Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 5.6s

Alternatives: 14

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{-1 - \mathsf{fma}\left(\sqrt{x}, 4, x\right)}{1 - x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (- -1.0 (fma (sqrt x) 4.0 x)) (- 1.0 x))))

C

double code(double x) {
	return 6.0 / ((-1.0 - fma(sqrt(x), 4.0, x)) / (1.0 - x));
}

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(-1.0 - fma(sqrt(x), 4.0, x)) / Float64(1.0 - x)))
end

Wolfram

code[x_] := N[(6.0 / N[(N[(-1.0 - N[(N[Sqrt[x], $MachinePrecision] * 4.0 + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{1 - x}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \cdot -6} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \cdot -6 \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}{1 - x}}} \cdot -6 \]

   3. lower-special-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}{1 - x}}} \cdot -6 \]

   4. lower-special-/.f64 99.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}{1 - x}}} \cdot -6 \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}}} \cdot -6 \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}} \cdot -6} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}}} \cdot -6 \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot -6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-6}}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}} \]

   5. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-6\right)}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{6}}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}\right)}} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x - -1\right)}{1 - x}}\right)} \]

   9. lift-fma.f64 N/A

      \[\leadsto \frac{6}{\mathsf{neg}\left(\frac{\color{blue}{4 \cdot \sqrt{x} + \left(x - -1\right)}}{1 - x}\right)} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{6}{\mathsf{neg}\left(\frac{4 \cdot \sqrt{x} + \color{blue}{\left(x - -1\right)}}{1 - x}\right)} \]

   11. associate--l+ N/A

       \[\leadsto \frac{6}{\mathsf{neg}\left(\frac{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) - -1}}{1 - x}\right)} \]

   12. lift-fma.f64 N/A

       \[\leadsto \frac{6}{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)} - -1}{1 - x}\right)} \]

   13. lift--.f64 N/A

       \[\leadsto \frac{6}{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) - -1}}{1 - x}\right)} \]

   14. distribute-neg-frac N/A

       \[\leadsto \frac{6}{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) - -1\right)\right)}{1 - x}}} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{6}{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{fma}\left(4, \sqrt{x}, x\right) - -1\right)\right)}{1 - x}}} \]

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{6}{\frac{-1 - \mathsf{fma}\left(\sqrt{x}, 4, x\right)}{1 - x}}} \]
8. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \cdot -6 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (- 1.0 x) (fma (sqrt x) 4.0 (- x -1.0))) -6.0))

C

double code(double x) {
	return ((1.0 - x) / fma(sqrt(x), 4.0, (x - -1.0))) * -6.0;
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 - x) / fma(sqrt(x), 4.0, Float64(x - -1.0))) * -6.0)
end

Wolfram

code[x_] := N[(N[(N[(1.0 - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -6.0), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{1 - x}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \cdot -6} \]
4. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, x\right) - -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma x 6.0 -6.0) (- (fma 4.0 (sqrt x) x) -1.0)))

C

double code(double x) {
	return fma(x, 6.0, -6.0) / (fma(4.0, sqrt(x), x) - -1.0);
}

Julia

function code(x)
	return Float64(fma(x, 6.0, -6.0) / Float64(fma(4.0, sqrt(x), x) - -1.0))
end

Wolfram

code[x_] := N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   3. sub-flip N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   4. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{x \cdot 6 + \left(\mathsf{neg}\left(1\right)\right) \cdot 6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

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

      \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(1 \cdot 6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{x \cdot 6 + \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   8. metadata-eval 99.7

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   10. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

   11. add-flip N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}} \]

   12. sub-flip-reverse N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

   15. remove-double-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x + 1\right)}} \]

   16. lower-fma.f64 99.7

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

   18. add-flip N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

   19. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

   20. metadata-eval 99.7

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - \color{blue}{-1}\right)} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \]
4. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4 + \left(x - -1\right)}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x - -1\right)}} \]

   3. associate-+r- N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(\sqrt{x} \cdot 4 + x\right) - -1}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(\sqrt{x} \cdot 4 + x\right) - -1}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(\color{blue}{4 \cdot \sqrt{x}} + x\right) - -1} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(\color{blue}{4 \cdot \sqrt{x}} + x\right) - -1} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 4 \cdot \sqrt{x}\right)} - -1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(x + \color{blue}{4 \cdot \sqrt{x}}\right) - -1} \]

