Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 2.9s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

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

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

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

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

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%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. distribute-frac-neg N/A

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

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

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-*.f64 N/A

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

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 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. metadata-eval N/A

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

   5. sub-flip-reverse N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. add-flip N/A

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

   11. sub-flip-reverse N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. add-flip N/A

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

   16. sub-negate-rev N/A

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

   17. sub-negate-rev N/A

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

   18. add-flip N/A

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

   19. lift-+.f64 N/A

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

   20. lower-fma.f64 99.7

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

   21. lift-+.f64 N/A

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

   22. add-flip N/A

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

   23. lower--.f64 N/A

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

   24. 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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. add-flip N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. add-flip N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. lift-*.f64 N/A

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

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

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

   19. metadata-eval 99.7

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

5. Applied rewrites 99.7%

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

Alternative 4: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 42:\\ \;\;\;\;\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.16666666666666666 \cdot \left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 42.0)
   (/ (fma 6.0 x -6.0) (fma 4.0 (sqrt x) 1.0))
   (/ 1.0 (* 0.16666666666666666 (+ 1.0 (* 4.0 (sqrt (/ 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 42.0) {
		tmp = fma(6.0, x, -6.0) / fma(4.0, sqrt(x), 1.0);
	} else {
		tmp = 1.0 / (0.16666666666666666 * (1.0 + (4.0 * sqrt((1.0 / x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 42.0)
		tmp = Float64(fma(6.0, x, -6.0) / fma(4.0, sqrt(x), 1.0));
	else
		tmp = Float64(1.0 / Float64(0.16666666666666666 * Float64(1.0 + Float64(4.0 * sqrt(Float64(1.0 / x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 42.0], N[(N[(6.0 * x + -6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.16666666666666666 * N[(1.0 + N[(4.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 42

   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

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 51.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 51.2

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

   6. Applied rewrites 51.2%

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

   if 42 < 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

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 99.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. sub-flip-reverse N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate-rev N/A

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

      14. sub-negate-rev N/A

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

      15. add-flip N/A

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

      16. lift-+.f64 N/A

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

      17. lower-fma.f64 99.5

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

      18. lift-+.f64 N/A

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

      19. add-flip N/A

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

      20. lower--.f64 N/A

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

      21. metadata-eval 99.5

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

      22. lift-*.f64 N/A

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

      23. lift--.f64 N/A

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

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

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

      25. metadata-eval N/A

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

      26. sub-flip-reverse N/A

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

      27. lower-fma.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 50.7

         \[\leadsto \frac{1}{0.16666666666666666 \cdot \left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)} \]

   6. Applied rewrites 50.7%

      \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 \cdot \left(1 + 4 \cdot \sqrt{\frac{1}{x}}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 42:\\ \;\;\;\;\frac{\mathsf{fma}\left(6, x, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 - -4 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 42.0)
   (/ (fma 6.0 x -6.0) (fma 4.0 (sqrt x) 1.0))
   (/ 6.0 (- 1.0 (* -4.0 (sqrt (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 42.0) {
		tmp = fma(6.0, x, -6.0) / fma(4.0, sqrt(x), 1.0);
	} else {
		tmp = 6.0 / (1.0 - (-4.0 * sqrt((1.0 / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 42.0)
		tmp = Float64(fma(6.0, x, -6.0) / fma(4.0, sqrt(x), 1.0));
	else
		tmp = Float64(6.0 / Float64(1.0 - Float64(-4.0 * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 42.0], N[(N[(6.0 * x + -6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 - N[(-4.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 42

   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

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

   4. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 51.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 51.2

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

   6. Applied rewrites 51.2%

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

   if 42 < 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. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. sub-flip-reverse N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. sub-flip-reverse N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate-rev N/A

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

      17. sub-negate-rev N/A

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

      18. add-flip N/A

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

      19. lift-+.f64 N/A

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

      20. lower-fma.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. add-flip N/A

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

      23. lower--.f64 N/A

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

      24. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 - -4 \cdot \sqrt{\frac{1}{x}}}} \]
   7. 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 50.7

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

   8. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{6}{1 - -4 \cdot \sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 19:\\ \;\;\;\;\frac{-6}{x - \mathsf{fma}\left(-4, \sqrt{x}, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{1 - -4 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 19.0)
   (/ -6.0 (- x (fma -4.0 (sqrt x) -1.0)))
   (/ 6.0 (- 1.0 (* -4.0 (sqrt (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 19.0) {
		tmp = -6.0 / (x - fma(-4.0, sqrt(x), -1.0));
	} else {
		tmp = 6.0 / (1.0 - (-4.0 * sqrt((1.0 / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 19.0)
		tmp = Float64(-6.0 / Float64(x - fma(-4.0, sqrt(x), -1.0)));
	else
		tmp = Float64(6.0 / Float64(1.0 - Float64(-4.0 * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 19.0], N[(-6.0 / N[(x - N[(-4.0 * N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(1.0 - N[(-4.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 19

   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. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. sub-flip-reverse N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. sub-flip-reverse N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate-rev N/A

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

      17. sub-negate-rev N/A

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

      18. add-flip N/A

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

      19. lift-+.f64 N/A

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

      20. lower-fma.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. add-flip N/A

