FastMath dist4

Percentage Accurate: 87.9% → 100.0%

Time: 3.8s

Alternatives: 11

Speedup: 1.7×

• 33.9% 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: 87.9% 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: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(-1, d1, d4\right) + d2\right) - d3\right) \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (* (- (+ (fma -1.0 d1 d4) d2) d3) d1))

C

double code(double d1, double d2, double d3, double d4) {
	return ((fma(-1.0, d1, d4) + d2) - d3) * d1;
}

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(fma(-1.0, d1, d4) + d2) - d3) * d1)
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(-1.0 * d1 + d4), $MachinePrecision] + d2), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in d1 around 0

   \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

      \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

   6. +-commutative N/A

      \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

   7. lower-fma.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 89.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -2.9 \cdot 10^{+121} \lor \neg \left(d1 \leq 2.15 \cdot 10^{+105}\right):\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(d2 - d3, d1, d4 \cdot d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (or (<= d1 -2.9e+121) (not (<= d1 2.15e+105)))
   (* (- (- d1) d3) d1)
   (fma (- d2 d3) d1 (* d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d1 <= -2.9e+121) || !(d1 <= 2.15e+105)) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = fma((d2 - d3), d1, (d4 * d1));
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if ((d1 <= -2.9e+121) || !(d1 <= 2.15e+105))
		tmp = Float64(Float64(Float64(-d1) - d3) * d1);
	else
		tmp = fma(Float64(d2 - d3), d1, Float64(d4 * d1));
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[Or[LessEqual[d1, -2.9e+121], N[Not[LessEqual[d1, 2.15e+105]], $MachinePrecision]], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d2 - d3), $MachinePrecision] * d1 + N[(d4 * d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d1 < -2.8999999999999999e121 or 2.1500000000000001e105 < d1

   1. Initial program 60.8%

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

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d1 around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 86.5

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

   8. Applied rewrites 86.5%

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

   if -2.8999999999999999e121 < d1 < 2.1500000000000001e105

   1. Initial program 99.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 92.8

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

   5. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. associate--l+ N/A

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

      6. distribute-lft-out N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 92.8

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

   7. Applied rewrites 92.8%

      \[\leadsto \mathsf{fma}\left(d2 - d3, \color{blue}{d1}, d4 \cdot d1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d1 \leq -2.9 \cdot 10^{+121} \lor \neg \left(d1 \leq 2.15 \cdot 10^{+105}\right):\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(d2 - d3, d1, d4 \cdot d1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -2.9 \cdot 10^{+121} \lor \neg \left(d1 \leq 2.7 \cdot 10^{+105}\right):\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (or (<= d1 -2.9e+121) (not (<= d1 2.7e+105)))
   (* (- (- d1) d3) d1)
   (* (- (+ d4 d2) d3) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d1 <= -2.9e+121) || !(d1 <= 2.7e+105)) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = ((d4 + d2) - 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 ((d1 <= (-2.9d+121)) .or. (.not. (d1 <= 2.7d+105))) then
        tmp = (-d1 - d3) * d1
    else
        tmp = ((d4 + d2) - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d1 <= -2.9e+121) || !(d1 <= 2.7e+105)) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = ((d4 + d2) - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if (d1 <= -2.9e+121) or not (d1 <= 2.7e+105):
		tmp = (-d1 - d3) * d1
	else:
		tmp = ((d4 + d2) - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if ((d1 <= -2.9e+121) || !(d1 <= 2.7e+105))
		tmp = Float64(Float64(Float64(-d1) - d3) * d1);
	else
		tmp = Float64(Float64(Float64(d4 + d2) - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if ((d1 <= -2.9e+121) || ~((d1 <= 2.7e+105)))
		tmp = (-d1 - d3) * d1;
	else
		tmp = ((d4 + d2) - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[Or[LessEqual[d1, -2.9e+121], N[Not[LessEqual[d1, 2.7e+105]], $MachinePrecision]], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(N[(d4 + d2), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d1 < -2.8999999999999999e121 or 2.70000000000000016e105 < d1

   1. Initial program 60.8%

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

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d1 around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 86.5

