Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Percentage Accurate: 99.7% → 99.8%

Time: 6.3s

Alternatives: 7

Speedup: 1.3×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 * (2.0d0 - (x * 3.0d0))) * x
end function

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 * (2.0d0 - (x * 3.0d0))) * x
end function

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 6.0 (* -9.0 (pow x 2.0))))

C

double code(double x) {
	return fma(x, 6.0, (-9.0 * pow(x, 2.0)));
}

Julia

function code(x)
	return fma(x, 6.0, Float64(-9.0 * (x ^ 2.0)))
end

Wolfram

code[x_] := N[(x * 6.0 + N[(-9.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

   3. distribute-rgt-in 99.7%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-lft-in 99.8%

      \[\leadsto \color{blue}{x \cdot 6 + x \cdot \left(x \cdot -9\right)} \]

   2. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, x \cdot \left(x \cdot -9\right)\right)} \]

   3. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{\left(x \cdot -9\right) \cdot x}\right) \]

   4. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{\left(-9 \cdot x\right)} \cdot x\right) \]

   5. associate-*l* 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{-9 \cdot \left(x \cdot x\right)}\right) \]

   6. pow2 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, -9 \cdot \color{blue}{{x}^{2}}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right)} \]

7. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right) \]
8. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \frac{x}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.66)
   (* x (* x -9.0))
   (if (<= x 0.66) (* x 6.0) (* -9.0 (/ x (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.66) {
		tmp = x * (x * -9.0);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = -9.0 * (x / (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.66d0)) then
        tmp = x * (x * (-9.0d0))
    else if (x <= 0.66d0) then
        tmp = x * 6.0d0
    else
        tmp = (-9.0d0) * (x / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.66) {
		tmp = x * (x * -9.0);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = -9.0 * (x / (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.66:
		tmp = x * (x * -9.0)
	elif x <= 0.66:
		tmp = x * 6.0
	else:
		tmp = -9.0 * (x / (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.66)
		tmp = Float64(x * Float64(x * -9.0));
	elseif (x <= 0.66)
		tmp = Float64(x * 6.0);
	else
		tmp = Float64(-9.0 * Float64(x / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = x * (x * -9.0);
	elseif (x <= 0.66)
		tmp = x * 6.0;
	else
		tmp = -9.0 * (x / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.66], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.66], N[(x * 6.0), $MachinePrecision], N[(-9.0 * N[(x / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.660000000000000031

   1. Initial program 99.9%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. cancel-sign-sub-inv 99.9%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.9%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. associate-*l* 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.2%

      \[\leadsto x \cdot \color{blue}{\left(-9 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   7. Simplified 97.2%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   if -0.660000000000000031 < x < 0.660000000000000031

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. associate-*l* 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 96.3%

      \[\leadsto x \cdot \color{blue}{6} \]

   if 0.660000000000000031 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.6%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.6%

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

      5. distribute-lft-neg-out 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. associate-*l* 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\left(6 + x \cdot -9\right) \cdot x} \]

      2. flip-+ 98.1%

         \[\leadsto \color{blue}{\frac{6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}{6 - x \cdot -9}} \cdot x \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{\left(6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9}} \]

      4. metadata-eval 83.2%

         \[\leadsto \frac{\left(\color{blue}{36} - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

      5. swap-sqr 83.2%

         \[\leadsto \frac{\left(36 - \color{blue}{\left(x \cdot x\right) \cdot \left(-9 \cdot -9\right)}\right) \cdot x}{6 - x \cdot -9} \]

      6. pow2 83.2%

         \[\leadsto \frac{\left(36 - \color{blue}{{x}^{2}} \cdot \left(-9 \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

      7. metadata-eval 83.2%

         \[\leadsto \frac{\left(36 - {x}^{2} \cdot \color{blue}{81}\right) \cdot x}{6 - x \cdot -9} \]

      8. *-commutative 83.2%

         \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 - \color{blue}{-9 \cdot x}} \]

      9. cancel-sign-sub-inv 83.2%

         \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{\color{blue}{6 + \left(--9\right) \cdot x}} \]

      10. metadata-eval 83.2%

          \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + \color{blue}{9} \cdot x} \]

   6. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + 9 \cdot x}} \]
   7. Step-by-step derivation

