Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.8% → 99.9%

Time: 8.9s

Alternatives: 7

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: 99.8% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ -6 \cdot \sqrt[3]{{\left(\frac{-1 + x}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right) - x}\right)}^{3}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* -6.0 (cbrt (pow (/ (+ -1.0 x) (- (fma (sqrt x) -4.0 -1.0) x)) 3.0))))

C

double code(double x) {
	return -6.0 * cbrt(pow(((-1.0 + x) / (fma(sqrt(x), -4.0, -1.0) - x)), 3.0));
}

Julia

function code(x)
	return Float64(-6.0 * cbrt((Float64(Float64(-1.0 + x) / Float64(fma(sqrt(x), -4.0, -1.0) - x)) ^ 3.0)))
end

Wolfram

code[x_] := N[(-6.0 * N[Power[N[Power[N[(N[(-1.0 + x), $MachinePrecision] / N[(N[(N[Sqrt[x], $MachinePrecision] * -4.0 + -1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. metadata-eval 99.9%

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

   5. sub-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. +-commutative 99.9%

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

   8. fma-def 99.9%

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

   9. sub-neg 99.9%

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

   10. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. associate-/r/ 99.8%

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

   2. associate-*l/ 99.9%

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

   3. *-lft-identity 99.9%

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

   4. +-commutative 99.9%

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

   5. neg-sub0 99.9%

      \[\leadsto -6 \cdot \frac{-1 + x}{\color{blue}{0 - \mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

   6. fma-udef 99.9%

      \[\leadsto -6 \cdot \frac{-1 + x}{0 - \color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)}} \]

   7. associate--r+ 99.9%

      \[\leadsto -6 \cdot \frac{-1 + x}{\color{blue}{\left(0 - 4 \cdot \sqrt{x}\right) - \left(x + 1\right)}} \]

   8. neg-sub0 99.9%

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

   9. distribute-lft-neg-in 99.9%

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

   10. metadata-eval 99.9%

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

   11. *-commutative 99.9%

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

   12. +-commutative 99.9%

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

7. Simplified 99.9%

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

   1. +-commutative 99.9%

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

   2. metadata-eval 99.9%

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

   3. sub-neg 99.9%

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

   4. add-cbrt-cube 99.9%

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

   5. pow3 99.9%

      \[\leadsto -6 \cdot \sqrt[3]{\color{blue}{{\left(\frac{x - 1}{\sqrt{x} \cdot -4 - \left(1 + x\right)}\right)}^{3}}} \]

   6. sub-neg 99.9%

      \[\leadsto -6 \cdot \sqrt[3]{{\left(\frac{\color{blue}{x + \left(-1\right)}}{\sqrt{x} \cdot -4 - \left(1 + x\right)}\right)}^{3}} \]

   7. metadata-eval 99.9%

      \[\leadsto -6 \cdot \sqrt[3]{{\left(\frac{x + \color{blue}{-1}}{\sqrt{x} \cdot -4 - \left(1 + x\right)}\right)}^{3}} \]

   8. +-commutative 99.9%

      \[\leadsto -6 \cdot \sqrt[3]{{\left(\frac{\color{blue}{-1 + x}}{\sqrt{x} \cdot -4 - \left(1 + x\right)}\right)}^{3}} \]

   9. associate--r+ 99.9%

      \[\leadsto -6 \cdot \sqrt[3]{{\left(\frac{-1 + x}{\color{blue}{\left(\sqrt{x} \cdot -4 - 1\right) - x}}\right)}^{3}} \]

   10. fma-neg 99.9%

       \[\leadsto -6 \cdot \sqrt[3]{{\left(\frac{-1 + x}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)} - x}\right)}^{3}} \]

   11. metadata-eval 99.9%

       \[\leadsto -6 \cdot \sqrt[3]{{\left(\frac{-1 + x}{\mathsf{fma}\left(\sqrt{x}, -4, \color{blue}{-1}\right) - x}\right)}^{3}} \]

