Expression, p6

Percentage Accurate: 94.3% → 99.6%

Time: 2.3min

Alternatives: 5

Speedup: N/A×

Specification

\[\left(\left(\left(-14 \leq a \land a \leq -13\right) \land \left(-3 \leq b \land b \leq -2\right)\right) \land \left(3 \leq c \land c \leq 3.5\right)\right) \land \left(12.5 \leq d \land d \leq 13.5\right)\]

Math

\[\begin{array}{l} \\ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

C

double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Initial Program: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

C

double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Alternative 1: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(b + c \cdot \left(1 + \frac{a + d}{c}\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (* (+ b (* c (+ 1.0 (/ (+ a d) c)))) 2.0))

C

double code(double a, double b, double c, double d) {
	return (b + (c * (1.0 + ((a + d) / c)))) * 2.0;
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (b + (c * (1.0d0 + ((a + d) / c)))) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (b + (c * (1.0 + ((a + d) / c)))) * 2.0;
}

Python

def code(a, b, c, d):
	return (b + (c * (1.0 + ((a + d) / c)))) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(b + Float64(c * Float64(1.0 + Float64(Float64(a + d) / c)))) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (b + (c * (1.0 + ((a + d) / c)))) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(b + N[(c * N[(1.0 + N[(N[(a + d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 94.3%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\left(c \cdot \left(1 + \left(\frac{a}{c} + \left(\frac{b}{c} + \frac{d}{c}\right)\right)\right)\right)} \cdot 2 \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \left(\frac{a}{c} + \left(\frac{b}{c} + \frac{d}{c}\right)\right)\right) \cdot \color{blue}{c}\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \left(\frac{a}{c} + \left(\frac{b}{c} + \frac{d}{c}\right)\right)\right) \cdot \color{blue}{c}\right) \cdot 2 \]

   3. div-add-rev N/A

      \[\leadsto \left(\left(1 + \left(\frac{a}{c} + \frac{b + d}{c}\right)\right) \cdot c\right) \cdot 2 \]

   4. div-add N/A

      \[\leadsto \left(\left(1 + \frac{a + \left(b + d\right)}{c}\right) \cdot c\right) \cdot 2 \]

   5. +-commutative N/A

      \[\leadsto \left(\left(\frac{a + \left(b + d\right)}{c} + 1\right) \cdot c\right) \cdot 2 \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(\frac{a + \left(b + d\right)}{c} + 1 \cdot 1\right) \cdot c\right) \cdot 2 \]

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

      \[\leadsto \left(\left(\frac{a + \left(b + d\right)}{c} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \cdot c\right) \cdot 2 \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\frac{a + \left(b + d\right)}{c} - -1 \cdot 1\right) \cdot c\right) \cdot 2 \]

   9. metadata-eval N/A

      \[\leadsto \left(\left(\frac{a + \left(b + d\right)}{c} - -1\right) \cdot c\right) \cdot 2 \]

   10. lower--.f64 N/A

       \[\leadsto \left(\left(\frac{a + \left(b + d\right)}{c} - -1\right) \cdot c\right) \cdot 2 \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\left(\frac{a + \left(b + d\right)}{c} - -1\right) \cdot c\right) \cdot 2 \]

   12. +-commutative N/A

       \[\leadsto \left(\left(\frac{\left(b + d\right) + a}{c} - -1\right) \cdot c\right) \cdot 2 \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\left(\frac{\left(b + d\right) + a}{c} - -1\right) \cdot c\right) \cdot 2 \]

   14. +-commutative N/A

       \[\leadsto \left(\left(\frac{\left(d + b\right) + a}{c} - -1\right) \cdot c\right) \cdot 2 \]

   15. lower-+.f64 95.2

       \[\leadsto \left(\left(\frac{\left(d + b\right) + a}{c} - -1\right) \cdot c\right) \cdot 2 \]

5. Applied rewrites 95.2%

   \[\leadsto \color{blue}{\left(\left(\frac{\left(d + b\right) + a}{c} - -1\right) \cdot c\right)} \cdot 2 \]

6. Taylor expanded in b around 0

   \[\leadsto \left(b + \color{blue}{c \cdot \left(1 + \left(\frac{a}{c} + \frac{d}{c}\right)\right)}\right) \cdot 2 \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(b + c \cdot \color{blue}{\left(1 + \left(\frac{a}{c} + \frac{d}{c}\right)\right)}\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(b + c \cdot \left(1 + \color{blue}{\left(\frac{a}{c} + \frac{d}{c}\right)}\right)\right) \cdot 2 \]

   3. lower-+.f64 N/A

      \[\leadsto \left(b + c \cdot \left(1 + \left(\frac{a}{c} + \color{blue}{\frac{d}{c}}\right)\right)\right) \cdot 2 \]