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

   8. Applied rewrites 86.5%

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

   if -2.8999999999999999e121 < d1 < 2.70000000000000016e105

   1. Initial program 99.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 92.8

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

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d1 \leq -2.9 \cdot 10^{+121} \lor \neg \left(d1 \leq 2.7 \cdot 10^{+105}\right):\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -62000:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{elif}\;d2 \leq -8.8 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(-d1\right) - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -62000.0)
   (* (- d2 d3) d1)
   (if (<= d2 -8.8e-160) (* (- (- d1) d3) d1) (* (- d4 d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -62000.0) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -8.8e-160) {
		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 <= (-62000.0d0)) then
        tmp = (d2 - d3) * d1
    else if (d2 <= (-8.8d-160)) 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 <= -62000.0) {
		tmp = (d2 - d3) * d1;
	} else if (d2 <= -8.8e-160) {
		tmp = (-d1 - d3) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -62000.0:
		tmp = (d2 - d3) * d1
	elif d2 <= -8.8e-160:
		tmp = (-d1 - d3) * d1
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -62000.0)
		tmp = Float64(Float64(d2 - d3) * d1);
	elseif (d2 <= -8.8e-160)
		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 <= -62000.0)
		tmp = (d2 - d3) * d1;
	elseif (d2 <= -8.8e-160)
		tmp = (-d1 - d3) * d1;
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -62000.0], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d2, -8.8e-160], N[(N[((-d1) - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -62000

   1. Initial program 78.3%

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

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d2 around inf

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

   8. Applied rewrites 80.5%

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

   if -62000 < d2 < -8.8e-160

   1. Initial program 93.9%

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

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d1 around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 72.4

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

   8. Applied rewrites 72.4%

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

   if -8.8e-160 < d2

   1. Initial program 89.5%

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

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d4 around inf

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

   8. Applied rewrites 57.8%

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

Alternative 5: 68.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d3 \leq -3.35 \cdot 10^{+108} \lor \neg \left(d3 \leq 8 \cdot 10^{+142}\right):\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 + d2\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (or (<= d3 -3.35e+108) (not (<= d3 8e+142)))
   (* (- d3) d1)
   (* (+ d4 d2) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if ((d3 <= -3.35e+108) || !(d3 <= 8e+142)) {
		tmp = -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 ((d3 <= (-3.35d+108)) .or. (.not. (d3 <= 8d+142))) then
        tmp = -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 ((d3 <= -3.35e+108) || !(d3 <= 8e+142)) {
		tmp = -d3 * d1;
	} else {
		tmp = (d4 + d2) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if (d3 <= -3.35e+108) or not (d3 <= 8e+142):
		tmp = -d3 * d1
	else:
		tmp = (d4 + d2) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if ((d3 <= -3.35e+108) || !(d3 <= 8e+142))
		tmp = Float64(Float64(-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 ((d3 <= -3.35e+108) || ~((d3 <= 8e+142)))
		tmp = -d3 * d1;
	else
		tmp = (d4 + d2) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[Or[LessEqual[d3, -3.35e+108], N[Not[LessEqual[d3, 8e+142]], $MachinePrecision]], N[((-d3) * d1), $MachinePrecision], N[(N[(d4 + d2), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d3 < -3.35000000000000007e108 or 8.00000000000000041e142 < d3

   1. Initial program 78.5%

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

   3. Taylor expanded in d3 around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if -3.35000000000000007e108 < d3 < 8.00000000000000041e142

   1. Initial program 90.8%

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

   3. Taylor expanded in d1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in d3 around 0

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

      1. +-commutative N/A

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

      2. lift-+.f64 71.1

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

   8. Applied rewrites 71.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d3 \leq -3.35 \cdot 10^{+108} \lor \neg \left(d3 \leq 8 \cdot 10^{+142}\right):\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 + d2\right) \cdot d1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq -4.3 \cdot 10^{-288}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d4 \leq 5.2 \cdot 10^{+52}:\\ \;\;\;\;\left(-d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 -4.3e-288) (* d2 d1) (if (<= d4 5.2e+52) (* (- d3) d1) (* d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -4.3e-288) {
		tmp = d2 * d1;
	} else if (d4 <= 5.2e+52) {
		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 <= (-4.3d-288)) then
        tmp = d2 * d1
    else if (d4 <= 5.2d+52) 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 <= -4.3e-288) {
		tmp = d2 * d1;
	} else if (d4 <= 5.2e+52) {
		tmp = -d3 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= -4.3e-288:
		tmp = d2 * d1
	elif d4 <= 5.2e+52:
		tmp = -d3 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= -4.3e-288)
		tmp = Float64(d2 * d1);
	elseif (d4 <= 5.2e+52)
		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 <= -4.3e-288)
		tmp = d2 * d1;
	elseif (d4 <= 5.2e+52)
		tmp = -d3 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, -4.3e-288], N[(d2 * d1), $MachinePrecision], If[LessEqual[d4, 5.2e+52], N[((-d3) * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d4 < -4.29999999999999976e-288

