d1 * d2
	elif d2 <= 6.2e-306:
		tmp = -d1 * d1
	else:
		tmp = -d1 * d3
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -3.3e-21)
		tmp = Float64(d1 * d2);
	elseif (d2 <= 6.2e-306)
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(Float64(-d1) * d3);
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -3.3e-21)
		tmp = d1 * d2;
	elseif (d2 <= 6.2e-306)
		tmp = -d1 * d1;
	else
		tmp = -d1 * d3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -3.3e-21], N[(d1 * d2), $MachinePrecision], If[LessEqual[d2, 6.2e-306], N[((-d1) * d1), $MachinePrecision], N[((-d1) * d3), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -3.30000000000000009e-21

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{{d1}^{2}} \]

      9. associate--l+ N/A

         \[\leadsto \color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right) + \left(d4 \cdot d1 - {d1}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) + \left(\color{blue}{d1 \cdot d4} - {d1}^{2}\right) \]

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

          \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(d2 - d3\right) \cdot d1} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - {d1}^{2}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{d2 - d3}, d1, d1 \cdot d4 - {d1}^{2}\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - \color{blue}{d1 \cdot d1}\right) \]

      16. distribute-lft-out-- N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 - d1\right)}\right) \]

      17. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 + \left(\mathsf{neg}\left(d1\right)\right)\right)}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 + -1 \cdot d1\right)}\right) \]

      20. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right)\right) \]

      21. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

      22. lower--.f64 97.0

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)} \]

   4. Taylor expanded in d2 around inf

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

   6. Applied rewrites 31.5%

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

   if -3.30000000000000009e-21 < d2 < 6.19999999999999995e-306

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 31.5

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

   4. Applied rewrites 31.5%

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

   if 6.19999999999999995e-306 < d2

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{{d1}^{2}} \]

      9. associate--l+ N/A

         \[\leadsto \color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right) + \left(d4 \cdot d1 - {d1}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) + \left(\color{blue}{d1 \cdot d4} - {d1}^{2}\right) \]

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

          \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(d2 - d3\right) \cdot d1} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - {d1}^{2}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{d2 - d3}, d1, d1 \cdot d4 - {d1}^{2}\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - \color{blue}{d1 \cdot d1}\right) \]

      16. distribute-lft-out-- N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 - d1\right)}\right) \]

      17. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 + \left(\mathsf{neg}\left(d1\right)\right)\right)}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 + -1 \cdot d1\right)}\right) \]

      20. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right)\right) \]

      21. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

      22. lower--.f64 97.0

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)} \]

   4. Taylor expanded in d3 around inf

      \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 31.3%

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

Alternative 11: 46.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ \begin{array}{l} \mathbf{if}\;d2 \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -3.3e-21) (* d1 d2) (* (- d1) d1)))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -3.3e-21) {
		tmp = d1 * d2;
	} else {
		tmp = -d1 * d1;
	}
	return tmp;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-3.3d-21)) then
        tmp = d1 * d2
    else
        tmp = -d1 * d1
    end if
    code = tmp
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -3.3e-21) {
		tmp = d1 * d2;
	} else {
		tmp = -d1 * d1;
	}
	return tmp;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -3.3e-21:
		tmp = d1 * d2
	else:
		tmp = -d1 * d1
	return tmp

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -3.3e-21)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(Float64(-d1) * d1);
	end
	return tmp
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -3.3e-21)
		tmp = d1 * d2;
	else
		tmp = -d1 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -3.3e-21], N[(d1 * d2), $MachinePrecision], N[((-d1) * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -3.30000000000000009e-21

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{{d1}^{2}} \]

      9. associate--l+ N/A

         \[\leadsto \color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right) + \left(d4 \cdot d1 - {d1}^{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) + \left(\color{blue}{d1 \cdot d4} - {d1}^{2}\right) \]

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

          \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \color{blue}{\left(d2 - d3\right) \cdot d1} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - {d1}^{2}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{d2 - d3}, d1, d1 \cdot d4 - {d1}^{2}\right) \]

      15. pow2 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - \color{blue}{d1 \cdot d1}\right) \]

      16. distribute-lft-out-- N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 - d1\right)}\right) \]

      17. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 + \left(\mathsf{neg}\left(d1\right)\right)\right)}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 + -1 \cdot d1\right)}\right) \]

      20. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right)\right) \]

