FastMath dist4

Percentage Accurate: 88.6% → 100.0%

Time: 2.5s

Alternatives: 12

Speedup: 1.8×

• 32.1% 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: 88.6% 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, 1.8× speedup

Math

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

FPCore

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

C

double code(double d1, double d2, double d3, double d4) {
	return d1 * ((d4 - (d3 - 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 = d1 * ((d4 - (d3 - d2)) - d1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. add-flip N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. remove-double-neg N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. +-commutative N/A

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

   18. sub-negate-rev N/A

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

   19. sub-flip-reverse N/A

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

   20. lower--.f64 N/A

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

   21. lower--.f64 100.0

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

3. Applied rewrites 100.0%

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

Alternative 2: 92.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -1.9 \cdot 10^{+74}:\\ \;\;\;\;\left(d2 - \left(d1 - d4\right)\right) \cdot d1\\ \mathbf{elif}\;d1 \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(d4 - d3, d1, d1 \cdot d2\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(\left(d4 - d3\right) - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d1 -1.9e+74)
   (* (- d2 (- d1 d4)) d1)
   (if (<= d1 1.75e+81) (fma (- d4 d3) d1 (* d1 d2)) (* d1 (- (- d4 d3) d1)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d1 <= -1.9e+74) {
		tmp = (d2 - (d1 - d4)) * d1;
	} else if (d1 <= 1.75e+81) {
		tmp = fma((d4 - d3), d1, (d1 * d2));
	} else {
		tmp = d1 * ((d4 - d3) - d1);
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d1 <= -1.9e+74)
		tmp = Float64(Float64(d2 - Float64(d1 - d4)) * d1);
	elseif (d1 <= 1.75e+81)
		tmp = fma(Float64(d4 - d3), d1, Float64(d1 * d2));
	else
		tmp = Float64(d1 * Float64(Float64(d4 - d3) - d1));
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d1, -1.9e+74], N[(N[(d2 - N[(d1 - d4), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d1, 1.75e+81], N[(N[(d4 - d3), $MachinePrecision] * d1 + N[(d1 * d2), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d1 < -1.8999999999999999e74

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.3

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. add-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 77.3

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

   8. Applied rewrites 77.3%

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

   if -1.8999999999999999e74 < d1 < 1.75e81

   1. Initial program 88.6%

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

   3. Applied rewrites 97.2%

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

   4. Taylor expanded in d1 around 0

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

   6. Applied rewrites 80.6%

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

   if 1.75e81 < d1

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d2 around 0

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

   6. Applied rewrites 76.3%

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

Alternative 3: 92.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d1 \leq -1.9 \cdot 10^{+74}:\\ \;\;\;\;\left(d2 - \left(d1 - d4\right)\right) \cdot d1\\ \mathbf{elif}\;d1 \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;d1 \cdot \left(\left(d2 + d4\right) - d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(\left(d4 - d3\right) - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d1 -1.9e+74)
   (* (- d2 (- d1 d4)) d1)
   (if (<= d1 1.75e+81) (* d1 (- (+ d2 d4) d3)) (* d1 (- (- d4 d3) d1)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d1 <= -1.9e+74) {
		tmp = (d2 - (d1 - d4)) * d1;
	} else if (d1 <= 1.75e+81) {
		tmp = d1 * ((d2 + d4) - d3);
	} else {
		tmp = d1 * ((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 (d1 <= (-1.9d+74)) then
        tmp = (d2 - (d1 - d4)) * d1
    else if (d1 <= 1.75d+81) then
        tmp = d1 * ((d2 + d4) - d3)
    else
        tmp = d1 * ((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 (d1 <= -1.9e+74) {
		tmp = (d2 - (d1 - d4)) * d1;
	} else if (d1 <= 1.75e+81) {
		tmp = d1 * ((d2 + d4) - d3);
	} else {
		tmp = d1 * ((d4 - d3) - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d1 <= -1.9e+74:
		tmp = (d2 - (d1 - d4)) * d1
	elif d1 <= 1.75e+81:
		tmp = d1 * ((d2 + d4) - d3)
	else:
		tmp = d1 * ((d4 - d3) - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d1 <= -1.9e+74)
		tmp = Float64(Float64(d2 - Float64(d1 - d4)) * d1);
	elseif (d1 <= 1.75e+81)
		tmp = Float64(d1 * Float64(Float64(d2 + d4) - d3));
	else
		tmp = Float64(d1 * Float64(Float64(d4 - d3) - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d1 <= -1.9e+74)
		tmp = (d2 - (d1 - d4)) * d1;
	elseif (d1 <= 1.75e+81)
		tmp = d1 * ((d2 + d4) - d3);
	else
		tmp = d1 * ((d4 - d3) - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d1, -1.9e+74], N[(N[(d2 - N[(d1 - d4), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d1, 1.75e+81], N[(d1 * N[(N[(d2 + d4), $MachinePrecision] - d3), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d1 < -1.8999999999999999e74