   4. div-add-rev N/A

      \[\leadsto \left(b + c \cdot \left(1 + \frac{a + d}{c}\right)\right) \cdot 2 \]

   5. lower-/.f64 N/A

      \[\leadsto \left(b + c \cdot \left(1 + \frac{a + d}{c}\right)\right) \cdot 2 \]

   6. lower-+.f64 99.4

      \[\leadsto \left(b + c \cdot \left(1 + \frac{a + d}{c}\right)\right) \cdot 2 \]

8. Applied rewrites 99.4%

   \[\leadsto \left(b + \color{blue}{c \cdot \left(1 + \frac{a + d}{c}\right)}\right) \cdot 2 \]
9. Add Preprocessing

Alternative 2: 97.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\\ t_1 := \frac{d \cdot d}{t\_0}\\ \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, t\_1, t\_1\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{t\_0}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (fma d d (pow (+ a b) 2.0)) (* d (+ a b))))
        (t_1 (/ (* d d) t_0)))
   (fma
    2.0
    (*
     c
     (-
      (fma 2.0 t_1 t_1)
      (/
       (* (+ (pow d 3.0) (pow (+ a b) 3.0)) (- (* 2.0 d) (+ a b)))
       (pow t_0 2.0))))
    (* b (+ 2.0 (* 2.0 (/ (+ a d) b)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, pow((a + b), 2.0)) - (d * (a + b));
	double t_1 = (d * d) / t_0;
	return fma(2.0, (c * (fma(2.0, t_1, t_1) - (((pow(d, 3.0) + pow((a + b), 3.0)) * ((2.0 * d) - (a + b))) / pow(t_0, 2.0)))), (b * (2.0 + (2.0 * ((a + d) / b)))));
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, d, (Float64(a + b) ^ 2.0)) - Float64(d * Float64(a + b)))
	t_1 = Float64(Float64(d * d) / t_0)
	return fma(2.0, Float64(c * Float64(fma(2.0, t_1, t_1) - Float64(Float64(Float64((d ^ 3.0) + (Float64(a + b) ^ 3.0)) * Float64(Float64(2.0 * d) - Float64(a + b))) / (t_0 ^ 2.0)))), Float64(b * Float64(2.0 + Float64(2.0 * Float64(Float64(a + d) / b)))))
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * d + N[Power[N[(a + b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(d * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * d), $MachinePrecision] / t$95$0), $MachinePrecision]}, N[(2.0 * N[(c * N[(N[(2.0 * t$95$1 + t$95$1), $MachinePrecision] - N[(N[(N[(N[Power[d, 3.0], $MachinePrecision] + N[Power[N[(a + b), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * d), $MachinePrecision] - N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(2.0 + N[(2.0 * N[(N[(a + d), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 94.3%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(a + \left(b + \left(c + d\right)\right)\right)} \cdot 2 \]

   2. lift-+.f64 N/A

      \[\leadsto \left(a + \left(b + \color{blue}{\left(c + d\right)}\right)\right) \cdot 2 \]

   3. lift-+.f64 N/A

      \[\leadsto \left(a + \color{blue}{\left(b + \left(c + d\right)\right)}\right) \cdot 2 \]

   4. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(a + b\right) + \left(c + d\right)\right)} \cdot 2 \]

   5. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(a + b\right)}^{3} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)}} \cdot 2 \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(a + b\right)}^{3} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)}} \cdot 2 \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(a + b\right)}^{3} + {\left(c + d\right)}^{3}}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(a + b\right)}^{3}} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(a + b\right)}}^{3} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + \color{blue}{{\left(c + d\right)}^{3}}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   11. +-commutative N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\color{blue}{\left(d + c\right)}}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\color{blue}{\left(d + c\right)}}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\color{blue}{\mathsf{fma}\left(a + b, a + b, \left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)}} \cdot 2 \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(\color{blue}{a + b}, a + b, \left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   15. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, \color{blue}{a + b}, \left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   16. lower--.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, \color{blue}{\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)}\right)} \cdot 2 \]

   17. pow2 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, \color{blue}{{\left(c + d\right)}^{2}} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   18. lower-pow.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, \color{blue}{{\left(c + d\right)}^{2}} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   19. +-commutative N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\color{blue}{\left(d + c\right)}}^{2} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   20. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\color{blue}{\left(d + c\right)}}^{2} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\left(d + c\right)}^{2} - \color{blue}{\left(a + b\right) \cdot \left(c + d\right)}\right)} \cdot 2 \]

4. Applied rewrites 93.6%

   \[\leadsto \color{blue}{\frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\left(d + c\right)}^{2} - \left(a + b\right) \cdot \left(d + c\right)\right)}} \cdot 2 \]

