FastMath dist4

Percentage Accurate: 87.4% → 98.4%

Time: 4.9s

Alternatives: 13

Speedup: 1.3×

• 33.6% 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: 87.4% 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: 98.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)) INFINITY)
   (fma d1 (- d4 d1) (* (- d2 d3) d1))
   (* (- (- d4 d3) d1) d1)))

C

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(d1, (d4 - d1), ((d2 - d3) * d1));
	} else {
		tmp = ((d4 - d3) - d1) * d1;
	}
	return tmp;
}

Julia

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(d1, Float64(d4 - d1), Float64(Float64(d2 - d3) * d1));
	else
		tmp = Float64(Float64(Float64(d4 - d3) - d1) * d1);
	end
	return tmp
end

Wolfram

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[(d1 * N[(d4 - d1), $MachinePrecision] + N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision] * d1), $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 100.0%

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

      1. lift--.f64 N/A

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

      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. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 100.0

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

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

   1. Initial program 0.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 99.6%

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

Alternative 2: 39.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq -1.45 \cdot 10^{-110}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d4 \leq 1.65 \cdot 10^{-298}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{elif}\;d4 \leq 10^{+56}:\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 -1.45e-110)
   (* d1 d2)
   (if (<= d4 1.65e-298)
     (* (- d1) d1)
     (if (<= d4 1e+56) (* (- d3) d1) (* d4 d1)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -1.45e-110) {
		tmp = d1 * d2;
	} else if (d4 <= 1.65e-298) {
		tmp = -d1 * d1;
	} else if (d4 <= 1e+56) {
		tmp = -d3 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-1.45d-110)) then
        tmp = d1 * d2
    else if (d4 <= 1.65d-298) then
        tmp = -d1 * d1
    else if (d4 <= 1d+56) then
        tmp = -d3 * d1
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -1.45e-110) {
		tmp = d1 * d2;
	} else if (d4 <= 1.65e-298) {
		tmp = -d1 * d1;
	} else if (d4 <= 1e+56) {
		tmp = -d3 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= -1.45e-110:
		tmp = d1 * d2
	elif d4 <= 1.65e-298:
		tmp = -d1 * d1
	elif d4 <= 1e+56:
		tmp = -d3 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= -1.45e-110)
		tmp = Float64(d1 * d2);
	elseif (d4 <= 1.65e-298)
		tmp = Float64(Float64(-d1) * d1);
	elseif (d4 <= 1e+56)
		tmp = Float64(Float64(-d3) * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= -1.45e-110)
		tmp = d1 * d2;
	elseif (d4 <= 1.65e-298)
		tmp = -d1 * d1;
	elseif (d4 <= 1e+56)
		tmp = -d3 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, -1.45e-110], N[(d1 * d2), $MachinePrecision], If[LessEqual[d4, 1.65e-298], N[((-d1) * d1), $MachinePrecision], If[LessEqual[d4, 1e+56], N[((-d3) * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d4 < -1.4500000000000001e-110

   1. Initial program 84.2%

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

      1. lift--.f64 N/A

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

      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. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in d2 around inf

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

      1. lower-*.f64 32.3

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

   7. Applied rewrites 32.3%

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

   if -1.4500000000000001e-110 < d4 < 1.6500000000000001e-298

   1. Initial program 88.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 49.5

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

   5. Applied rewrites 49.5%

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

   if 1.6500000000000001e-298 < d4 < 1.00000000000000009e56

   1. Initial program 90.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in d3 around inf

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

   8. Applied rewrites 45.6%

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

   if 1.00000000000000009e56 < d4

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      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. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 94.4

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

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2, d1, \mathsf{fma}\left(-d1, d3, d1 \cdot \left(d4 - d1\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-neg.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 94.4

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{d4 \cdot d1} \]

      2. lower-*.f64 61.1

         \[\leadsto \color{blue}{d4 \cdot d1} \]