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.3

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. add-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 77.3

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

   8. Applied rewrites 77.3%

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

   if -1.8999999999999999e74 < d1 < 1.75e81

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   if 1.75e81 < d1

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d2 around 0

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

   6. Applied rewrites 76.3%

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

Alternative 4: 84.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -2.95 \cdot 10^{+88}:\\ \;\;\;\;\left(d2 - \left(d1 - d4\right)\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(\left(d4 - d3\right) - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -2.95e+88) (* (- d2 (- d1 d4)) d1) (* d1 (- (- d4 d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -2.95e+88) {
		tmp = (d2 - (d1 - d4)) * d1;
	} else {
		tmp = d1 * ((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 <= (-2.95d+88)) then
        tmp = (d2 - (d1 - d4)) * d1
    else
        tmp = d1 * ((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 <= -2.95e+88) {
		tmp = (d2 - (d1 - d4)) * d1;
	} else {
		tmp = d1 * ((d4 - d3) - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -2.95e+88:
		tmp = (d2 - (d1 - d4)) * d1
	else:
		tmp = d1 * ((d4 - d3) - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -2.95e+88)
		tmp = Float64(Float64(d2 - Float64(d1 - d4)) * d1);
	else
		tmp = Float64(d1 * Float64(Float64(d4 - d3) - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -2.95e+88)
		tmp = (d2 - (d1 - d4)) * d1;
	else
		tmp = d1 * ((d4 - d3) - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -2.95e+88], N[(N[(d2 - N[(d1 - d4), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision], N[(d1 * N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -2.94999999999999984e88

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.3

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. add-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 77.3

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

   8. Applied rewrites 77.3%

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

   if -2.94999999999999984e88 < d2

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d2 around 0

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

   6. Applied rewrites 76.3%

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

Alternative 5: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -7.5 \cdot 10^{+88}:\\ \;\;\;\;\left(d2 - d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(\left(d4 - d3\right) - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -7.5e+88) (* (- d2 d1) d1) (* d1 (- (- d4 d3) d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -7.5e+88) {
		tmp = (d2 - d1) * d1;
	} else {
		tmp = d1 * ((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 <= (-7.5d+88)) then
        tmp = (d2 - d1) * d1
    else
        tmp = d1 * ((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 <= -7.5e+88) {
		tmp = (d2 - d1) * d1;
	} else {
		tmp = d1 * ((d4 - d3) - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -7.5e+88:
		tmp = (d2 - d1) * d1
	else:
		tmp = d1 * ((d4 - d3) - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -7.5e+88)
		tmp = Float64(Float64(d2 - d1) * d1);
	else
		tmp = Float64(d1 * Float64(Float64(d4 - d3) - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -7.5e+88)
		tmp = (d2 - d1) * d1;
	else
		tmp = d1 * ((d4 - d3) - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -7.5e+88], N[(N[(d2 - d1), $MachinePrecision] * d1), $MachinePrecision], N[(d1 * N[(N[(d4 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -7.50000000000000031e88

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.3

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. add-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 77.3