      1. frac-2neg 83.2%

         \[\leadsto \color{blue}{\frac{-\left(36 - {x}^{2} \cdot 81\right) \cdot x}{-\left(6 + 9 \cdot x\right)}} \]

      2. div-inv 83.2%

         \[\leadsto \color{blue}{\left(-\left(36 - {x}^{2} \cdot 81\right) \cdot x\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)}} \]

      3. *-commutative 83.2%

         \[\leadsto \left(-\color{blue}{x \cdot \left(36 - {x}^{2} \cdot 81\right)}\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      4. distribute-rgt-neg-in 83.2%

         \[\leadsto \color{blue}{\left(x \cdot \left(-\left(36 - {x}^{2} \cdot 81\right)\right)\right)} \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      5. sub-neg 83.2%

         \[\leadsto \left(x \cdot \left(-\color{blue}{\left(36 + \left(-{x}^{2} \cdot 81\right)\right)}\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      6. distribute-rgt-neg-in 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + \color{blue}{{x}^{2} \cdot \left(-81\right)}\right)\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      7. metadata-eval 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot \color{blue}{-81}\right)\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      8. +-commutative 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\color{blue}{\left(9 \cdot x + 6\right)}} \]

      9. *-commutative 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\left(\color{blue}{x \cdot 9} + 6\right)} \]

      10. fma-def 83.2%

          \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\color{blue}{\mathsf{fma}\left(x, 9, 6\right)}} \]

   8. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(x, 9, 6\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.2%

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot 1}{-\mathsf{fma}\left(x, 9, 6\right)}} \]

      2. *-rgt-identity 83.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)}}{-\mathsf{fma}\left(x, 9, 6\right)} \]

      3. associate-/l* 98.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{-\mathsf{fma}\left(x, 9, 6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}}} \]

      4. fma-udef 98.0%

         \[\leadsto \frac{x}{\frac{-\color{blue}{\left(x \cdot 9 + 6\right)}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      5. distribute-neg-in 98.0%

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(-x \cdot 9\right) + \left(-6\right)}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      6. *-commutative 98.0%

         \[\leadsto \frac{x}{\frac{\left(-\color{blue}{9 \cdot x}\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      7. metadata-eval 98.0%

         \[\leadsto \frac{x}{\frac{\left(-\color{blue}{\left(--9\right)} \cdot x\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      8. distribute-lft-neg-in 98.0%

         \[\leadsto \frac{x}{\frac{\left(-\color{blue}{\left(--9 \cdot x\right)}\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      9. remove-double-neg 98.0%

         \[\leadsto \frac{x}{\frac{\color{blue}{-9 \cdot x} + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      10. *-commutative 98.0%

          \[\leadsto \frac{x}{\frac{\color{blue}{x \cdot -9} + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      11. metadata-eval 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + \color{blue}{-6}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      12. +-commutative 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{-\color{blue}{\left({x}^{2} \cdot -81 + 36\right)}}} \]

      13. distribute-neg-in 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{\left(-{x}^{2} \cdot -81\right) + \left(-36\right)}}} \]

      14. distribute-rgt-neg-in 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{{x}^{2} \cdot \left(--81\right)} + \left(-36\right)}} \]

      15. metadata-eval 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{{x}^{2} \cdot \color{blue}{81} + \left(-36\right)}} \]

      16. fma-def 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{\mathsf{fma}\left({x}^{2}, 81, -36\right)}}} \]

      17. metadata-eval 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\mathsf{fma}\left({x}^{2}, 81, \color{blue}{-36}\right)}} \]

   10. Simplified 98.0%

       \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot -9 + -6}{\mathsf{fma}\left({x}^{2}, 81, -36\right)}}} \]

   11. Taylor expanded in x around inf 97.3%

       \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 97.3%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{-0.1111111111111111}{x}} \]

       2. div-inv 97.2%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{-0.1111111111111111 \cdot \frac{1}{x}}} \]

       3. times-frac 97.3%

          \[\leadsto \color{blue}{\frac{1}{-0.1111111111111111} \cdot \frac{x}{\frac{1}{x}}} \]

       4. metadata-eval 97.3%

          \[\leadsto \color{blue}{-9} \cdot \frac{x}{\frac{1}{x}} \]

