Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 7.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-/l* 99.9%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. frac-2neg 99.9%

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

   2. metadata-eval 99.9%

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

   3. div-inv 99.9%

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

   4. distribute-neg-frac 99.9%

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

   5. +-commutative 99.9%

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

   6. associate-+r+ 99.9%

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

   7. fma-udef 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. associate-/r/ 99.9%

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

   2. associate-*l/ 100.0%

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

   3. *-lft-identity 100.0%

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

   4. +-commutative 100.0%

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

   5. unsub-neg 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + 1\right) + \sqrt{x} \cdot -4\\ \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;\left(x \cdot -78 - 6\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (+ x 1.0) (* (sqrt x) -4.0))))
   (if (<= x 0.07) (* (- (* x -78.0) 6.0) t_0) (* t_0 (/ 6.0 x)))))

C

double code(double x) {
	double t_0 = (x + 1.0) + (sqrt(x) * -4.0);
	double tmp;
	if (x <= 0.07) {
		tmp = ((x * -78.0) - 6.0) * t_0;
	} else {
		tmp = t_0 * (6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) + (sqrt(x) * (-4.0d0))
    if (x <= 0.07d0) then
        tmp = ((x * (-78.0d0)) - 6.0d0) * t_0
    else
        tmp = t_0 * (6.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x + 1.0) + (Math.sqrt(x) * -4.0);
	double tmp;
	if (x <= 0.07) {
		tmp = ((x * -78.0) - 6.0) * t_0;
	} else {
		tmp = t_0 * (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x + 1.0) + (math.sqrt(x) * -4.0)
	tmp = 0
	if x <= 0.07:
		tmp = ((x * -78.0) - 6.0) * t_0
	else:
		tmp = t_0 * (6.0 / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x + 1.0) + Float64(sqrt(x) * -4.0))
	tmp = 0.0
	if (x <= 0.07)
		tmp = Float64(Float64(Float64(x * -78.0) - 6.0) * t_0);
	else
		tmp = Float64(t_0 * Float64(6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x + 1.0) + (sqrt(x) * -4.0);
	tmp = 0.0;
	if (x <= 0.07)
		tmp = ((x * -78.0) - 6.0) * t_0;
	else
		tmp = t_0 * (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.07], N[(N[(N[(x * -78.0), $MachinePrecision] - 6.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.070000000000000007

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. fma-udef 99.9%

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

      5. flip-+ 99.9%

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

      6. associate-/r/ 100.0%

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

      7. pow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      8. *-commutative 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot 4\right)} \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \left(\sqrt{x} \cdot 4\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      10. swap-sqr 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      11. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{x} \cdot \left(4 \cdot 4\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot \color{blue}{16}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      13. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(\left(x + 1\right) + \left(-4\right) \cdot \sqrt{x}\right)} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + \color{blue}{-4} \cdot \sqrt{x}\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{\left(-78 \cdot x - 6\right)} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right) \]

   if 0.070000000000000007 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l* 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. metadata-eval 98.9%

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

      4. fma-udef 98.9%

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

      5. flip-+ 45.2%

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

      6. associate-/r/ 45.0%

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

      7. pow2 45.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      8. *-commutative 45.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot 4\right)} \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      9. *-commutative 45.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \left(\sqrt{x} \cdot 4\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      10. swap-sqr 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      11. add-sqr-sqrt 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{x} \cdot \left(4 \cdot 4\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      12. metadata-eval 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot \color{blue}{16}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      13. cancel-sign-sub-inv 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(\left(x + 1\right) + \left(-4\right) \cdot \sqrt{x}\right)} \]

      14. metadata-eval 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + \color{blue}{-4} \cdot \sqrt{x}\right) \]

   5. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]

   6. Taylor expanded in x around inf 97.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;\left(x \cdot -78 - 6\right) \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) + \sqrt{x} \cdot -4\right) \cdot \frac{6}{x}\\ \end{array} \]

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + 1\right) + \sqrt{x} \cdot -4\\ \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;-6 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (+ x 1.0) (* (sqrt x) -4.0))))
   (if (<= x 0.07) (* -6.0 t_0) (* t_0 (/ 6.0 x)))))