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in d4 around 0

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

       1. lower--.f64 54.1

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

   11. Applied rewrites 54.1%

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

   if -7.50000000000000031e88 < d2

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d2 around 0

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

   6. Applied rewrites 76.3%

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

Alternative 6: 64.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d1 \cdot \left(d2 - d3\right)\\ \mathbf{if}\;d4 \leq -3.7 \cdot 10^{-278}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d4 \leq 2.6 \cdot 10^{-119}:\\ \;\;\;\;\left(d2 - d1\right) \cdot d1\\ \mathbf{elif}\;d4 \leq 3.7 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (* d1 (- d2 d3))))
   (if (<= d4 -3.7e-278)
     t_0
     (if (<= d4 2.6e-119)
       (* (- d2 d1) d1)
       (if (<= d4 3.7e-23) t_0 (* d1 (- d4 d1)))))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d3);
	double tmp;
	if (d4 <= -3.7e-278) {
		tmp = t_0;
	} else if (d4 <= 2.6e-119) {
		tmp = (d2 - d1) * d1;
	} else if (d4 <= 3.7e-23) {
		tmp = t_0;
	} else {
		tmp = d1 * (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) :: t_0
    real(8) :: tmp
    t_0 = d1 * (d2 - d3)
    if (d4 <= (-3.7d-278)) then
        tmp = t_0
    else if (d4 <= 2.6d-119) then
        tmp = (d2 - d1) * d1
    else if (d4 <= 3.7d-23) then
        tmp = t_0
    else
        tmp = d1 * (d4 - d1)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d3);
	double tmp;
	if (d4 <= -3.7e-278) {
		tmp = t_0;
	} else if (d4 <= 2.6e-119) {
		tmp = (d2 - d1) * d1;
	} else if (d4 <= 3.7e-23) {
		tmp = t_0;
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	t_0 = d1 * (d2 - d3)
	tmp = 0
	if d4 <= -3.7e-278:
		tmp = t_0
	elif d4 <= 2.6e-119:
		tmp = (d2 - d1) * d1
	elif d4 <= 3.7e-23:
		tmp = t_0
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(d1 * Float64(d2 - d3))
	tmp = 0.0
	if (d4 <= -3.7e-278)
		tmp = t_0;
	elseif (d4 <= 2.6e-119)
		tmp = Float64(Float64(d2 - d1) * d1);
	elseif (d4 <= 3.7e-23)
		tmp = t_0;
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	t_0 = d1 * (d2 - d3);
	tmp = 0.0;
	if (d4 <= -3.7e-278)
		tmp = t_0;
	elseif (d4 <= 2.6e-119)
		tmp = (d2 - d1) * d1;
	elseif (d4 <= 3.7e-23)
		tmp = t_0;
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d4, -3.7e-278], t$95$0, If[LessEqual[d4, 2.6e-119], N[(N[(d2 - d1), $MachinePrecision] * d1), $MachinePrecision], If[LessEqual[d4, 3.7e-23], t$95$0, N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d4 < -3.70000000000000022e-278 or 2.60000000000000012e-119 < d4 < 3.7000000000000003e-23

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d4 around 0

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

      1. lower--.f64 56.8

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

   7. Applied rewrites 56.8%

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

   if -3.70000000000000022e-278 < d4 < 2.60000000000000012e-119

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.3

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. add-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 77.3

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in d4 around 0

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

       1. lower--.f64 54.1

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

   11. Applied rewrites 54.1%

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

   if 3.7000000000000003e-23 < d4

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

   7. Taylor expanded in d2 around 0

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

      1. lower--.f64 53.8

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

   9. Applied rewrites 53.8%

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

Alternative 7: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -1.1 \cdot 10^{-42}:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -1.1e-42) (* d1 (- d2 d3)) (* d1 (- d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -1.1e-42) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (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 <= (-1.1d-42)) then
        tmp = d1 * (d2 - d3)
    else
        tmp = d1 * (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 <= -1.1e-42) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -1.1e-42:
		tmp = d1 * (d2 - d3)
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -1.1e-42)
		tmp = Float64(d1 * Float64(d2 - d3));
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -1.1e-42)
		tmp = d1 * (d2 - d3);
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -1.1e-42], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -1.10000000000000003e-42

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d4 around 0

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

      1. lower--.f64 56.8

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

   7. Applied rewrites 56.8%

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

   if -1.10000000000000003e-42 < d2

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

   7. Taylor expanded in d2 around 0

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

      1. lower--.f64 53.8

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

   9. Applied rewrites 53.8%

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

Alternative 8: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -2600000000000:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -2600000000000.0) (* d1 (- d2 d3)) (* d1 (- d4 d3))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -2600000000000.0) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (d4 - d3);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d1, d2, d3, 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 <= (-2600000000000.0d0)) then
        tmp = d1 * (d2 - d3)
    else
        tmp = d1 * (d4 - d3)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -2600000000000.0) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (d4 - d3);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -2600000000000.0:
		tmp = d1 * (d2 - d3)
	else:
		tmp = d1 * (d4 - d3)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -2600000000000.0)
		tmp = Float64(d1 * Float64(d2 - d3));
	else
		tmp = Float64(d1 * Float64(d4 - d3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -2600000000000.0)
		tmp = d1 * (d2 - d3);
	else
		tmp = d1 * (d4 - d3);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -2600000000000.0], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - d3), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -2.6e12

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d4 around 0

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

      1. lower--.f64 56.8

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

   7. Applied rewrites 56.8%

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

   if -2.6e12 < d2

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d2 around 0

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

   7. Applied rewrites 56.1%

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

Alternative 9: 61.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 1.45 \cdot 10^{+122}:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d4\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 1.45e+122) (* d1 (- d2 d3)) (* d1 d4)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1.45e+122) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * d4;
	}
	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+122) then
        tmp = d1 * (d2 - d3)
    else
        tmp = d1 * d4
    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+122) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * d4;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 1.45e+122:
		tmp = d1 * (d2 - d3)
	else:
		tmp = d1 * d4
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 1.45e+122)
		tmp = Float64(d1 * Float64(d2 - d3));
	else
		tmp = Float64(d1 * d4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 1.45e+122)
		tmp = d1 * (d2 - d3);
	else
		tmp = d1 * d4;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 1.45e+122], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], N[(d1 * d4), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 1.45e122