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

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x - \left(\mathsf{neg}\left(4\right)\right) \cdot \sqrt{x}\right)} - -1} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(x - \color{blue}{-4} \cdot \sqrt{x}\right) - -1} \]

   11. sub-to-mult N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right) \cdot x} - -1} \]

   12. lower-special-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right) \cdot x} - -1} \]

   13. lower-special--.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right)} \cdot x - -1} \]

   14. lower-special-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(1 - \color{blue}{\frac{-4 \cdot \sqrt{x}}{x}}\right) \cdot x - -1} \]

   15. lower-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(1 - \color{blue}{\frac{-4 \cdot \sqrt{x}}{x}}\right) \cdot x - -1} \]

   16. lower--.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right)} \cdot x - -1} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right) \cdot x} - -1} \]

5. Applied rewrites 99.7%

   \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) - -1}} \]
6. Add Preprocessing

Alternative 4: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{6 \cdot \left(x - 1\right)}{1 - -4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)} \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (/ (* 6.0 (- x 1.0)) (- 1.0 (* -4.0 (sqrt x))))
   (* (/ -1.0 (fma (sqrt (/ 1.0 x)) 4.0 1.0)) -6.0)))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (6.0 * (x - 1.0)) / (1.0 - (-4.0 * sqrt(x)));
	} else {
		tmp = (-1.0 / fma(sqrt((1.0 / x)), 4.0, 1.0)) * -6.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(1.0 - Float64(-4.0 * sqrt(x))));
	else
		tmp = Float64(Float64(-1.0 / fma(sqrt(Float64(1.0 / x)), 4.0, 1.0)) * -6.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision] * -6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}} \]

      4. add-flip N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{x - \left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)}} \]

      5. sub-to-mult N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}\right) \cdot x}} \]

      6. lower-special-*.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}\right) \cdot x}} \]

      7. lower-special--.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}\right)} \cdot x} \]

      8. lower-special-/.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \color{blue}{\frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}}\right) \cdot x} \]

      9. +-commutative N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)}{x}\right) \cdot x} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}{x}\right) \cdot x} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \sqrt{x}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{x}\right) \cdot x} \]

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

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \sqrt{x}} + \left(\mathsf{neg}\left(1\right)\right)}{x}\right) \cdot x} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(4\right), \sqrt{x}, \mathsf{neg}\left(1\right)\right)}}{x}\right) \cdot x} \]

      14. metadata-eval N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\mathsf{fma}\left(\color{blue}{-4}, \sqrt{x}, \mathsf{neg}\left(1\right)\right)}{x}\right) \cdot x} \]

      15. metadata-eval 99.6

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\mathsf{fma}\left(-4, \sqrt{x}, \color{blue}{-1}\right)}{x}\right) \cdot x} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}{x}\right) \cdot x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 - -4 \cdot \sqrt{x}}} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 - \color{blue}{-4 \cdot \sqrt{x}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 - -4 \cdot \color{blue}{\sqrt{x}}} \]

      3. lower-sqrt.f64 51.2

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 - -4 \cdot \sqrt{x}} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 - -4 \cdot \sqrt{x}}} \]

   if 4 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(6\right)}{\color{blue}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-6}{\mathsf{neg}\left(\color{blue}{\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)}\right)} \]

      4. mult-flip N/A

         \[\leadsto -6 \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)\right)} \cdot \color{blue}{-6} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)\right)} \cdot \color{blue}{-6} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)\right)} \cdot -6 \]

      8. frac-2neg-rev N/A

         \[\leadsto \frac{-1}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \cdot -6 \]

      9. lower-/.f64 51.1

         \[\leadsto \frac{-1}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \cdot -6 \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{-1}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \cdot -6 \]

      11. +-commutative N/A

          \[\leadsto \frac{-1}{4 \cdot \sqrt{\frac{1}{x}} + 1} \cdot -6 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4 \cdot \sqrt{\frac{1}{x}} + 1} \cdot -6 \]

      13. *-commutative N/A

          \[\leadsto \frac{-1}{\sqrt{\frac{1}{x}} \cdot 4 + 1} \cdot -6 \]