   9. Applied rewrites 61.1%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d4 \leq -1.45 \cdot 10^{-110}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d4 \leq 1.65 \cdot 10^{-298}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{elif}\;d4 \leq 10^{+56}:\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -4 \cdot 10^{+112}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{elif}\;d1 \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d1\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d1 -4e+112)
   (* (- (- d1) d3) d1)
   (if (<= d1 3e+98) (* (- (+ d4 d2) d3) d1) (* (- d4 d1) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d1 <= -4e+112) {
		tmp = (-d1 - d3) * d1;
	} else if (d1 <= 3e+98) {
		tmp = ((d4 + d2) - d3) * d1;
	} else {
		tmp = (d4 - d1) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 (d1 <= (-4d+112)) then
        tmp = (-d1 - d3) * d1
    else if (d1 <= 3d+98) then
        tmp = ((d4 + d2) - d3) * d1
    else
        tmp = (d4 - d1) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d1 <= -4e+112) {
		tmp = (-d1 - d3) * d1;
	} else if (d1 <= 3e+98) {
		tmp = ((d4 + d2) - d3) * d1;
	} else {
		tmp = (d4 - d1) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d1 <= -4e+112:
		tmp = (-d1 - d3) * d1
	elif d1 <= 3e+98:
		tmp = ((d4 + d2) - d3) * d1
	else:
		tmp = (d4 - d1) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d1 <= -4e+112)
		tmp = Float64(Float64(Float64(-d1) - d3) * d1);
	elseif (d1 <= 3e+98)
		tmp = Float64(Float64(Float64(d4 + d2) - d3) * d1);
	else
		tmp = Float64(Float64(d4 - d1) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d1 <= -4e+112)
		tmp = (-d1 - d3) * d1;
	elseif (d1 <= 3e+98)
		tmp = ((d4 + d2) - d3) * d1;
	else
		tmp = (d4 - d1) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d1, -4e+112], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d1, 3e+98], N[(N[(N[(d4 + d2), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 - d1), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d1 < -3.9999999999999997e112

   1. Initial program 51.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d4 around 0

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

   8. Applied rewrites 92.9%

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

   if -3.9999999999999997e112 < d1 < 3.0000000000000001e98

   1. Initial program 100.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if 3.0000000000000001e98 < d1

   1. Initial program 64.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in d3 around 0

      \[\leadsto \left(d4 - d1\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 81.1%

      \[\leadsto \left(d4 - d1\right) \cdot d1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d1 \leq -4 \cdot 10^{+112}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{elif}\;d1 \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d1\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.25 \cdot 10^{+63}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq -7.8 \cdot 10^{-188}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -1.25e+63)
   (* (- d2 d3) d1)
   (if (<= d2 -7.8e-188) (* (- (- d1) d3) d1) (* (- d4 d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.25e+63) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -7.8e-188) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-1.25d+63)) then
        tmp = (d2 - d3) * d1
    else if (d2 <= (-7.8d-188)) then
        tmp = (-d1 - d3) * d1
    else
        tmp = (d4 - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.25e+63) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -7.8e-188) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -1.25e+63:
		tmp = (d2 - d3) * d1
	elif d2 <= -7.8e-188:
		tmp = (-d1 - d3) * d1
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -1.25e+63)
		tmp = Float64(Float64(d2 - d3) * d1);
	elseif (d2 <= -7.8e-188)
		tmp = Float64(Float64(Float64(-d1) - d3) * d1);
	else
		tmp = Float64(Float64(d4 - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -1.25e+63)
		tmp = (d2 - d3) * d1;
	elseif (d2 <= -7.8e-188)
		tmp = (-d1 - d3) * d1;
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -1.25e+63], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d2, -7.8e-188], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -1.25000000000000003e63