5. Taylor expanded in c around 0

   \[\leadsto \color{blue}{2 \cdot \left(c \cdot \left(\left(2 \cdot \frac{{d}^{2}}{\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)} + \frac{{d}^{2}}{\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right)\right) + 2 \cdot \frac{{d}^{3} + {\left(a + b\right)}^{3}}{\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}} \]

7. Applied rewrites 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), 2 \cdot \frac{{d}^{3} + {\left(a + b\right)}^{3}}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right)} \]

8. Taylor expanded in b around inf

   \[\leadsto \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \]

   2. lower-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \]

   5. lift-+.f64 97.1

      \[\leadsto \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \]

10. Applied rewrites 97.1%

    \[\leadsto \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), b \cdot \left(2 + 2 \cdot \frac{a + d}{b}\right)\right) \]
11. Add Preprocessing

Alternative 3: 96.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, a \cdot a, a \cdot a\right)\\ t_1 := {a}^{3} + {d}^{3}\\ t_2 := 2 \cdot a - d\\ t_3 := 2 \cdot d - a\\ t_4 := \mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\\ t_5 := {t\_4}^{2}\\ t_6 := \frac{d \cdot d}{t\_4}\\ t_7 := {t\_4}^{3}\\ t_8 := \frac{t\_1}{t\_5}\\ t_9 := \frac{t\_0 \cdot t\_3}{t\_5}\\ t_10 := \frac{{\left(d \cdot t\_2\right)}^{2}}{t\_7}\\ t_11 := \frac{\left(d \cdot d\right) \cdot t\_2}{t\_5}\\ t_12 := \frac{d \cdot d}{t\_5}\\ t_13 := 2 \cdot \frac{t\_1 \cdot \left(t\_2 \cdot t\_3\right)}{t\_7}\\ t_14 := \frac{a \cdot a}{t\_4}\\ t_15 := \mathsf{fma}\left(2, t\_14, t\_14\right) - \frac{t\_1 \cdot t\_2}{t\_5}\\ t_16 := \frac{a}{t\_4}\\ \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, t\_6, t\_6\right) - \frac{t\_1 \cdot t\_3}{t\_5}\right), \mathsf{fma}\left(2, \frac{t\_1}{t\_4}, b \cdot \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(-2, t\_11, t\_13\right) - \mathsf{fma}\left(-1, t\_8, t\_9 + t\_11\right)\right), \mathsf{fma}\left(2, t\_15, b \cdot \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(-2, \mathsf{fma}\left(-1, t\_10, t\_12\right), \mathsf{fma}\left(2, \frac{t\_2 \cdot \left(\mathsf{fma}\left(-1, t\_8, t\_9\right) - t\_13\right)}{t\_4}, \frac{t\_1 \cdot \left(t\_3 \cdot \left(\mathsf{fma}\left(2, a \cdot a, \mathsf{fma}\left(2, d \cdot d, {t\_2}^{2}\right)\right) - 2 \cdot \left(a \cdot d\right)\right)\right)}{{t\_4}^{4}}\right)\right) - \mathsf{fma}\left(-1, t\_10, \mathsf{fma}\left(-1, \frac{t\_0}{t\_5}, \frac{\left(a + 2 \cdot a\right) \cdot t\_3}{t\_5} + t\_12\right)\right)\right), 2 \cdot \left(\mathsf{fma}\left(2, t\_16, t\_16\right) - \left(\frac{t\_2 \cdot t\_15}{t\_4} + \left(\frac{{a}^{3}}{t\_5} + \frac{{d}^{3}}{t\_5}\right)\right)\right)\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma 2.0 (* a a) (* a a)))