      14. lower-fma.f64 51.1

          \[\leadsto \frac{-1}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)} \cdot -6 \]

   6. Applied rewrites 51.1%

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)} \cdot \color{blue}{-6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{6 \cdot \left(x - 1\right)}{1 - -4 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (/ (* 6.0 (- x 1.0)) (- 1.0 (* -4.0 (sqrt x))))
   (/ 6.0 (fma (sqrt (/ 1.0 x)) 4.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (6.0 * (x - 1.0)) / (1.0 - (-4.0 * sqrt(x)));
	} else {
		tmp = 6.0 / fma(sqrt((1.0 / x)), 4.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(1.0 - Float64(-4.0 * sqrt(x))));
	else
		tmp = Float64(6.0 / fma(sqrt(Float64(1.0 / x)), 4.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}} \]

      4. add-flip N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{x - \left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)}} \]

      5. sub-to-mult N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}\right) \cdot x}} \]

      6. lower-special-*.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}\right) \cdot x}} \]

      7. lower-special--.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}\right)} \cdot x} \]

      8. lower-special-/.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \color{blue}{\frac{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}{x}}\right) \cdot x} \]

      9. +-commutative N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)}{x}\right) \cdot x} \]

      10. distribute-neg-in N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}{x}\right) \cdot x} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \sqrt{x}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{x}\right) \cdot x} \]

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

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \sqrt{x}} + \left(\mathsf{neg}\left(1\right)\right)}{x}\right) \cdot x} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(4\right), \sqrt{x}, \mathsf{neg}\left(1\right)\right)}}{x}\right) \cdot x} \]

      14. metadata-eval N/A

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\mathsf{fma}\left(\color{blue}{-4}, \sqrt{x}, \mathsf{neg}\left(1\right)\right)}{x}\right) \cdot x} \]

      15. metadata-eval 99.6

          \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(1 - \frac{\mathsf{fma}\left(-4, \sqrt{x}, \color{blue}{-1}\right)}{x}\right) \cdot x} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(1 - \frac{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}{x}\right) \cdot x}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 - -4 \cdot \sqrt{x}}} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 - \color{blue}{-4 \cdot \sqrt{x}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 - -4 \cdot \color{blue}{\sqrt{x}}} \]

      3. lower-sqrt.f64 51.2

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 - -4 \cdot \sqrt{x}} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 - -4 \cdot \sqrt{x}}} \]

   if 4 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{6}{4 \cdot \sqrt{\frac{1}{x}} + \color{blue}{1}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{6}{4 \cdot \sqrt{\frac{1}{x}} + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{6}{\sqrt{\frac{1}{x}} \cdot 4 + 1} \]

      5. lower-fma.f64 51.1

         \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{4}, 1\right)} \]

   6. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 6, -6\right)}{4 \cdot \sqrt{x} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (/ (fma x 6.0 -6.0) (- (* 4.0 (sqrt x)) -1.0))
   (/ 6.0 (fma (sqrt (/ 1.0 x)) 4.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = fma(x, 6.0, -6.0) / ((4.0 * sqrt(x)) - -1.0);
	} else {
		tmp = 6.0 / fma(sqrt((1.0 / x)), 4.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(fma(x, 6.0, -6.0) / Float64(Float64(4.0 * sqrt(x)) - -1.0));
	else
		tmp = Float64(6.0 / fma(sqrt(Float64(1.0 / x)), 4.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. sub-flip N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6 + \left(\mathsf{neg}\left(1\right)\right) \cdot 6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

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

         \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(1 \cdot 6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot 6 + \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      8. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      11. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x + 1\right)}} \]

      16. lower-fma.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      20. metadata-eval 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - \color{blue}{-1}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4 + \left(x - -1\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x - -1\right)}} \]

      3. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(\sqrt{x} \cdot 4 + x\right) - -1}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(\sqrt{x} \cdot 4 + x\right) - -1}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(\color{blue}{4 \cdot \sqrt{x}} + x\right) - -1} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(\color{blue}{4 \cdot \sqrt{x}} + x\right) - -1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 4 \cdot \sqrt{x}\right)} - -1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(x + \color{blue}{4 \cdot \sqrt{x}}\right) - -1} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x - \left(\mathsf{neg}\left(4\right)\right) \cdot \sqrt{x}\right)} - -1} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(x - \color{blue}{-4} \cdot \sqrt{x}\right) - -1} \]