   13. Applied egg-rr 97.3%

       \[\leadsto \color{blue}{-9 \cdot \frac{x}{\frac{1}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;-9 \cdot \frac{x}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.66\right):\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.66) (not (<= x 0.66))) (* x (* x -9.0)) (* x 6.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.66) || !(x <= 0.66)) {
		tmp = x * (x * -9.0);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.66d0)) .or. (.not. (x <= 0.66d0))) then
        tmp = x * (x * (-9.0d0))
    else
        tmp = x * 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.66) || !(x <= 0.66)) {
		tmp = x * (x * -9.0);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.66) or not (x <= 0.66):
		tmp = x * (x * -9.0)
	else:
		tmp = x * 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.66) || !(x <= 0.66))
		tmp = Float64(x * Float64(x * -9.0));
	else
		tmp = Float64(x * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.66) || ~((x <= 0.66)))
		tmp = x * (x * -9.0);
	else
		tmp = x * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.66], N[Not[LessEqual[x, 0.66]], $MachinePrecision]], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision], N[(x * 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031 or 0.660000000000000031 < x

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. associate-*l* 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.2%

      \[\leadsto x \cdot \color{blue}{\left(-9 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   7. Simplified 97.2%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   if -0.660000000000000031 < x < 0.660000000000000031

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. associate-*l* 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 96.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.66\right):\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{-0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.66)
   (* x (* x -9.0))
   (if (<= x 0.66) (* x 6.0) (/ x (/ -0.1111111111111111 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.66) {
		tmp = x * (x * -9.0);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x / (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.66d0)) then
        tmp = x * (x * (-9.0d0))
    else if (x <= 0.66d0) then
        tmp = x * 6.0d0
    else
        tmp = x / ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.66) {
		tmp = x * (x * -9.0);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x / (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.66:
		tmp = x * (x * -9.0)
	elif x <= 0.66:
		tmp = x * 6.0
	else:
		tmp = x / (-0.1111111111111111 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.66)
		tmp = Float64(x * Float64(x * -9.0));
	elseif (x <= 0.66)
		tmp = Float64(x * 6.0);
	else
		tmp = Float64(x / Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = x * (x * -9.0);
	elseif (x <= 0.66)
		tmp = x * 6.0;
	else
		tmp = x / (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.66], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.66], N[(x * 6.0), $MachinePrecision], N[(x / N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.660000000000000031

   1. Initial program 99.9%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. cancel-sign-sub-inv 99.9%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.9%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. associate-*l* 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.2%

      \[\leadsto x \cdot \color{blue}{\left(-9 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   7. Simplified 97.2%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   if -0.660000000000000031 < x < 0.660000000000000031

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

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

      5. distribute-lft-neg-out 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. associate-*l* 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 96.3%

      \[\leadsto x \cdot \color{blue}{6} \]

   if 0.660000000000000031 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.6%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.6%

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

      5. distribute-lft-neg-out 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. associate-*l* 99.7%

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

      8. metadata-eval 99.7%

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

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\left(6 + x \cdot -9\right) \cdot x} \]

      2. flip-+ 98.1%

         \[\leadsto \color{blue}{\frac{6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}{6 - x \cdot -9}} \cdot x \]

      3. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{\left(6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9}} \]

      4. metadata-eval 83.2%

         \[\leadsto \frac{\left(\color{blue}{36} - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

      5. swap-sqr 83.2%

         \[\leadsto \frac{\left(36 - \color{blue}{\left(x \cdot x\right) \cdot \left(-9 \cdot -9\right)}\right) \cdot x}{6 - x \cdot -9} \]

      6. pow2 83.2%

         \[\leadsto \frac{\left(36 - \color{blue}{{x}^{2}} \cdot \left(-9 \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

      7. metadata-eval 83.2%

         \[\leadsto \frac{\left(36 - {x}^{2} \cdot \color{blue}{81}\right) \cdot x}{6 - x \cdot -9} \]

      8. *-commutative 83.2%

         \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 - \color{blue}{-9 \cdot x}} \]

      9. cancel-sign-sub-inv 83.2%

         \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{\color{blue}{6 + \left(--9\right) \cdot x}} \]

      10. metadata-eval 83.2%

          \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + \color{blue}{9} \cdot x} \]

   6. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + 9 \cdot x}} \]
   7. Step-by-step derivation

      1. frac-2neg 83.2%

         \[\leadsto \color{blue}{\frac{-\left(36 - {x}^{2} \cdot 81\right) \cdot x}{-\left(6 + 9 \cdot x\right)}} \]