        (t_1 (+ (pow a 3.0) (pow d 3.0)))
        (t_2 (- (* 2.0 a) d))
        (t_3 (- (* 2.0 d) a))
        (t_4 (- (fma a a (* d d)) (* a d)))
        (t_5 (pow t_4 2.0))
        (t_6 (/ (* d d) t_4))
        (t_7 (pow t_4 3.0))
        (t_8 (/ t_1 t_5))
        (t_9 (/ (* t_0 t_3) t_5))
        (t_10 (/ (pow (* d t_2) 2.0) t_7))
        (t_11 (/ (* (* d d) t_2) t_5))
        (t_12 (/ (* d d) t_5))
        (t_13 (* 2.0 (/ (* t_1 (* t_2 t_3)) t_7)))
        (t_14 (/ (* a a) t_4))
        (t_15 (- (fma 2.0 t_14 t_14) (/ (* t_1 t_2) t_5)))
        (t_16 (/ a t_4)))
   (fma
    2.0
    (* c (- (fma 2.0 t_6 t_6) (/ (* t_1 t_3) t_5)))
    (fma
     2.0
     (/ t_1 t_4)
     (*
      b
      (fma
       2.0
       (* c (- (fma -2.0 t_11 t_13) (fma -1.0 t_8 (+ t_9 t_11))))
       (fma
        2.0
        t_15
        (*
         b
         (fma
          2.0
          (*
           c
           (-
            (fma
             -2.0
             (fma -1.0 t_10 t_12)
             (fma
              2.0
              (/ (* t_2 (- (fma -1.0 t_8 t_9) t_13)) t_4)
              (/
               (*
                t_1
                (*
                 t_3
                 (-
                  (fma 2.0 (* a a) (fma 2.0 (* d d) (pow t_2 2.0)))
                  (* 2.0 (* a d)))))
               (pow t_4 4.0))))
            (fma
             -1.0
             t_10
             (fma -1.0 (/ t_0 t_5) (+ (/ (* (+ a (* 2.0 a)) t_3) t_5) t_12)))))
          (*
           2.0
           (-
            (fma 2.0 t_16 t_16)
            (+
             (/ (* t_2 t_15) t_4)
             (+ (/ (pow a 3.0) t_5) (/ (pow d 3.0) t_5))))))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(2.0, (a * a), (a * a));
	double t_1 = pow(a, 3.0) + pow(d, 3.0);
	double t_2 = (2.0 * a) - d;
	double t_3 = (2.0 * d) - a;
	double t_4 = fma(a, a, (d * d)) - (a * d);
	double t_5 = pow(t_4, 2.0);
	double t_6 = (d * d) / t_4;
	double t_7 = pow(t_4, 3.0);
	double t_8 = t_1 / t_5;
	double t_9 = (t_0 * t_3) / t_5;
	double t_10 = pow((d * t_2), 2.0) / t_7;
	double t_11 = ((d * d) * t_2) / t_5;
	double t_12 = (d * d) / t_5;
	double t_13 = 2.0 * ((t_1 * (t_2 * t_3)) / t_7);
	double t_14 = (a * a) / t_4;
	double t_15 = fma(2.0, t_14, t_14) - ((t_1 * t_2) / t_5);
	double t_16 = a / t_4;
	return fma(2.0, (c * (fma(2.0, t_6, t_6) - ((t_1 * t_3) / t_5))), fma(2.0, (t_1 / t_4), (b * fma(2.0, (c * (fma(-2.0, t_11, t_13) - fma(-1.0, t_8, (t_9 + t_11)))), fma(2.0, t_15, (b * fma(2.0, (c * (fma(-2.0, fma(-1.0, t_10, t_12), fma(2.0, ((t_2 * (fma(-1.0, t_8, t_9) - t_13)) / t_4), ((t_1 * (t_3 * (fma(2.0, (a * a), fma(2.0, (d * d), pow(t_2, 2.0))) - (2.0 * (a * d))))) / pow(t_4, 4.0)))) - fma(-1.0, t_10, fma(-1.0, (t_0 / t_5), ((((a + (2.0 * a)) * t_3) / t_5) + t_12))))), (2.0 * (fma(2.0, t_16, t_16) - (((t_2 * t_15) / t_4) + ((pow(a, 3.0) / t_5) + (pow(d, 3.0) / t_5))))))))))));
}