      11. sub-to-mult N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right) \cdot x} - -1} \]

      12. lower-special-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right) \cdot x} - -1} \]

      13. lower-special--.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right)} \cdot x - -1} \]

      14. lower-special-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(1 - \color{blue}{\frac{-4 \cdot \sqrt{x}}{x}}\right) \cdot x - -1} \]

      15. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\left(1 - \color{blue}{\frac{-4 \cdot \sqrt{x}}{x}}\right) \cdot x - -1} \]

      16. lower--.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right)} \cdot x - -1} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(1 - \frac{-4 \cdot \sqrt{x}}{x}\right) \cdot x} - -1} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) - -1}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} - -1} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{4 \cdot \color{blue}{\sqrt{x}} - -1} \]

      2. lower-sqrt.f64 51.2

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{4 \cdot \sqrt{x} - -1} \]

   8. Applied rewrites 51.2%

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} - -1} \]

   if 4 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{6}{4 \cdot \sqrt{\frac{1}{x}} + \color{blue}{1}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{6}{4 \cdot \sqrt{\frac{1}{x}} + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{6}{\sqrt{\frac{1}{x}} \cdot 4 + 1} \]

      5. lower-fma.f64 51.1

         \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{4}, 1\right)} \]

   6. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (fma (sqrt x) 4.0 (- x -1.0)))
   (/ 6.0 (fma (sqrt (/ 1.0 x)) 4.0 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (x - -1.0));
	} else {
		tmp = 6.0 / fma(sqrt((1.0 / x)), 4.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(x - -1.0)));
	else
		tmp = Float64(6.0 / fma(sqrt(Float64(1.0 / x)), 4.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. sub-flip N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6 + \left(\mathsf{neg}\left(1\right)\right) \cdot 6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

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

         \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(1 \cdot 6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot 6 + \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      8. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      11. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x + 1\right)}} \]

      16. lower-fma.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      20. metadata-eval 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - \color{blue}{-1}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 48.8%

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{6}{4 \cdot \sqrt{\frac{1}{x}} + \color{blue}{1}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{6}{4 \cdot \sqrt{\frac{1}{x}} + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{6}{\sqrt{\frac{1}{x}} \cdot 4 + 1} \]

      5. lower-fma.f64 51.1

         \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{4}, 1\right)} \]

   6. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 4, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (fma (sqrt x) 4.0 (- x -1.0)))
   (/ 6.0 (/ (fma 4.0 (sqrt x) x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (x - -1.0));
	} else {
		tmp = 6.0 / (fma(4.0, sqrt(x), x) / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(x - -1.0)));
	else
		tmp = Float64(6.0 / Float64(fma(4.0, sqrt(x), x) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. sub-flip N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6 + \left(\mathsf{neg}\left(1\right)\right) \cdot 6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

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

         \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(1 \cdot 6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot 6 + \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      8. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      11. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x + 1\right)}} \]

      16. lower-fma.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      20. metadata-eval 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - \color{blue}{-1}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 48.8%

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{6}{\frac{x + 4 \cdot \sqrt{x}}{\color{blue}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\frac{x + 4 \cdot \sqrt{x}}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{\frac{x + 4 \cdot \sqrt{x}}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{\frac{x + 4 \cdot \sqrt{x}}{x}} \]

      4. lower-sqrt.f64 51.1

         \[\leadsto \frac{6}{\frac{x + 4 \cdot \sqrt{x}}{x}} \]

   7. Applied rewrites 51.1%

      \[\leadsto \frac{6}{\frac{x + 4 \cdot \sqrt{x}}{\color{blue}{x}}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6}{\frac{x + 4 \cdot \sqrt{x}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{6}{\frac{4 \cdot \sqrt{x} + x}{x}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{6}{\frac{4 \cdot \sqrt{x} + x}{x}} \]

      4. lower-fma.f64 51.1

         \[\leadsto \frac{6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right)}{x}} \]