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 37.6

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

   5. Applied rewrites 37.6%

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

   if -4.29999999999999976e-288 < d4 < 5.2e52

   1. Initial program 90.9%

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

   3. Taylor expanded in d3 around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 39.1

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

   5. Applied rewrites 39.1%

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

   if 5.2e52 < d4

   1. Initial program 81.0%

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

   3. Taylor expanded in d4 around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

Alternative 7: 39.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -235000:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d2 \leq -1.95 \cdot 10^{-244}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -235000.0)
   (* d2 d1)
   (if (<= d2 -1.95e-244) (* (- d1) d1) (* d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -235000.0) {
		tmp = d2 * d1;
	} else if (d2 <= -1.95e-244) {
		tmp = -d1 * 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 (d2 <= (-235000.0d0)) then
        tmp = d2 * d1
    else if (d2 <= (-1.95d-244)) then
        tmp = -d1 * 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 (d2 <= -235000.0) {
		tmp = d2 * d1;
	} else if (d2 <= -1.95e-244) {
		tmp = -d1 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -235000.0:
		tmp = d2 * d1
	elif d2 <= -1.95e-244:
		tmp = -d1 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -235000.0)
		tmp = Float64(d2 * d1);
	elseif (d2 <= -1.95e-244)
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -235000.0)
		tmp = d2 * d1;
	elseif (d2 <= -1.95e-244)
		tmp = -d1 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -235000.0], N[(d2 * d1), $MachinePrecision], If[LessEqual[d2, -1.95e-244], N[((-d1) * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -235000

   1. Initial program 78.3%

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

   3. Taylor expanded in d2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -235000 < d2 < -1.9499999999999999e-244

   1. Initial program 92.3%

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

   3. Taylor expanded in d1 around inf

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

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 44.5

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

   5. Applied rewrites 44.5%

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

   if -1.9499999999999999e-244 < d2

   1. Initial program 89.6%

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

   3. Taylor expanded in d4 around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 28.7

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

   5. Applied rewrites 28.7%

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

Alternative 8: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -0.00092:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -0.00092) (* (- d2 d3) d1) (* (- d4 d3) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -0.00092) {
		tmp = (d2 - 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 <= (-0.00092d0)) then
        tmp = (d2 - 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 <= -0.00092) {
		tmp = (d2 - d3) * d1;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -0.00092:
		tmp = (d2 - d3) * d1
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -0.00092)
		tmp = Float64(Float64(d2 - 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 <= -0.00092)
		tmp = (d2 - d3) * d1;
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -0.00092], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -9.2000000000000003e-4

   1. Initial program 78.3%

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

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d2 around inf

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

   8. Applied rewrites 80.5%

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

   if -9.2000000000000003e-4 < d2

   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

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d4 around inf

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

   8. Applied rewrites 61.0%

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

Alternative 9: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 5.2 \cdot 10^{+52}:\\ \;\;\;\;\left(d2 - d3\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;\left(d4 + d2\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 5.2e+52) (* (- d2 d3) d1) (* (+ d4 d2) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 5.2e+52) {
		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 <= 5.2d+52) 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 <= 5.2e+52) {
		tmp = (d2 - d3) * d1;
	} else {
		tmp = (d4 + d2) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 5.2e+52:
		tmp = (d2 - d3) * d1
	else:
		tmp = (d4 + d2) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 5.2e+52)
		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 <= 5.2e+52)
		tmp = (d2 - d3) * d1;
	else
		tmp = (d4 + d2) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 5.2e+52], N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision], N[(N[(d4 + d2), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 5.2e52

   1. Initial program 89.4%

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

   3. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + \left(d4 + -1 \cdot d1\right)\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(d4 + -1 \cdot d1\right) + d2\right) - d3\right) \cdot d1 \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(-1 \cdot d1 + d4\right) + d2\right) - d3\right) \cdot d1 \]

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in d2 around inf

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

   8. Applied rewrites 66.2%

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

   if 5.2e52 < d4

   1. Initial program 81.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in d3 around 0

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

      1. +-commutative N/A

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

      2. lift-+.f64 77.5

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

   8. Applied rewrites 77.5%

      \[\leadsto \left(d4 + d2\right) \cdot d1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 39.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -0.00082:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -0.00082) (* d2 d1) (* d4 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -0.00082) {
		tmp = d2 * 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 (d2 <= (-0.00082d0)) then
        tmp = d2 * 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 (d2 <= -0.00082) {
		tmp = d2 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -0.00082:
		tmp = d2 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -0.00082)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -0.00082)
		tmp = d2 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -0.00082], N[(d2 * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -8.1999999999999998e-4

   1. Initial program 78.6%

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

   3. Taylor expanded in d2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if -8.1999999999999998e-4 < d2

   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 d4 around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 32.4

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

   5. Applied rewrites 32.4%

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

Alternative 11: 31.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 = d2 * d1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in d2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 37.5

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

5. Applied rewrites 37.5%

   \[\leadsto \color{blue}{d2 \cdot d1} \]
6. 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 2025040 
(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)))