9. Applied egg-rr 99.9%

   \[\leadsto -6 \cdot \color{blue}{\sqrt[3]{{\left(\frac{-1 + x}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right) - x}\right)}^{3}}} \]

10. Final simplification 99.9%

    \[\leadsto -6 \cdot \sqrt[3]{{\left(\frac{-1 + x}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right) - x}\right)}^{3}} \]

Alternative 2: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;x \cdot 6 - 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\left(x + 1\right) + \sqrt{x} \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (- (* x 6.0) 6.0)
   (/ (* x 6.0) (+ (+ x 1.0) (* (sqrt x) 4.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = (x * 6.0) - 6.0;
	} else {
		tmp = (x * 6.0) / ((x + 1.0) + (sqrt(x) * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = (x * 6.0d0) - 6.0d0
    else
        tmp = (x * 6.0d0) / ((x + 1.0d0) + (sqrt(x) * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = (x * 6.0) - 6.0;
	} else {
		tmp = (x * 6.0) / ((x + 1.0) + (Math.sqrt(x) * 4.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = (x * 6.0) - 6.0
	else:
		tmp = (x * 6.0) / ((x + 1.0) + (math.sqrt(x) * 4.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64(Float64(x * 6.0) - 6.0);
	else
		tmp = Float64(Float64(x * 6.0) / Float64(Float64(x + 1.0) + Float64(sqrt(x) * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = (x * 6.0) - 6.0;
	else
		tmp = (x * 6.0) / ((x + 1.0) + (sqrt(x) * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(N[(x * 6.0), $MachinePrecision] - 6.0), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   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. metadata-eval 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-udef 99.9%

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

      9. frac-2neg 99.9%

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

      10. associate-+l+ 99.9%

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

      11. +-commutative 99.9%

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

      12. fma-udef 99.9%

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

      13. div-inv 99.9%

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

      14. neg-sub0 99.9%

          \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(6, x, -6\right)\right)} \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      15. fma-udef 99.9%

          \[\leadsto \left(0 - \color{blue}{\left(6 \cdot x + -6\right)}\right) \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      16. +-commutative 99.9%

          \[\leadsto \left(0 - \color{blue}{\left(-6 + 6 \cdot x\right)}\right) \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      17. associate--r+ 99.9%

          \[\leadsto \color{blue}{\left(\left(0 - -6\right) - 6 \cdot x\right)} \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      18. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. sqrt-unprod 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. swap-sqr 99.9%

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

      8. metadata-eval 99.9%

         \[\leadsto \left(6 - 6 \cdot x\right) \cdot \frac{1}{-\left(\left(x + 1\right) + \sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{16}}\right)} \]

      9. metadata-eval 99.9%

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

      10. swap-sqr 99.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 96.0%

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

      13. *-commutative 96.0%

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

      14. metadata-eval 96.0%

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

      15. cancel-sign-sub-inv 96.0%

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

      16. add-sqr-sqrt 96.0%

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

      17. sqrt-unprod 96.0%

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

      18. *-commutative 96.0%

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

      19. *-commutative 96.0%

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

      20. swap-sqr 96.0%

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

      21. metadata-eval 96.0%

          \[\leadsto \left(6 - 6 \cdot x\right) \cdot \frac{1}{-\left(\left(x + 1\right) - \sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{16}}\right)} \]

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 96.2%

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

   if 1.19999999999999996 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 97.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;x \cdot 6 - 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\left(x + 1\right) + \sqrt{x} \cdot 4}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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. metadata-eval 99.9%

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

   5. sub-neg 99.9%

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

   6. distribute-neg-frac 99.9%

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

   7. +-commutative 99.9%

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

   8. fma-def 99.9%

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

   9. sub-neg 99.9%

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

   10. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. associate-/r/ 99.8%

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

   2. associate-*l/ 99.9%

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

   3. *-lft-identity 99.9%

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

   4. +-commutative 99.9%

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

   5. neg-sub0 99.9%

      \[\leadsto -6 \cdot \frac{-1 + x}{\color{blue}{0 - \mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}} \]