   9. Applied rewrites 51.1%

      \[\leadsto \frac{6}{\frac{\mathsf{fma}\left(4, \sqrt{x}, x\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot x}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
   (/ -6.0 (fma (sqrt x) 4.0 (- x -1.0)))
   (/ (* 6.0 x) (fma (sqrt x) 4.0 1.0))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (x - -1.0));
	} else {
		tmp = (6.0 * x) / fma(sqrt(x), 4.0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(x - -1.0)));
	else
		tmp = Float64(Float64(6.0 * x) / fma(sqrt(x), 4.0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 * x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. sub-flip N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6 + \left(\mathsf{neg}\left(1\right)\right) \cdot 6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

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

         \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(1 \cdot 6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot 6 + \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      8. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      11. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x + 1\right)}} \]

      16. lower-fma.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      20. metadata-eval 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - \color{blue}{-1}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 48.8%

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. sub-flip N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6 + \left(\mathsf{neg}\left(1\right)\right) \cdot 6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

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

         \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(1 \cdot 6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot 6 + \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      8. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      11. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x + 1\right)}} \]

      16. lower-fma.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      20. metadata-eval 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - \color{blue}{-1}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{6 \cdot x}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 51.1

         \[\leadsto \frac{6 \cdot \color{blue}{x}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]

   6. Applied rewrites 51.1%

      \[\leadsto \frac{\color{blue}{6 \cdot x}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot x}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{1}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.7%

      \[\leadsto \frac{6 \cdot x}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
   (/ -6.0 (fma (sqrt x) 4.0 (- x -1.0)))
   (* (* x (sqrt (/ 1.0 x))) 1.5)))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (x - -1.0));
	} else {
		tmp = (x * sqrt((1.0 / x))) * 1.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(x - -1.0)));
	else
		tmp = Float64(Float64(x * sqrt(Float64(1.0 / x))) * 1.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      3. sub-flip N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6 + \left(\mathsf{neg}\left(1\right)\right) \cdot 6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

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

         \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(1 \cdot 6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot 6 + \left(\mathsf{neg}\left(\color{blue}{6}\right)\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      8. metadata-eval 99.7

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      11. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)}} \]

      12. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{4 \cdot \sqrt{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\sqrt{x} \cdot 4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right)\right)\right)} \]

      15. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\sqrt{x} \cdot 4 + \color{blue}{\left(x + 1\right)}} \]

      16. lower-fma.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x + 1}\right)} \]

      18. add-flip N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, \color{blue}{x - \left(\mathsf{neg}\left(1\right)\right)}\right)} \]

      20. metadata-eval 99.7

          \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - \color{blue}{-1}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 48.8%

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(\sqrt{x}, 4, x - -1\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{3}{2} \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\color{blue}{\sqrt{x}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      3. lower-sqrt.f64 4.5

         \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      2. mult-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left(x \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      5. pow1/2 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{{x}^{\frac{1}{2}}} \cdot x\right) \]

      6. pow-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x\right) \]

      7. pow-plus N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{1}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\frac{-1}{2} + 1\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\frac{1}{2}} \]

      10. pow1/2 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      11. lift-sqrt.f64 4.5

          \[\leadsto 1.5 \cdot \sqrt{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} \]

      14. lower-*.f64 4.5

          \[\leadsto \sqrt{x} \cdot 1.5 \]

   9. Applied rewrites 4.5%

      \[\leadsto \sqrt{x} \cdot 1.5 \]

   10. Taylor expanded in x around inf

       \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{3}{2} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{3}{2} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{3}{2} \]

       3. lower-/.f64 4.5

          \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot 1.5 \]

   12. Applied rewrites 4.5%

       \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
   (/ 6.0 (fma -4.0 (sqrt x) -1.0))
   (* (sqrt x) 1.5)))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = 6.0 / fma(-4.0, sqrt(x), -1.0);
	} else {
		tmp = sqrt(x) * 1.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(6.0 / fma(-4.0, sqrt(x), -1.0));
	else
		tmp = Float64(sqrt(x) * 1.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(6.0 / N[(-4.0 * N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-6}{1 + \color{blue}{4 \cdot \sqrt{x}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-6}{1 + 4 \cdot \color{blue}{\sqrt{x}}} \]

      4. lower-sqrt.f64 48.6

         \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]