   1. Initial program 81.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in d4 around 0

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

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

   if -1.25000000000000003e63 < d2 < -7.79999999999999954e-188

   1. Initial program 91.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in d4 around 0

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

   8. Applied rewrites 80.0%

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

   if -7.79999999999999954e-188 < d2

   1. Initial program 87.4%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in d1 around 0

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 59.7%

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -1.25 \cdot 10^{+63}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq -7.8 \cdot 10^{-188}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double d1, double d2, double d3, double d4) {
	return fma(d2, d1, fma(-d1, d3, (d1 * (d4 - d1))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

   1. lift--.f64 N/A

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

   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. associate--l+ N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-rgt-out-- N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 97.6

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

4. Applied rewrites 97.6%

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

Alternative 6: 65.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -6.3 \cdot 10^{+69} \lor \neg \left(d1 \leq 1.65 \cdot 10^{+114}\right):\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 + d2\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (or (<= d1 -6.3e+69) (not (<= d1 1.65e+114)))
   (* (- d1) d1)
   (* (+ d4 d2) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d1 <= -6.3e+69) || !(d1 <= 1.65e+114)) {
		tmp = -d1 * d1;
	} else {
		tmp = (d4 + d2) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 ((d1 <= (-6.3d+69)) .or. (.not. (d1 <= 1.65d+114))) then
        tmp = -d1 * d1
    else
        tmp = (d4 + d2) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d1 <= -6.3e+69) || !(d1 <= 1.65e+114)) {
		tmp = -d1 * d1;
	} else {
		tmp = (d4 + d2) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if (d1 <= -6.3e+69) or not (d1 <= 1.65e+114):
		tmp = -d1 * d1
	else:
		tmp = (d4 + d2) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if ((d1 <= -6.3e+69) || !(d1 <= 1.65e+114))
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(Float64(d4 + d2) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if ((d1 <= -6.3e+69) || ~((d1 <= 1.65e+114)))
		tmp = -d1 * d1;
	else
		tmp = (d4 + d2) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[Or[LessEqual[d1, -6.3e+69], N[Not[LessEqual[d1, 1.65e+114]], $MachinePrecision]], N[((-d1) * d1), $MachinePrecision], N[(N[(d4 + d2), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d1 < -6.30000000000000007e69 or 1.65e114 < d1

   1. Initial program 59.8%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -6.30000000000000007e69 < d1 < 1.65e114

   1. Initial program 100.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 93.4

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in d3 around 0

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

   8. Applied rewrites 64.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d1 \leq -6.3 \cdot 10^{+69} \lor \neg \left(d1 \leq 1.65 \cdot 10^{+114}\right):\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 + d2\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq -3.8 \cdot 10^{-140}:\\ \;\;\;\;\left(d4 - d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -9.8e+27)
   (* (- d2 d3) d1)
   (if (<= d2 -3.8e-140) (* (- d4 d1) d1) (* (- d4 d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -9.8e+27) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -3.8e-140) {
		tmp = (d4 - d1) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-9.8d+27)) then
        tmp = (d2 - d3) * d1
    else if (d2 <= (-3.8d-140)) then
        tmp = (d4 - d1) * d1
    else
        tmp = (d4 - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -9.8e+27) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -3.8e-140) {
		tmp = (d4 - d1) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -9.8e+27:
		tmp = (d2 - d3) * d1
	elif d2 <= -3.8e-140:
		tmp = (d4 - d1) * d1
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -9.8e+27)
		tmp = Float64(Float64(d2 - d3) * d1);
	elseif (d2 <= -3.8e-140)
		tmp = Float64(Float64(d4 - d1) * d1);
	else
		tmp = Float64(Float64(d4 - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -9.8e+27)
		tmp = (d2 - d3) * d1;
	elseif (d2 <= -3.8e-140)
		tmp = (d4 - d1) * d1;
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -9.8e+27], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d2, -3.8e-140], N[(N[(d4 - d1), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -9.8000000000000003e27