Julia

function code(a, b, c, d)
	t_0 = fma(2.0, Float64(a * a), Float64(a * a))
	t_1 = Float64((a ^ 3.0) + (d ^ 3.0))
	t_2 = Float64(Float64(2.0 * a) - d)
	t_3 = Float64(Float64(2.0 * d) - a)
	t_4 = Float64(fma(a, a, Float64(d * d)) - Float64(a * d))
	t_5 = t_4 ^ 2.0
	t_6 = Float64(Float64(d * d) / t_4)
	t_7 = t_4 ^ 3.0
	t_8 = Float64(t_1 / t_5)
	t_9 = Float64(Float64(t_0 * t_3) / t_5)
	t_10 = Float64((Float64(d * t_2) ^ 2.0) / t_7)
	t_11 = Float64(Float64(Float64(d * d) * t_2) / t_5)
	t_12 = Float64(Float64(d * d) / t_5)
	t_13 = Float64(2.0 * Float64(Float64(t_1 * Float64(t_2 * t_3)) / t_7))
	t_14 = Float64(Float64(a * a) / t_4)
	t_15 = Float64(fma(2.0, t_14, t_14) - Float64(Float64(t_1 * t_2) / t_5))
	t_16 = Float64(a / t_4)
	return fma(2.0, Float64(c * Float64(fma(2.0, t_6, t_6) - Float64(Float64(t_1 * t_3) / t_5))), fma(2.0, Float64(t_1 / t_4), Float64(b * fma(2.0, Float64(c * Float64(fma(-2.0, t_11, t_13) - fma(-1.0, t_8, Float64(t_9 + t_11)))), fma(2.0, t_15, Float64(b * fma(2.0, Float64(c * Float64(fma(-2.0, fma(-1.0, t_10, t_12), fma(2.0, Float64(Float64(t_2 * Float64(fma(-1.0, t_8, t_9) - t_13)) / t_4), Float64(Float64(t_1 * Float64(t_3 * Float64(fma(2.0, Float64(a * a), fma(2.0, Float64(d * d), (t_2 ^ 2.0))) - Float64(2.0 * Float64(a * d))))) / (t_4 ^ 4.0)))) - fma(-1.0, t_10, fma(-1.0, Float64(t_0 / t_5), Float64(Float64(Float64(Float64(a + Float64(2.0 * a)) * t_3) / t_5) + t_12))))), Float64(2.0 * Float64(fma(2.0, t_16, t_16) - Float64(Float64(Float64(t_2 * t_15) / t_4) + Float64(Float64((a ^ 3.0) / t_5) + Float64((d ^ 3.0) / t_5))))))))))))
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(2.0 * N[(a * a), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[a, 3.0], $MachinePrecision] + N[Power[d, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * a), $MachinePrecision] - d), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * d), $MachinePrecision] - a), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a * a + N[(d * d), $MachinePrecision]), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$4, 2.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[(d * d), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[Power[t$95$4, 3.0], $MachinePrecision]}, Block[{t$95$8 = N[(t$95$1 / t$95$5), $MachinePrecision]}, Block[{t$95$9 = N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$5), $MachinePrecision]}, Block[{t$95$10 = N[(N[Power[N[(d * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / t$95$7), $MachinePrecision]}, Block[{t$95$11 = N[(N[(N[(d * d), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$5), $MachinePrecision]}, Block[{t$95$12 = N[(N[(d * d), $MachinePrecision] / t$95$5), $MachinePrecision]}, Block[{t$95$13 = N[(2.0 * N[(N[(t$95$1 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[(N[(a * a), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$15 = N[(N[(2.0 * t$95$14 + t$95$14), $MachinePrecision] - N[(N[(t$95$1 * t$95$2), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$16 = N[(a / t$95$4), $MachinePrecision]}, N[(2.0 * N[(c * N[(N[(2.0 * t$95$6 + t$95$6), $MachinePrecision] - N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$1 / t$95$4), $MachinePrecision] + N[(b * N[(2.0 * N[(c * N[(N[(-2.0 * t$95$11 + t$95$13), $MachinePrecision] - N[(-1.0 * t$95$8 + N[(t$95$9 + t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$15 + N[(b * N[(2.0 * N[(c * N[(N[(-2.0 * N[(-1.0 * t$95$10 + t$95$12), $MachinePrecision] + N[(2.0 * N[(N[(t$95$2 * N[(N[(-1.0 * t$95$8 + t$95$9), $MachinePrecision] - t$95$13), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(N[(t$95$1 * N[(t$95$3 * N[(N[(2.0 * N[(a * a), $MachinePrecision] + N[(2.0 * N[(d * d), $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(a * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$4, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * t$95$10 + N[(-1.0 * N[(t$95$0 / t$95$5), $MachinePrecision] + N[(N[(N[(N[(a + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$5), $MachinePrecision] + t$95$12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(2.0 * t$95$16 + t$95$16), $MachinePrecision] - N[(N[(N[(t$95$2 * t$95$15), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(N[(N[Power[a, 3.0], $MachinePrecision] / t$95$5), $MachinePrecision] + N[(N[Power[d, 3.0], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]

Derivation

1. Initial program 94.3%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(a + \left(b + \left(c + d\right)\right)\right)} \cdot 2 \]

   2. lift-+.f64 N/A

      \[\leadsto \left(a + \left(b + \color{blue}{\left(c + d\right)}\right)\right) \cdot 2 \]

   3. lift-+.f64 N/A

      \[\leadsto \left(a + \color{blue}{\left(b + \left(c + d\right)\right)}\right) \cdot 2 \]

   4. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(a + b\right) + \left(c + d\right)\right)} \cdot 2 \]

   5. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(a + b\right)}^{3} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)}} \cdot 2 \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(a + b\right)}^{3} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)}} \cdot 2 \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(a + b\right)}^{3} + {\left(c + d\right)}^{3}}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(a + b\right)}^{3}} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(a + b\right)}}^{3} + {\left(c + d\right)}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + \color{blue}{{\left(c + d\right)}^{3}}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   11. +-commutative N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\color{blue}{\left(d + c\right)}}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\color{blue}{\left(d + c\right)}}^{3}}{\left(a + b\right) \cdot \left(a + b\right) + \left(\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\color{blue}{\mathsf{fma}\left(a + b, a + b, \left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)}} \cdot 2 \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(\color{blue}{a + b}, a + b, \left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   15. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, \color{blue}{a + b}, \left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   16. lower--.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, \color{blue}{\left(c + d\right) \cdot \left(c + d\right) - \left(a + b\right) \cdot \left(c + d\right)}\right)} \cdot 2 \]