      2. div-inv 83.2%

         \[\leadsto \color{blue}{\left(-\left(36 - {x}^{2} \cdot 81\right) \cdot x\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)}} \]

      3. *-commutative 83.2%

         \[\leadsto \left(-\color{blue}{x \cdot \left(36 - {x}^{2} \cdot 81\right)}\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      4. distribute-rgt-neg-in 83.2%

         \[\leadsto \color{blue}{\left(x \cdot \left(-\left(36 - {x}^{2} \cdot 81\right)\right)\right)} \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      5. sub-neg 83.2%

         \[\leadsto \left(x \cdot \left(-\color{blue}{\left(36 + \left(-{x}^{2} \cdot 81\right)\right)}\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      6. distribute-rgt-neg-in 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + \color{blue}{{x}^{2} \cdot \left(-81\right)}\right)\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      7. metadata-eval 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot \color{blue}{-81}\right)\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

      8. +-commutative 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\color{blue}{\left(9 \cdot x + 6\right)}} \]

      9. *-commutative 83.2%

         \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\left(\color{blue}{x \cdot 9} + 6\right)} \]

      10. fma-def 83.2%

          \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\color{blue}{\mathsf{fma}\left(x, 9, 6\right)}} \]

   8. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(x, 9, 6\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.2%

         \[\leadsto \color{blue}{\frac{\left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot 1}{-\mathsf{fma}\left(x, 9, 6\right)}} \]

      2. *-rgt-identity 83.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)}}{-\mathsf{fma}\left(x, 9, 6\right)} \]

      3. associate-/l* 98.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{-\mathsf{fma}\left(x, 9, 6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}}} \]

      4. fma-udef 98.0%

         \[\leadsto \frac{x}{\frac{-\color{blue}{\left(x \cdot 9 + 6\right)}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      5. distribute-neg-in 98.0%

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(-x \cdot 9\right) + \left(-6\right)}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      6. *-commutative 98.0%

         \[\leadsto \frac{x}{\frac{\left(-\color{blue}{9 \cdot x}\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      7. metadata-eval 98.0%

         \[\leadsto \frac{x}{\frac{\left(-\color{blue}{\left(--9\right)} \cdot x\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      8. distribute-lft-neg-in 98.0%

         \[\leadsto \frac{x}{\frac{\left(-\color{blue}{\left(--9 \cdot x\right)}\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      9. remove-double-neg 98.0%

         \[\leadsto \frac{x}{\frac{\color{blue}{-9 \cdot x} + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      10. *-commutative 98.0%

          \[\leadsto \frac{x}{\frac{\color{blue}{x \cdot -9} + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      11. metadata-eval 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + \color{blue}{-6}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

      12. +-commutative 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{-\color{blue}{\left({x}^{2} \cdot -81 + 36\right)}}} \]

      13. distribute-neg-in 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{\left(-{x}^{2} \cdot -81\right) + \left(-36\right)}}} \]

      14. distribute-rgt-neg-in 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{{x}^{2} \cdot \left(--81\right)} + \left(-36\right)}} \]

      15. metadata-eval 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{{x}^{2} \cdot \color{blue}{81} + \left(-36\right)}} \]

      16. fma-def 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{\mathsf{fma}\left({x}^{2}, 81, -36\right)}}} \]

      17. metadata-eval 98.0%

          \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\mathsf{fma}\left({x}^{2}, 81, \color{blue}{-36}\right)}} \]

   10. Simplified 98.0%

       \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot -9 + -6}{\mathsf{fma}\left({x}^{2}, 81, -36\right)}}} \]

   11. Taylor expanded in x around inf 97.3%

       \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{-0.1111111111111111}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(6 + x \cdot -9\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 6.0 (* x -9.0))))

C

double code(double x) {
	return x * (6.0 + (x * -9.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (6.0d0 + (x * (-9.0d0)))
end function

Java

public static double code(double x) {
	return x * (6.0 + (x * -9.0));
}

Python

def code(x):
	return x * (6.0 + (x * -9.0))

Julia

function code(x)
	return Float64(x * Float64(6.0 + Float64(x * -9.0)))
end

MATLAB

function tmp = code(x)
	tmp = x * (6.0 + (x * -9.0));
end

Wolfram

code[x_] := N[(x * N[(6.0 + N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