      21. sub-flip-reverse N/A

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

      22. lower--.f64 97.0

          \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

   3. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)} \]

   4. Taylor expanded in d2 around inf

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

   6. Applied rewrites 31.5%

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

   if -3.30000000000000009e-21 < d2

   1. Initial program 87.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 31.5

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

   4. Applied rewrites 31.5%

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

Alternative 12: 31.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} [d1, d2, d3, d4] = \mathsf{sort}([d1, d2, d3, d4])\\ \\ d1 \cdot d2 \end{array} \]

FPCore

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
(FPCore (d1 d2 d3 d4) :precision binary64 (* d1 d2))

C

assert(d1 < d2 && d2 < d3 && d3 < d4);
double code(double d1, double d2, double d3, double d4) {
	return d1 * d2;
}

Fortran

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d1 * d2
end function

Java

assert d1 < d2 && d2 < d3 && d3 < d4;
public static double code(double d1, double d2, double d3, double d4) {
	return d1 * d2;
}

Python

[d1, d2, d3, d4] = sort([d1, d2, d3, d4])
def code(d1, d2, d3, d4):
	return d1 * d2

Julia

d1, d2, d3, d4 = sort([d1, d2, d3, d4])
function code(d1, d2, d3, d4)
	return Float64(d1 * d2)
end

MATLAB

d1, d2, d3, d4 = num2cell(sort([d1, d2, d3, d4])){:}
function tmp = code(d1, d2, d3, d4)
	tmp = d1 * d2;
end

Wolfram

NOTE: d1, d2, d3, and d4 should be sorted in increasing order before calling this function.
code[d1_, d2_, d3_, d4_] := N[(d1 * d2), $MachinePrecision]

Derivation

1. Initial program 87.3%

   \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

   4. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   7. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

   8. pow2 N/A

      \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{{d1}^{2}} \]

   9. associate--l+ N/A

      \[\leadsto \color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right) + \left(d4 \cdot d1 - {d1}^{2}\right)} \]

   10. *-commutative N/A

       \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) + \left(\color{blue}{d1 \cdot d4} - {d1}^{2}\right) \]

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

       \[\leadsto \color{blue}{d1 \cdot \left(d2 - d3\right)} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

   12. *-commutative N/A

       \[\leadsto \color{blue}{\left(d2 - d3\right) \cdot d1} + \left(d1 \cdot d4 - {d1}^{2}\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - {d1}^{2}\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{d2 - d3}, d1, d1 \cdot d4 - {d1}^{2}\right) \]

   15. pow2 N/A

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot d4 - \color{blue}{d1 \cdot d1}\right) \]

   16. distribute-lft-out-- N/A

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 - d1\right)}\right) \]

   17. sub-flip-reverse N/A

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 + \left(\mathsf{neg}\left(d1\right)\right)\right)}\right) \]

   18. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{-1 \cdot d1}\right)\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, \color{blue}{d1 \cdot \left(d4 + -1 \cdot d1\right)}\right) \]

   20. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 + \color{blue}{\left(\mathsf{neg}\left(d1\right)\right)}\right)\right) \]

   21. sub-flip-reverse N/A

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

   22. lower--.f64 97.0

       \[\leadsto \mathsf{fma}\left(d2 - d3, d1, d1 \cdot \color{blue}{\left(d4 - d1\right)}\right) \]

3. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(d2 - d3, d1, d1 \cdot \left(d4 - d1\right)\right)} \]

4. Taylor expanded in d2 around inf

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

6. Applied rewrites 31.5%

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

Developer Target 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(\left(\left(d2 - d3\right) + d4\right) - d1\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4) :precision binary64 (* d1 (- (+ (- d2 d3) d4) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return d1 * (((d2 - d3) + d4) - 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, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d1 * (((d2 - d3) + d4) - d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return d1 * (((d2 - d3) + d4) - d1);
}

Python

def code(d1, d2, d3, d4):
	return d1 * (((d2 - d3) + d4) - d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(d1 * Float64(Float64(Float64(d2 - d3) + d4) - d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = d1 * (((d2 - d3) + d4) - d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(d1 * N[(N[(N[(d2 - d3), $MachinePrecision] + d4), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025132 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  :precision binary64

  :alt
  (! :herbie-platform c (* d1 (- (+ (- d2 d3) d4) d1)))

  (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))