   17. pow2 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, \color{blue}{{\left(c + d\right)}^{2}} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   18. lower-pow.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, \color{blue}{{\left(c + d\right)}^{2}} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   19. +-commutative N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\color{blue}{\left(d + c\right)}}^{2} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   20. lower-+.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\color{blue}{\left(d + c\right)}}^{2} - \left(a + b\right) \cdot \left(c + d\right)\right)} \cdot 2 \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\left(d + c\right)}^{2} - \color{blue}{\left(a + b\right) \cdot \left(c + d\right)}\right)} \cdot 2 \]

4. Applied rewrites 93.6%

   \[\leadsto \color{blue}{\frac{{\left(a + b\right)}^{3} + {\left(d + c\right)}^{3}}{\mathsf{fma}\left(a + b, a + b, {\left(d + c\right)}^{2} - \left(a + b\right) \cdot \left(d + c\right)\right)}} \cdot 2 \]

5. Taylor expanded in c around 0

   \[\leadsto \color{blue}{2 \cdot \left(c \cdot \left(\left(2 \cdot \frac{{d}^{2}}{\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)} + \frac{{d}^{2}}{\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right)\right) + 2 \cdot \frac{{d}^{3} + {\left(a + b\right)}^{3}}{\left({d}^{2} + {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}} \]

7. Applied rewrites 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}, \frac{d \cdot d}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right) - \frac{\left({d}^{3} + {\left(a + b\right)}^{3}\right) \cdot \left(2 \cdot d - \left(a + b\right)\right)}{{\left(\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)\right)}^{2}}\right), 2 \cdot \frac{{d}^{3} + {\left(a + b\right)}^{3}}{\mathsf{fma}\left(d, d, {\left(a + b\right)}^{2}\right) - d \cdot \left(a + b\right)}\right)} \]

8. Taylor expanded in b around 0

   \[\leadsto 2 \cdot \left(c \cdot \left(\left(2 \cdot \frac{{d}^{2}}{\left({a}^{2} + {d}^{2}\right) - a \cdot d} + \frac{{d}^{2}}{\left({a}^{2} + {d}^{2}\right) - a \cdot d}\right) - \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(2 \cdot d - a\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right)\right) + \color{blue}{\left(2 \cdot \frac{{a}^{3} + {d}^{3}}{\left({a}^{2} + {d}^{2}\right) - a \cdot d} + b \cdot \left(2 \cdot \left(c \cdot \left(\left(-2 \cdot \frac{{d}^{2} \cdot \left(2 \cdot a - d\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}} + 2 \cdot \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(\left(2 \cdot a - d\right) \cdot \left(2 \cdot d - a\right)\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{3}}\right) - \left(-1 \cdot \frac{{a}^{3} + {d}^{3}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}} + \left(\frac{\left(2 \cdot {a}^{2} + {a}^{2}\right) \cdot \left(2 \cdot d - a\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}} + \frac{{d}^{2} \cdot \left(2 \cdot a - d\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right)\right)\right)\right) + \left(2 \cdot \left(\left(2 \cdot \frac{{a}^{2}}{\left({a}^{2} + {d}^{2}\right) - a \cdot d} + \frac{{a}^{2}}{\left({a}^{2} + {d}^{2}\right) - a \cdot d}\right) - \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(2 \cdot a - d\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right) + b \cdot \left(2 \cdot \left(c \cdot \left(\left(-2 \cdot \left(-1 \cdot \frac{{d}^{2} \cdot {\left(2 \cdot a - d\right)}^{2}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{3}} + \frac{{d}^{2}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right) + \left(2 \cdot \frac{\left(2 \cdot a - d\right) \cdot \left(\left(-1 \cdot \frac{{a}^{3} + {d}^{3}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}} + \frac{\left(2 \cdot {a}^{2} + {a}^{2}\right) \cdot \left(2 \cdot d - a\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right) - 2 \cdot \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(\left(2 \cdot a - d\right) \cdot \left(2 \cdot d - a\right)\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{3}}\right)}{\left({a}^{2} + {d}^{2}\right) - a \cdot d} + \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(\left(2 \cdot d - a\right) \cdot \left(\left(2 \cdot {a}^{2} + \left(2 \cdot {d}^{2} + {\left(2 \cdot a - d\right)}^{2}\right)\right) - 2 \cdot \left(a \cdot d\right)\right)\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{4}}\right)\right) - \left(-1 \cdot \frac{{d}^{2} \cdot {\left(2 \cdot a - d\right)}^{2}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{3}} + \left(-1 \cdot \frac{2 \cdot {a}^{2} + {a}^{2}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}} + \left(\frac{\left(a + 2 \cdot a\right) \cdot \left(2 \cdot d - a\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}} + \frac{{d}^{2}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right)\right)\right)\right)\right) + 2 \cdot \left(\left(2 \cdot \frac{a}{\left({a}^{2} + {d}^{2}\right) - a \cdot d} + \frac{a}{\left({a}^{2} + {d}^{2}\right) - a \cdot d}\right) - \left(\frac{\left(2 \cdot a - d\right) \cdot \left(\left(2 \cdot \frac{{a}^{2}}{\left({a}^{2} + {d}^{2}\right) - a \cdot d} + \frac{{a}^{2}}{\left({a}^{2} + {d}^{2}\right) - a \cdot d}\right) - \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(2 \cdot a - d\right)}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right)}{\left({a}^{2} + {d}^{2}\right) - a \cdot d} + \left(\frac{{a}^{3}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}} + \frac{{d}^{3}}{{\left(\left({a}^{2} + {d}^{2}\right) - a \cdot d\right)}^{2}}\right)\right)\right)\right)\right)\right)\right)} \]