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d4 around 0

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

      1. lower--.f64 56.8

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

   7. Applied rewrites 56.8%

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

   if 1.45e122 < d4

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d4 around inf

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

      1. lower-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

Alternative 10: 39.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq -3.5 \cdot 10^{-272}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d4 \leq 1.06 \cdot 10^{+135}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d4\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 -3.5e-272)
   (* d1 d2)
   (if (<= d4 1.06e+135) (* (- d1) d1) (* d1 d4))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -3.5e-272) {
		tmp = d1 * d2;
	} else if (d4 <= 1.06e+135) {
		tmp = -d1 * d1;
	} else {
		tmp = d1 * d4;
	}
	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 <= (-3.5d-272)) then
        tmp = d1 * d2
    else if (d4 <= 1.06d+135) then
        tmp = -d1 * d1
    else
        tmp = d1 * d4
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= -3.5e-272) {
		tmp = d1 * d2;
	} else if (d4 <= 1.06e+135) {
		tmp = -d1 * d1;
	} else {
		tmp = d1 * d4;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= -3.5e-272:
		tmp = d1 * d2
	elif d4 <= 1.06e+135:
		tmp = -d1 * d1
	else:
		tmp = d1 * d4
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= -3.5e-272)
		tmp = Float64(d1 * d2);
	elseif (d4 <= 1.06e+135)
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(d1 * d4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= -3.5e-272)
		tmp = d1 * d2;
	elseif (d4 <= 1.06e+135)
		tmp = -d1 * d1;
	else
		tmp = d1 * d4;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, -3.5e-272], N[(d1 * d2), $MachinePrecision], If[LessEqual[d4, 1.06e+135], N[((-d1) * d1), $MachinePrecision], N[(d1 * d4), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d4 < -3.4999999999999997e-272

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.3

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. add-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 77.3

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in d2 around inf

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

       1. lower-*.f64 31.8

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

   11. Applied rewrites 31.8%

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

   if -3.4999999999999997e-272 < d4 < 1.06e135

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d1 around inf

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

      1. lower-*.f64 30.6

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

   6. Applied rewrites 30.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 30.6

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 30.6

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

   8. Applied rewrites 30.6%

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

   if 1.06e135 < d4

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d4 around inf

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

      1. lower-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

Alternative 11: 39.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d4\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -8.8e+15) (* d1 d2) (* d1 d4)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -8.8e+15) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * d4;
	}
	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 <= (-8.8d+15)) then
        tmp = d1 * d2
    else
        tmp = d1 * d4
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -8.8e+15) {
		tmp = d1 * d2;
	} else {
		tmp = d1 * d4;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -8.8e+15:
		tmp = d1 * d2
	else:
		tmp = d1 * d4
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -8.8e+15)
		tmp = Float64(d1 * d2);
	else
		tmp = Float64(d1 * d4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -8.8e+15)
		tmp = d1 * d2;
	else
		tmp = d1 * d4;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -8.8e+15], N[(d1 * d2), $MachinePrecision], N[(d1 * d4), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -8.8e15

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. remove-double-neg N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. +-commutative N/A

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

      18. sub-negate-rev N/A

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

      19. sub-flip-reverse N/A

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

      20. lower--.f64 N/A

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

      21. lower--.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in d3 around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.3

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. add-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 77.3

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in d2 around inf

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

       1. lower-*.f64 31.8

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

   11. Applied rewrites 31.8%

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

   if -8.8e15 < d2

   1. Initial program 88.6%

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

   2. Taylor expanded in d1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in d4 around inf

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

      1. lower-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

Alternative 12: 31.8% accurate, 5.3× 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 88.6%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. add-flip N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. remove-double-neg N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. +-commutative N/A

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

   18. sub-negate-rev N/A

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

   19. sub-flip-reverse N/A

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

   20. lower--.f64 N/A

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

   21. lower--.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in d3 around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 77.3

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

6. Applied rewrites 77.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 77.3

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. add-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lower--.f64 N/A

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

   10. lower--.f64 77.3

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

8. Applied rewrites 77.3%

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

9. Taylor expanded in d2 around inf

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

    1. lower-*.f64 31.8

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

11. Applied rewrites 31.8%

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

Developer Target 1: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025159 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  :precision binary64

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

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