C

double code(double x) {
	double t_0 = (x + 1.0) + (sqrt(x) * -4.0);
	double tmp;
	if (x <= 0.07) {
		tmp = -6.0 * t_0;
	} else {
		tmp = t_0 * (6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) + (sqrt(x) * (-4.0d0))
    if (x <= 0.07d0) then
        tmp = (-6.0d0) * t_0
    else
        tmp = t_0 * (6.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x + 1.0) + (Math.sqrt(x) * -4.0);
	double tmp;
	if (x <= 0.07) {
		tmp = -6.0 * t_0;
	} else {
		tmp = t_0 * (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x + 1.0) + (math.sqrt(x) * -4.0)
	tmp = 0
	if x <= 0.07:
		tmp = -6.0 * t_0
	else:
		tmp = t_0 * (6.0 / x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x + 1.0) + Float64(sqrt(x) * -4.0))
	tmp = 0.0
	if (x <= 0.07)
		tmp = Float64(-6.0 * t_0);
	else
		tmp = Float64(t_0 * Float64(6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x + 1.0) + (sqrt(x) * -4.0);
	tmp = 0.0;
	if (x <= 0.07)
		tmp = -6.0 * t_0;
	else
		tmp = t_0 * (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.07], N[(-6.0 * t$95$0), $MachinePrecision], N[(t$95$0 * N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.070000000000000007

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. fma-udef 99.9%

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

      5. flip-+ 99.9%

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

      6. associate-/r/ 100.0%

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

      7. pow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      8. *-commutative 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot 4\right)} \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \left(\sqrt{x} \cdot 4\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      10. swap-sqr 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      11. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{x} \cdot \left(4 \cdot 4\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot \color{blue}{16}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      13. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(\left(x + 1\right) + \left(-4\right) \cdot \sqrt{x}\right)} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + \color{blue}{-4} \cdot \sqrt{x}\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0 99.1%

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

   if 0.070000000000000007 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l* 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. metadata-eval 98.9%

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

      4. fma-udef 98.9%

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

      5. flip-+ 45.2%

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

      6. associate-/r/ 45.0%

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

      7. pow2 45.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      8. *-commutative 45.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot 4\right)} \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      9. *-commutative 45.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \left(\sqrt{x} \cdot 4\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      10. swap-sqr 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      11. add-sqr-sqrt 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{x} \cdot \left(4 \cdot 4\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      12. metadata-eval 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot \color{blue}{16}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      13. cancel-sign-sub-inv 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(\left(x + 1\right) + \left(-4\right) \cdot \sqrt{x}\right)} \]

      14. metadata-eval 45.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + \color{blue}{-4} \cdot \sqrt{x}\right) \]

   5. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]

   6. Taylor expanded in x around inf 97.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.07:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + 1\right) + \sqrt{x} \cdot -4\right) \cdot \frac{6}{x}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-/l* 99.9%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.068:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{-1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.068)
   (* -6.0 (+ (+ x 1.0) (* (sqrt x) -4.0)))
   (/ 6.0 (/ x (+ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.068) {
		tmp = -6.0 * ((x + 1.0) + (sqrt(x) * -4.0));
	} else {
		tmp = 6.0 / (x / (-1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.068d0) then
        tmp = (-6.0d0) * ((x + 1.0d0) + (sqrt(x) * (-4.0d0)))
    else
        tmp = 6.0d0 / (x / ((-1.0d0) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.068) {
		tmp = -6.0 * ((x + 1.0) + (Math.sqrt(x) * -4.0));
	} else {
		tmp = 6.0 / (x / (-1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.068:
		tmp = -6.0 * ((x + 1.0) + (math.sqrt(x) * -4.0))
	else:
		tmp = 6.0 / (x / (-1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.068)
		tmp = Float64(-6.0 * Float64(Float64(x + 1.0) + Float64(sqrt(x) * -4.0)));
	else
		tmp = Float64(6.0 / Float64(x / Float64(-1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.068)
		tmp = -6.0 * ((x + 1.0) + (sqrt(x) * -4.0));
	else
		tmp = 6.0 / (x / (-1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.068], N[(-6.0 * N[(N[(x + 1.0), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.068000000000000005