   3. distribute-rgt-in 99.7%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto x \cdot \left(6 + x \cdot -9\right) \]
6. Add Preprocessing

Alternative 6: 50.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 6.0))

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 6.0

Julia

function code(x)
	return Float64(x * 6.0)
end

MATLAB

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

Wolfram

code[x_] := N[(x * 6.0), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

   3. distribute-rgt-in 99.7%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.3%

   \[\leadsto x \cdot \color{blue}{6} \]

6. Final simplification 52.3%

   \[\leadsto x \cdot 6 \]
7. Add Preprocessing

Alternative 7: 2.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x) :precision binary64 4.0)

C

double code(double x) {
	return 4.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 4.0d0
end function

Java

public static double code(double x) {
	return 4.0;
}

Python

def code(x):
	return 4.0

Julia

function code(x)
	return 4.0
end

MATLAB

function tmp = code(x)
	tmp = 4.0;
end

Wolfram

code[x_] := 4.0

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. cancel-sign-sub-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

   3. distribute-rgt-in 99.7%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

   4. metadata-eval 99.7%

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

   5. distribute-lft-neg-out 99.7%

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

   6. distribute-rgt-neg-in 99.7%

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

   7. associate-*l* 99.8%

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

   8. metadata-eval 99.8%

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

   9. metadata-eval 99.8%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(6 + x \cdot -9\right) \cdot x} \]

   2. flip-+ 99.4%

      \[\leadsto \color{blue}{\frac{6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}{6 - x \cdot -9}} \cdot x \]

   3. associate-*l/ 93.8%

      \[\leadsto \color{blue}{\frac{\left(6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9}} \]

   4. metadata-eval 93.8%

      \[\leadsto \frac{\left(\color{blue}{36} - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

   5. swap-sqr 93.8%

      \[\leadsto \frac{\left(36 - \color{blue}{\left(x \cdot x\right) \cdot \left(-9 \cdot -9\right)}\right) \cdot x}{6 - x \cdot -9} \]

   6. pow2 93.8%

      \[\leadsto \frac{\left(36 - \color{blue}{{x}^{2}} \cdot \left(-9 \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

   7. metadata-eval 93.8%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot \color{blue}{81}\right) \cdot x}{6 - x \cdot -9} \]

   8. *-commutative 93.8%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 - \color{blue}{-9 \cdot x}} \]

   9. cancel-sign-sub-inv 93.8%

      \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{\color{blue}{6 + \left(--9\right) \cdot x}} \]

   10. metadata-eval 93.8%

       \[\leadsto \frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + \color{blue}{9} \cdot x} \]

6. Applied egg-rr 93.8%

   \[\leadsto \color{blue}{\frac{\left(36 - {x}^{2} \cdot 81\right) \cdot x}{6 + 9 \cdot x}} \]
7. Step-by-step derivation

   1. frac-2neg 93.8%

      \[\leadsto \color{blue}{\frac{-\left(36 - {x}^{2} \cdot 81\right) \cdot x}{-\left(6 + 9 \cdot x\right)}} \]

   2. div-inv 93.7%

      \[\leadsto \color{blue}{\left(-\left(36 - {x}^{2} \cdot 81\right) \cdot x\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)}} \]

   3. *-commutative 93.7%

      \[\leadsto \left(-\color{blue}{x \cdot \left(36 - {x}^{2} \cdot 81\right)}\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

   4. distribute-rgt-neg-in 93.7%

      \[\leadsto \color{blue}{\left(x \cdot \left(-\left(36 - {x}^{2} \cdot 81\right)\right)\right)} \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

   5. sub-neg 93.7%

      \[\leadsto \left(x \cdot \left(-\color{blue}{\left(36 + \left(-{x}^{2} \cdot 81\right)\right)}\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

   6. distribute-rgt-neg-in 93.7%

      \[\leadsto \left(x \cdot \left(-\left(36 + \color{blue}{{x}^{2} \cdot \left(-81\right)}\right)\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

   7. metadata-eval 93.7%

      \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot \color{blue}{-81}\right)\right)\right) \cdot \frac{1}{-\left(6 + 9 \cdot x\right)} \]

   8. +-commutative 93.7%

      \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\color{blue}{\left(9 \cdot x + 6\right)}} \]

   9. *-commutative 93.7%

      \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\left(\color{blue}{x \cdot 9} + 6\right)} \]