10. Applied rewrites 96.4%

    \[\leadsto \mathsf{fma}\left(2, \color{blue}{c \cdot \left(\mathsf{fma}\left(2, \frac{d \cdot d}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}, \frac{d \cdot d}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}\right) - \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(2 \cdot d - a\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}\right)}, \mathsf{fma}\left(2, \frac{{a}^{3} + {d}^{3}}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}, b \cdot \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(-2, \frac{\left(d \cdot d\right) \cdot \left(2 \cdot a - d\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}, 2 \cdot \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(\left(2 \cdot a - d\right) \cdot \left(2 \cdot d - a\right)\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{3}}\right) - \mathsf{fma}\left(-1, \frac{{a}^{3} + {d}^{3}}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}, \frac{\mathsf{fma}\left(2, a \cdot a, a \cdot a\right) \cdot \left(2 \cdot d - a\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}} + \frac{\left(d \cdot d\right) \cdot \left(2 \cdot a - d\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}\right)\right), \mathsf{fma}\left(2, \mathsf{fma}\left(2, \frac{a \cdot a}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}, \frac{a \cdot a}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}\right) - \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(2 \cdot a - d\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}, b \cdot \mathsf{fma}\left(2, c \cdot \left(\mathsf{fma}\left(-2, \mathsf{fma}\left(-1, \frac{{\left(d \cdot \left(2 \cdot a - d\right)\right)}^{2}}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{3}}, \frac{d \cdot d}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}\right), \mathsf{fma}\left(2, \frac{\left(2 \cdot a - d\right) \cdot \left(\mathsf{fma}\left(-1, \frac{{a}^{3} + {d}^{3}}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}, \frac{\mathsf{fma}\left(2, a \cdot a, a \cdot a\right) \cdot \left(2 \cdot d - a\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}\right) - 2 \cdot \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(\left(2 \cdot a - d\right) \cdot \left(2 \cdot d - a\right)\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{3}}\right)}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}, \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(\left(2 \cdot d - a\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \mathsf{fma}\left(2, d \cdot d, {\left(2 \cdot a - d\right)}^{2}\right)\right) - 2 \cdot \left(a \cdot d\right)\right)\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{4}}\right)\right) - \mathsf{fma}\left(-1, \frac{{\left(d \cdot \left(2 \cdot a - d\right)\right)}^{2}}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{3}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, a \cdot a, a \cdot a\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}, \frac{\left(a + 2 \cdot a\right) \cdot \left(2 \cdot d - a\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}} + \frac{d \cdot d}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}\right)\right)\right), 2 \cdot \left(\mathsf{fma}\left(2, \frac{a}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}, \frac{a}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}\right) - \left(\frac{\left(2 \cdot a - d\right) \cdot \left(\mathsf{fma}\left(2, \frac{a \cdot a}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}, \frac{a \cdot a}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d}\right) - \frac{\left({a}^{3} + {d}^{3}\right) \cdot \left(2 \cdot a - d\right)}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}\right)}{\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d} + \left(\frac{{a}^{3}}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}} + \frac{{d}^{3}}{{\left(\mathsf{fma}\left(a, a, d \cdot d\right) - a \cdot d\right)}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 95.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(c + \left(\left(d + b\right) + a\right)\right) \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* 2.0 (+ c (+ (+ d b) a))))

C

double code(double a, double b, double c, double d) {
	return 2.0 * (c + ((d + b) + a));
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = 2.0d0 * (c + ((d + b) + a))
end function

Java

public static double code(double a, double b, double c, double d) {
	return 2.0 * (c + ((d + b) + a));
}

Python

def code(a, b, c, d):
	return 2.0 * (c + ((d + b) + a))