   1. Initial program 82.1%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in d4 around 0

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 70.2%

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

   if -9.8000000000000003e27 < d2 < -3.79999999999999998e-140

   1. Initial program 91.1%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in d3 around 0

      \[\leadsto \left(d4 - d1\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 65.7%

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

   if -3.79999999999999998e-140 < d2

   1. Initial program 87.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in d1 around 0

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 60.2%

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq -3.8 \cdot 10^{-140}:\\ \;\;\;\;\left(d4 - d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d4 \leq 10^{+56}:\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 -1.05e-63) (* d1 d2) (if (<= d4 1e+56) (* (- d3) d1) (* d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -1.05e-63) {
		tmp = d1 * d2;
	} else if (d4 <= 1e+56) {
		tmp = -d3 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-1.05d-63)) then
        tmp = d1 * d2
    else if (d4 <= 1d+56) then
        tmp = -d3 * d1
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -1.05e-63) {
		tmp = d1 * d2;
	} else if (d4 <= 1e+56) {
		tmp = -d3 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= -1.05e-63:
		tmp = d1 * d2
	elif d4 <= 1e+56:
		tmp = -d3 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= -1.05e-63)
		tmp = Float64(d1 * d2);
	elseif (d4 <= 1e+56)
		tmp = Float64(Float64(-d3) * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= -1.05e-63)
		tmp = d1 * d2;
	elseif (d4 <= 1e+56)
		tmp = -d3 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, -1.05e-63], N[(d1 * d2), $MachinePrecision], If[LessEqual[d4, 1e+56], N[((-d3) * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d4 < -1.05e-63

   1. Initial program 82.1%

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

      1. lift--.f64 N/A

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

      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. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in d2 around inf

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

      1. lower-*.f64 30.4

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

   7. Applied rewrites 30.4%

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

   if -1.05e-63 < d4 < 1.00000000000000009e56

   1. Initial program 90.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in d3 around inf

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

   8. Applied rewrites 41.7%

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

   if 1.00000000000000009e56 < d4

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      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. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 94.4

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

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2, d1, \mathsf{fma}\left(-d1, d3, d1 \cdot \left(d4 - d1\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-neg.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 94.4

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{d4 \cdot d1} \]

      2. lower-*.f64 61.1

         \[\leadsto \color{blue}{d4 \cdot d1} \]

   9. Applied rewrites 61.1%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d4 \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d4 \leq 10^{+56}:\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -5.8 \cdot 10^{+62}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(d4 - d3\right) - d1\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -5.8e+62) (* (- (+ d4 d2) d3) d1) (* (- (- d4 d3) d1) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -5.8e+62) {
		tmp = ((d4 + d2) - d3) * d1;
	} else {
		tmp = ((d4 - d3) - d1) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-5.8d+62)) then
        tmp = ((d4 + d2) - d3) * d1
    else
        tmp = ((d4 - d3) - d1) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -5.8e+62) {
		tmp = ((d4 + d2) - d3) * d1;
	} else {
		tmp = ((d4 - d3) - d1) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -5.8e+62:
		tmp = ((d4 + d2) - d3) * d1
	else:
		tmp = ((d4 - d3) - d1) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -5.8e+62)
		tmp = Float64(Float64(Float64(d4 + d2) - d3) * d1);
	else
		tmp = Float64(Float64(Float64(d4 - d3) - d1) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -5.8e+62)
		tmp = ((d4 + d2) - d3) * d1;
	else
		tmp = ((d4 - d3) - d1) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -5.8e+62], N[(N[(N[(d4 + d2), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -5.79999999999999968e62