   6. fma-udef 99.9%

      \[\leadsto -6 \cdot \frac{-1 + x}{0 - \color{blue}{\left(4 \cdot \sqrt{x} + \left(x + 1\right)\right)}} \]

   7. associate--r+ 99.9%

      \[\leadsto -6 \cdot \frac{-1 + x}{\color{blue}{\left(0 - 4 \cdot \sqrt{x}\right) - \left(x + 1\right)}} \]

   8. neg-sub0 99.9%

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

   9. distribute-lft-neg-in 99.9%

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

   10. metadata-eval 99.9%

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

   11. *-commutative 99.9%

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

   12. +-commutative 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 4: 95.3% 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 96.2%

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

   if 0.5 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 94.7%

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

   3. Taylor expanded in x around 0 95.5%

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

      1. associate-*r/ 95.5%

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

      2. metadata-eval 95.5%

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

   5. Simplified 95.5%

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

4. Final simplification 95.8%

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

Alternative 5: 95.3% accurate, 16.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x * 6.0) - 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 <= 1.0d0) then
        tmp = (x * 6.0d0) - 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 <= 1.0) {
		tmp = (x * 6.0) - 6.0;
	} else {
		tmp = 6.0 - (6.0 / x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x * 6.0) - 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 <= 1.0)
		tmp = (x * 6.0) - 6.0;
	else
		tmp = 6.0 - (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

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. Step-by-step derivation

      1. metadata-eval 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-udef 99.9%

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

      9. frac-2neg 99.9%

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

      10. associate-+l+ 99.9%

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

      11. +-commutative 99.9%

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

      12. fma-udef 99.9%

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

      13. div-inv 99.9%

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

      14. neg-sub0 99.9%

          \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(6, x, -6\right)\right)} \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      15. fma-udef 99.9%

          \[\leadsto \left(0 - \color{blue}{\left(6 \cdot x + -6\right)}\right) \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      16. +-commutative 99.9%

          \[\leadsto \left(0 - \color{blue}{\left(-6 + 6 \cdot x\right)}\right) \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      17. associate--r+ 99.9%

          \[\leadsto \color{blue}{\left(\left(0 - -6\right) - 6 \cdot x\right)} \cdot \frac{1}{-\left(x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)\right)} \]

      18. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. sqrt-unprod 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. swap-sqr 99.9%

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

      8. metadata-eval 99.9%

         \[\leadsto \left(6 - 6 \cdot x\right) \cdot \frac{1}{-\left(\left(x + 1\right) + \sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{16}}\right)} \]

      9. metadata-eval 99.9%

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

      10. swap-sqr 99.9%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 96.0%

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

      13. *-commutative 96.0%

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

      14. metadata-eval 96.0%

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

      15. cancel-sign-sub-inv 96.0%

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

      16. add-sqr-sqrt 96.0%

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

      17. sqrt-unprod 96.0%

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

      18. *-commutative 96.0%

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

      19. *-commutative 96.0%

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

      20. swap-sqr 96.0%

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

      21. metadata-eval 96.0%

          \[\leadsto \left(6 - 6 \cdot x\right) \cdot \frac{1}{-\left(\left(x + 1\right) - \sqrt{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{16}}\right)} \]

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0 96.2%

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

   if 1 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 94.7%

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

   3. Taylor expanded in x around 0 95.5%

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

      1. associate-*r/ 95.5%

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

      2. metadata-eval 95.5%

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

   5. Simplified 95.5%

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

4. Final simplification 95.8%

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

Alternative 6: 95.3% 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 96.2%

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

   if 1 < x

   1. Initial program 99.1%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   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 95.5%

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

4. Final simplification 95.8%

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

Alternative 7: 49.3% 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 99.5%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
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 48.9%

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

5. Final simplification 48.9%

   \[\leadsto -6 \]

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 2023301 
(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)))))