Julia

function code(a, b, c, d)
	return Float64(2.0 * Float64(c + Float64(Float64(d + b) + a)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = 2.0 * (c + ((d + b) + a));
end

Wolfram

code[a_, b_, c_, d_] := N[(2.0 * N[(c + N[(N[(d + b), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{2 \cdot c + 2 \cdot \left(a + \left(b + d\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto 2 \cdot \color{blue}{\left(c + \left(a + \left(b + d\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(c + \left(a + \left(b + d\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto 2 \cdot \left(c + \color{blue}{\left(a + \left(b + d\right)\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto 2 \cdot \left(c + \left(\left(b + d\right) + \color{blue}{a}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto 2 \cdot \left(c + \left(\left(b + d\right) + \color{blue}{a}\right)\right) \]

   6. +-commutative N/A

      \[\leadsto 2 \cdot \left(c + \left(\left(d + b\right) + a\right)\right) \]

   7. lower-+.f64 95.7

      \[\leadsto 2 \cdot \left(c + \left(\left(d + b\right) + a\right)\right) \]

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{2 \cdot \left(c + \left(\left(d + b\right) + a\right)\right)} \]
6. Add Preprocessing

Alternative 5: 95.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(b + d \cdot \left(1 + \frac{a + c}{d}\right)\right) \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (* 2.0 (+ b (* d (+ 1.0 (/ (+ a c) d))))))

C

double code(double a, double b, double c, double d) {
	return 2.0 * (b + (d * (1.0 + ((a + c) / d))));
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = 2.0d0 * (b + (d * (1.0d0 + ((a + c) / d))))
end function

Java

public static double code(double a, double b, double c, double d) {
	return 2.0 * (b + (d * (1.0 + ((a + c) / d))));
}

Python

def code(a, b, c, d):
	return 2.0 * (b + (d * (1.0 + ((a + c) / d))))

Julia

function code(a, b, c, d)
	return Float64(2.0 * Float64(b + Float64(d * Float64(1.0 + Float64(Float64(a + c) / d)))))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = 2.0 * (b + (d * (1.0 + ((a + c) / d))));
end

Wolfram

code[a_, b_, c_, d_] := N[(2.0 * N[(b + N[(d * N[(1.0 + N[(N[(a + c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{2 \cdot b + 2 \cdot \left(a + \left(c + d\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto 2 \cdot \color{blue}{\left(b + \left(a + \left(c + d\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(b + \left(a + \left(c + d\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto 2 \cdot \left(b + \color{blue}{\left(a + \left(c + d\right)\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto 2 \cdot \left(b + \left(\left(c + d\right) + \color{blue}{a}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto 2 \cdot \left(b + \left(\left(c + d\right) + \color{blue}{a}\right)\right) \]

   6. +-commutative N/A

      \[\leadsto 2 \cdot \left(b + \left(\left(d + c\right) + a\right)\right) \]

   7. lower-+.f64 94.5

      \[\leadsto 2 \cdot \left(b + \left(\left(d + c\right) + a\right)\right) \]

5. Applied rewrites 94.5%

   \[\leadsto \color{blue}{2 \cdot \left(b + \left(\left(d + c\right) + a\right)\right)} \]

6. Taylor expanded in d around inf

   \[\leadsto 2 \cdot \left(b + d \cdot \color{blue}{\left(1 + \left(\frac{a}{d} + \frac{c}{d}\right)\right)}\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(b + d \cdot \left(1 + \color{blue}{\left(\frac{a}{d} + \frac{c}{d}\right)}\right)\right) \]

   2. lower-+.f64 N/A

      \[\leadsto 2 \cdot \left(b + d \cdot \left(1 + \left(\frac{a}{d} + \color{blue}{\frac{c}{d}}\right)\right)\right) \]

   3. div-add-rev N/A

      \[\leadsto 2 \cdot \left(b + d \cdot \left(1 + \frac{a + c}{d}\right)\right) \]

   4. lower-/.f64 N/A

      \[\leadsto 2 \cdot \left(b + d \cdot \left(1 + \frac{a + c}{d}\right)\right) \]

   5. lower-+.f64 95.2

      \[\leadsto 2 \cdot \left(b + d \cdot \left(1 + \frac{a + c}{d}\right)\right) \]

8. Applied rewrites 95.2%

   \[\leadsto 2 \cdot \left(b + d \cdot \color{blue}{\left(1 + \frac{a + c}{d}\right)}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (and (and (and (<= -14.0 a) (<= a -13.0)) (and (<= -3.0 b) (<= b -2.0))) (and (<= 3.0 c) (<= c 3.5))) (and (<= 12.5 d) (<= d 13.5)))

  :alt
  (! :herbie-platform default (let ((e 2)) (+ (* (+ a b) e) (* (+ c d) e))))

  (* (+ a (+ b (+ c d))) 2.0))