   4. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(-6\right)}{\color{blue}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)} \]

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

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 - \left(\mathsf{neg}\left(4\right)\right) \cdot \sqrt{x}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 - -4 \cdot \sqrt{x}\right)\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto \frac{6}{-4 \cdot \sqrt{x} - \color{blue}{1}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{-4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(-1\right)\right)} \]

      10. add-flip N/A

          \[\leadsto \frac{6}{-4 \cdot \sqrt{x} + \color{blue}{-1}} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(-4, \color{blue}{\sqrt{x}}, -1\right)} \]

      12. lower-/.f64 48.6

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}} \]

   6. Applied rewrites 48.6%

      \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{3}{2} \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\color{blue}{\sqrt{x}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      3. lower-sqrt.f64 4.5

         \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      2. mult-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left(x \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      5. pow1/2 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{{x}^{\frac{1}{2}}} \cdot x\right) \]

      6. pow-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x\right) \]

      7. pow-plus N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{1}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\frac{-1}{2} + 1\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\frac{1}{2}} \]

      10. pow1/2 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      11. lift-sqrt.f64 4.5

          \[\leadsto 1.5 \cdot \sqrt{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} \]

      14. lower-*.f64 4.5

          \[\leadsto \sqrt{x} \cdot 1.5 \]

   9. Applied rewrites 4.5%

      \[\leadsto \sqrt{x} \cdot 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
   (/ 6.0 (fma -4.0 (sqrt x) -1.0))
   (* (* x (sqrt (/ 1.0 x))) 1.5)))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = 6.0 / fma(-4.0, sqrt(x), -1.0);
	} else {
		tmp = (x * sqrt((1.0 / x))) * 1.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(6.0 / fma(-4.0, sqrt(x), -1.0));
	else
		tmp = Float64(Float64(x * sqrt(Float64(1.0 / x))) * 1.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(6.0 / N[(-4.0 * N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-6}{1 + \color{blue}{4 \cdot \sqrt{x}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-6}{1 + 4 \cdot \color{blue}{\sqrt{x}}} \]

      4. lower-sqrt.f64 48.6

         \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]

   4. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(-6\right)}{\color{blue}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(1 + 4 \cdot \sqrt{x}\right)}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)} \]

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

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 - \left(\mathsf{neg}\left(4\right)\right) \cdot \sqrt{x}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\left(1 - -4 \cdot \sqrt{x}\right)\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto \frac{6}{-4 \cdot \sqrt{x} - \color{blue}{1}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{-4 \cdot \sqrt{x} - \left(\mathsf{neg}\left(-1\right)\right)} \]

      10. add-flip N/A

          \[\leadsto \frac{6}{-4 \cdot \sqrt{x} + \color{blue}{-1}} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(-4, \color{blue}{\sqrt{x}}, -1\right)} \]

      12. lower-/.f64 48.6

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}} \]

   6. Applied rewrites 48.6%

      \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{3}{2} \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\color{blue}{\sqrt{x}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      3. lower-sqrt.f64 4.5

         \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      2. mult-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left(x \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      5. pow1/2 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{{x}^{\frac{1}{2}}} \cdot x\right) \]

      6. pow-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x\right) \]

      7. pow-plus N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{1}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\frac{-1}{2} + 1\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\frac{1}{2}} \]

      10. pow1/2 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      11. lift-sqrt.f64 4.5

          \[\leadsto 1.5 \cdot \sqrt{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} \]

      14. lower-*.f64 4.5

          \[\leadsto \sqrt{x} \cdot 1.5 \]

   9. Applied rewrites 4.5%

      \[\leadsto \sqrt{x} \cdot 1.5 \]

   10. Taylor expanded in x around inf

       \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{3}{2} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{3}{2} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{3}{2} \]

       3. lower-/.f64 4.5

          \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot 1.5 \]