   10. fma-def 93.7%

       \[\leadsto \left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\color{blue}{\mathsf{fma}\left(x, 9, 6\right)}} \]

8. Applied egg-rr 93.7%

   \[\leadsto \color{blue}{\left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(x, 9, 6\right)}} \]
9. Step-by-step derivation

   1. associate-*r/ 93.8%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)\right) \cdot 1}{-\mathsf{fma}\left(x, 9, 6\right)}} \]

   2. *-rgt-identity 93.8%

      \[\leadsto \frac{\color{blue}{x \cdot \left(-\left(36 + {x}^{2} \cdot -81\right)\right)}}{-\mathsf{fma}\left(x, 9, 6\right)} \]

   3. associate-/l* 99.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{-\mathsf{fma}\left(x, 9, 6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}}} \]

   4. fma-udef 99.2%

      \[\leadsto \frac{x}{\frac{-\color{blue}{\left(x \cdot 9 + 6\right)}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   5. distribute-neg-in 99.2%

      \[\leadsto \frac{x}{\frac{\color{blue}{\left(-x \cdot 9\right) + \left(-6\right)}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   6. *-commutative 99.2%

      \[\leadsto \frac{x}{\frac{\left(-\color{blue}{9 \cdot x}\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   7. metadata-eval 99.2%

      \[\leadsto \frac{x}{\frac{\left(-\color{blue}{\left(--9\right)} \cdot x\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   8. distribute-lft-neg-in 99.2%

      \[\leadsto \frac{x}{\frac{\left(-\color{blue}{\left(--9 \cdot x\right)}\right) + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   9. remove-double-neg 99.2%

      \[\leadsto \frac{x}{\frac{\color{blue}{-9 \cdot x} + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   10. *-commutative 99.2%

       \[\leadsto \frac{x}{\frac{\color{blue}{x \cdot -9} + \left(-6\right)}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   11. metadata-eval 99.2%

       \[\leadsto \frac{x}{\frac{x \cdot -9 + \color{blue}{-6}}{-\left(36 + {x}^{2} \cdot -81\right)}} \]

   12. +-commutative 99.2%

       \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{-\color{blue}{\left({x}^{2} \cdot -81 + 36\right)}}} \]

   13. distribute-neg-in 99.2%

       \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{\left(-{x}^{2} \cdot -81\right) + \left(-36\right)}}} \]

   14. distribute-rgt-neg-in 99.2%

       \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{{x}^{2} \cdot \left(--81\right)} + \left(-36\right)}} \]

   15. metadata-eval 99.2%

       \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{{x}^{2} \cdot \color{blue}{81} + \left(-36\right)}} \]

   16. fma-def 99.2%

       \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\color{blue}{\mathsf{fma}\left({x}^{2}, 81, -36\right)}}} \]

   17. metadata-eval 99.2%

       \[\leadsto \frac{x}{\frac{x \cdot -9 + -6}{\mathsf{fma}\left({x}^{2}, 81, \color{blue}{-36}\right)}} \]

10. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot -9 + -6}{\mathsf{fma}\left({x}^{2}, 81, -36\right)}}} \]

11. Taylor expanded in x around 0 52.0%

    \[\leadsto \frac{x}{\color{blue}{0.16666666666666666 + 0.25 \cdot x}} \]
12. Step-by-step derivation

    1. *-commutative 52.0%

       \[\leadsto \frac{x}{0.16666666666666666 + \color{blue}{x \cdot 0.25}} \]

13. Simplified 52.0%

    \[\leadsto \frac{x}{\color{blue}{0.16666666666666666 + x \cdot 0.25}} \]

14. Taylor expanded in x around inf 2.5%

    \[\leadsto \color{blue}{4} \]

15. Final simplification 2.5%

    \[\leadsto 4 \]
16. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 6 \cdot x - 9 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 6.0 x) (* 9.0 (* x x))))

C

double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * x) - (9.0d0 * (x * x))
end function

Java

public static double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Python

def code(x):
	return (6.0 * x) - (9.0 * (x * x))

Julia

function code(x)
	return Float64(Float64(6.0 * x) - Float64(9.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * x) - (9.0 * (x * x));
end

Wolfram

code[x_] := N[(N[(6.0 * x), $MachinePrecision] - N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024021 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

  (* (* 3.0 (- 2.0 (* x 3.0))) x))