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

      23. lower--.f64 N/A

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

      24. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.7%

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

   if 19 < 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. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. sub-flip-reverse N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. sub-flip-reverse N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate-rev N/A

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

      17. sub-negate-rev N/A

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

      18. add-flip N/A

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

      19. lift-+.f64 N/A

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

      20. lower-fma.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. add-flip N/A

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

      23. lower--.f64 N/A

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

      24. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{1 - -4 \cdot \sqrt{\frac{1}{x}}}} \]
   7. 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 50.7

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

   8. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{6}{1 - -4 \cdot \sqrt{\frac{1}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;\frac{-6}{x - \mathsf{fma}\left(-4, \sqrt{x}, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(-4, \sqrt{\frac{1}{x}}, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8)
   (/ -6.0 (- x (fma -4.0 (sqrt x) -1.0)))
   (/ 6.0 (fma -4.0 (sqrt (/ 1.0 x)) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = -6.0 / (x - fma(-4.0, sqrt(x), -1.0));
	} else {
		tmp = 6.0 / fma(-4.0, sqrt((1.0 / x)), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.8)
		tmp = Float64(-6.0 / Float64(x - fma(-4.0, sqrt(x), -1.0)));
	else
		tmp = Float64(6.0 / fma(-4.0, sqrt(Float64(1.0 / x)), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.8], N[(-6.0 / N[(x - N[(-4.0 * N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(-4.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   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. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. sub-flip-reverse N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. sub-flip-reverse N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate-rev N/A

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

      17. sub-negate-rev N/A

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

      18. add-flip N/A

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

      19. lift-+.f64 N/A

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

      20. lower-fma.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. add-flip N/A

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

      23. lower--.f64 N/A

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

      24. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.7%

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

   if 0.80000000000000004 < 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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 51.7

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

   4. Applied rewrites 51.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-negate-rev N/A

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

      7. sub-flip N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval 51.7

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

   6. Applied rewrites 51.7%

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

Alternative 8: 51.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 6

   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. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. sub-flip-reverse N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. sub-flip-reverse N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate-rev N/A

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

      17. sub-negate-rev N/A

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

      18. add-flip N/A

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

      19. lift-+.f64 N/A

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

      20. lower-fma.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. add-flip N/A

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

      23. lower--.f64 N/A

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

      24. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. add-flip N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.7%

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

   if 6 < 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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 51.7

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

   4. Applied rewrites 51.7%

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

   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 4.1

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

   7. Applied rewrites 4.1%

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

   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 4.5

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

   10. Applied rewrites 4.5%

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

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-sqrt.f64 4.5

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

   13. Applied rewrites 4.5%

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

Alternative 9: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -0.2:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(-4, \sqrt{x}, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot \frac{x}{\sqrt{x}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -0.2)
   (/ 6.0 (fma -4.0 (sqrt x) -1.0))
   (* 1.5 (/ x (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -0.2) {
		tmp = 6.0 / fma(-4.0, sqrt(x), -1.0);
	} else {
		tmp = 1.5 * (x / sqrt(x));
	}
	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)))) <= -0.2)
		tmp = Float64(6.0 / fma(-4.0, sqrt(x), -1.0));
	else
		tmp = Float64(1.5 * Float64(x / sqrt(x)));
	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], -0.2], N[(6.0 / N[(-4.0 * N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(1.5 * N[(x / N[Sqrt[x], $MachinePrecision]), $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)))) < -0.20000000000000001

   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 48.5

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

   4. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   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. lower-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval 48.5

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

   6. Applied rewrites 48.5%

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

   if -0.20000000000000001 < (/.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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 51.7

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

   4. Applied rewrites 51.7%

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

   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 4.1

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

   7. Applied rewrites 4.1%

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

   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 4.5

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

   10. Applied rewrites 4.5%

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

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-sqrt.f64 4.5

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

   13. Applied rewrites 4.5%

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

Alternative 10: 7.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = sqrt(x) * -1.5;
	} else {
		tmp = 1.5 * (x / sqrt(x));
	}
	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 = sqrt(x) * (-1.5d0)
    else
        tmp = 1.5d0 * (x / sqrt(x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -1.5), $MachinePrecision], N[(1.5 * N[(x / N[Sqrt[x], $MachinePrecision]), $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. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 51.7

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

   4. Applied rewrites 51.7%

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

   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 4.1

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

   7. Applied rewrites 4.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-flip N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-*.f64 4.1

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

   9. Applied rewrites 4.1%

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

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 51.7

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

   4. Applied rewrites 51.7%

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

   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 4.1

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

   7. Applied rewrites 4.1%

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

   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 4.5

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

   10. Applied rewrites 4.5%

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

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-sqrt.f64 4.5

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

   13. Applied rewrites 4.5%

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

Alternative 11: 4.1% accurate, 3.5× speedup

Math

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

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

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 51.7

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

4. Applied rewrites 51.7%

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

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 4.1

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

7. Applied rewrites 4.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. lift-sqrt.f64 N/A

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

   7. pow1/2 N/A

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

   8. pow-flip N/A

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

   9. pow-plus N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. pow1/2 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lower-*.f64 4.1

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

9. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2025172 
(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)))))