   12. Applied rewrites 4.5%

       \[\leadsto \left(x \cdot \sqrt{\frac{1}{x}}\right) \cdot 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 6.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-1.5}{\sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -1.5 (sqrt (/ 1.0 x))) (* (sqrt x) 1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / sqrt((1.0 / x));
	} else {
		tmp = sqrt(x) * 1.5;
	}
	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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-1.5d0) / sqrt((1.0d0 / x))
    else
        tmp = sqrt(x) * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / Math.sqrt((1.0 / x));
	} else {
		tmp = Math.sqrt(x) * 1.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -1.5 / math.sqrt((1.0 / x))
	else:
		tmp = math.sqrt(x) * 1.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-1.5 / sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(sqrt(x) * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -1.5 / sqrt((1.0 / x));
	else
		tmp = sqrt(x) * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-1.5 / N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{3}{2} \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\color{blue}{\sqrt{x}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      3. lower-sqrt.f64 4.5

         \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]

   8. Taylor expanded in x around -inf

      \[\leadsto \frac{\frac{-3}{2}}{\sqrt{\frac{1}{x}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2}}{\sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-3}{2}}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 4.0

         \[\leadsto \frac{-1.5}{\sqrt{\frac{1}{x}}} \]

   10. Applied rewrites 4.0%

       \[\leadsto \frac{-1.5}{\sqrt{\frac{1}{x}}} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

      5. lower-/.f64 51.1

         \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{3}{2} \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\color{blue}{\sqrt{x}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      3. lower-sqrt.f64 4.5

         \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

      2. mult-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left(x \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

      5. pow1/2 N/A

         \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{{x}^{\frac{1}{2}}} \cdot x\right) \]

      6. pow-flip N/A

         \[\leadsto \frac{3}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x\right) \]

      7. pow-plus N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{1}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\left(\frac{-1}{2} + 1\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{3}{2} \cdot {x}^{\frac{1}{2}} \]

      10. pow1/2 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      11. lift-sqrt.f64 4.5

          \[\leadsto 1.5 \cdot \sqrt{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \frac{3}{2} \]

      14. lower-*.f64 4.5

          \[\leadsto \sqrt{x} \cdot 1.5 \]

   9. Applied rewrites 4.5%

      \[\leadsto \sqrt{x} \cdot 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 4.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt x) 1.5))

C

double code(double x) {
	return sqrt(x) * 1.5;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = sqrt(x) * 1.5d0
end function

Java

public static double code(double x) {
	return Math.sqrt(x) * 1.5;
}

Python

def code(x):
	return math.sqrt(x) * 1.5

Julia

function code(x)
	return Float64(sqrt(x) * 1.5)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(x) * 1.5;
end

Wolfram

code[x_] := N[(N[Sqrt[x], $MachinePrecision] * 1.5), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{6}{\color{blue}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{6}{1 + \color{blue}{4 \cdot \sqrt{\frac{1}{x}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{6}{1 + 4 \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   5. lower-/.f64 51.1

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{3}{2} \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{3}{2} \cdot \frac{x}{\color{blue}{\sqrt{x}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

   3. lower-sqrt.f64 4.5

      \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]

7. Applied rewrites 4.5%

   \[\leadsto 1.5 \cdot \color{blue}{\frac{x}{\sqrt{x}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{3}{2} \cdot \frac{x}{\sqrt{x}} \]

   2. mult-flip N/A

      \[\leadsto \frac{3}{2} \cdot \left(x \cdot \frac{1}{\color{blue}{\sqrt{x}}}\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{\sqrt{x}} \cdot x\right) \]

   5. pow1/2 N/A

      \[\leadsto \frac{3}{2} \cdot \left(\frac{1}{{x}^{\frac{1}{2}}} \cdot x\right) \]

   6. pow-flip N/A

      \[\leadsto \frac{3}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot x\right) \]

   7. pow-plus N/A

      \[\leadsto \frac{3}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \color{blue}{1}\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{3}{2} \cdot {x}^{\left(\frac{-1}{2} + 1\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{3}{2} \cdot {x}^{\frac{1}{2}} \]

   10. pow1/2 N/A

       \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

   11. lift-sqrt.f64 4.5

       \[\leadsto 1.5 \cdot \sqrt{x} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{3}{2} \cdot \sqrt{x} \]

   13. *-commutative N/A

       \[\leadsto \sqrt{x} \cdot \frac{3}{2} \]

   14. lower-*.f64 4.5

       \[\leadsto \sqrt{x} \cdot 1.5 \]

9. Applied rewrites 4.5%

   \[\leadsto \sqrt{x} \cdot 1.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64
  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))