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. fma-udef 99.9%

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

      5. flip-+ 99.9%

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

      6. associate-/r/ 100.0%

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

      7. pow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{\color{blue}{{\left(x + 1\right)}^{2}} - \left(4 \cdot \sqrt{x}\right) \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      8. *-commutative 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot 4\right)} \cdot \left(4 \cdot \sqrt{x}\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \left(\sqrt{x} \cdot 4\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      10. swap-sqr 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(4 \cdot 4\right)}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      11. add-sqr-sqrt 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - \color{blue}{x} \cdot \left(4 \cdot 4\right)} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot \color{blue}{16}} \cdot \left(\left(x + 1\right) - 4 \cdot \sqrt{x}\right) \]

      13. cancel-sign-sub-inv 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \color{blue}{\left(\left(x + 1\right) + \left(-4\right) \cdot \sqrt{x}\right)} \]

      14. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + \color{blue}{-4} \cdot \sqrt{x}\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(6, x, -6\right)}{{\left(x + 1\right)}^{2} - x \cdot 16} \cdot \left(\left(x + 1\right) + -4 \cdot \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0 99.1%

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

   if 0.068000000000000005 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 96.0%

      \[\leadsto \frac{6}{\frac{\color{blue}{x}}{x + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.068:\\ \;\;\;\;-6 \cdot \left(\left(x + 1\right) + \sqrt{x} \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{-1 + x}}\\ \end{array} \]

Alternative 6: 95.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{-1 + x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.5) -6.0 (/ 6.0 (/ x (+ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 / (x / (-1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0 / (x / ((-1.0d0) + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 / (x / (-1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = -6.0
	else:
		tmp = 6.0 / (x / (-1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = Float64(6.0 / Float64(x / Float64(-1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = 6.0 / (x / (-1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], -6.0, N[(6.0 / N[(x / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 96.3%

      \[\leadsto \color{blue}{-6} \]

   if 0.5 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 96.6%

      \[\leadsto \frac{6}{\frac{\color{blue}{x}}{x + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{x}{-1 + x}}\\ \end{array} \]

Alternative 7: 95.4% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{6}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.5) -6.0 (- 6.0 (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0 - (6.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = -6.0;
	} else {
		tmp = 6.0 - (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = -6.0
	else:
		tmp = 6.0 - (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = Float64(6.0 - Float64(6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = -6.0;
	else
		tmp = 6.0 - (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], -6.0, N[(6.0 - N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 96.3%

      \[\leadsto \color{blue}{-6} \]

   if 0.5 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 96.6%

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

   5. Taylor expanded in x around 0 96.6%

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

      1. associate-*r/ 96.6%

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

      2. metadata-eval 96.6%

         \[\leadsto 6 - \frac{\color{blue}{6}}{x} \]

   7. Simplified 96.6%

      \[\leadsto \color{blue}{6 - \frac{6}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6 - \frac{6}{x}\\ \end{array} \]

Alternative 8: 95.4% accurate, 36.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) -6.0 6.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = -6.0d0
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -6.0
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -6.0;
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], -6.0, 6.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 96.3%

      \[\leadsto \color{blue}{-6} \]

   if 1 < x

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-6\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

Alternative 9: 48.7% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ -6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -6.0)

C

double code(double x) {
	return -6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -6.0d0
end function

Java

public static double code(double x) {
	return -6.0;
}

Python

def code(x):
	return -6.0

Julia

function code(x)
	return -6.0
end

MATLAB

function tmp = code(x)
	tmp = -6.0;
end

Wolfram

code[x_] := -6.0

Derivation

1. Initial program 99.4%

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

   1. associate-/l* 99.9%

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

   2. sub-neg 99.9%

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

   3. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 50.0%

   \[\leadsto \color{blue}{-6} \]

5. Final simplification 50.0%

   \[\leadsto -6 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023299 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))