   1. Initial program 81.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -5.79999999999999968e62 < d2

   1. Initial program 88.4%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -5.8 \cdot 10^{+62}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(d4 - d3\right) - d1\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d1\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -9.8e+27) (* (- d2 d3) d1) (* (- d4 d1) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -9.8e+27) {
		tmp = (d2 - d3) * d1;
	} else {
		tmp = (d4 - d1) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-9.8d+27)) then
        tmp = (d2 - d3) * d1
    else
        tmp = (d4 - d1) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -9.8e+27) {
		tmp = (d2 - d3) * d1;
	} else {
		tmp = (d4 - d1) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -9.8e+27:
		tmp = (d2 - d3) * d1
	else:
		tmp = (d4 - d1) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -9.8e+27)
		tmp = Float64(Float64(d2 - d3) * d1);
	else
		tmp = Float64(Float64(d4 - d1) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -9.8e+27)
		tmp = (d2 - d3) * d1;
	else
		tmp = (d4 - d1) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -9.8e+27], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 - d1), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -9.8000000000000003e27

   1. Initial program 82.1%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in d4 around 0

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 70.2%

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

   if -9.8000000000000003e27 < d2

   1. Initial program 88.5%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d2 around 0

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

      1. associate--r+ N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in d3 around 0

      \[\leadsto \left(d4 - d1\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 56.8%

      \[\leadsto \left(d4 - d1\right) \cdot d1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d1\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 10^{+56}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 + d2\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 1e+56) (* (- d2 d3) d1) (* (+ d4 d2) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1e+56) {
		tmp = (d2 - d3) * d1;
	} else {
		tmp = (d4 + d2) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= 1d+56) then
        tmp = (d2 - d3) * d1
    else
        tmp = (d4 + d2) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1e+56) {
		tmp = (d2 - d3) * d1;
	} else {
		tmp = (d4 + d2) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 1e+56:
		tmp = (d2 - d3) * d1
	else:
		tmp = (d4 + d2) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 1e+56)
		tmp = Float64(Float64(d2 - d3) * d1);
	else
		tmp = Float64(Float64(d4 + d2) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 1e+56)
		tmp = (d2 - d3) * d1;
	else
		tmp = (d4 + d2) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 1e+56], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 + d2), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 1.00000000000000009e56

   1. Initial program 87.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in d4 around 0

      \[\leadsto \left(d2 - d3\right) \cdot d1 \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

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

   if 1.00000000000000009e56 < d4

   1. Initial program 85.2%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Add Preprocessing

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in d3 around 0

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

   8. Applied rewrites 83.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d4 \leq 10^{+56}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 + d2\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.35 \cdot 10^{+63}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -1.35e+63) (* d1 d2) (* d4 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.35e+63) {
		tmp = d1 * d2;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-1.35d+63)) then
        tmp = d1 * d2
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.35e+63) {
		tmp = d1 * d2;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -1.35e+63:
		tmp = d1 * d2
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -1.35e+63)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -1.35e+63)
		tmp = d1 * d2;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -1.35e+63], N[(d1 * d2), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -1.35000000000000009e63

   1. Initial program 81.6%

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

      1. lift--.f64 N/A

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

      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. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in d2 around inf

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

      1. lower-*.f64 65.3

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

   7. Applied rewrites 65.3%

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

   if -1.35000000000000009e63 < d2

   1. Initial program 88.4%

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

      1. lift--.f64 N/A

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

      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. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-out-- N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 97.1

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

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d2, d1, \mathsf{fma}\left(-d1, d3, d1 \cdot \left(d4 - d1\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-neg.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 96.1

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

   6. Applied rewrites 96.1%

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

   7. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{d4 \cdot d1} \]

      2. lower-*.f64 31.1

         \[\leadsto \color{blue}{d4 \cdot d1} \]

   9. Applied rewrites 31.1%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d2 \leq -1.35 \cdot 10^{+63}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

   1. lift--.f64 N/A

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

   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. associate--l+ N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-rgt-out-- N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 97.6

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in d2 around inf

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

   1. lower-*.f64 35.5

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

7. Applied rewrites 35.5%

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

8. Final simplification 35.5%

   \[\leadsto d1 \cdot d2 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.0× 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 2025017 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  :precision binary64

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

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