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

Percentage Accurate: 99.8% → 99.9%

Time: 9.2s

Alternatives: 9

Speedup: 1.2×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

C

double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 * ((((x * 3.0d0) * x) - (x * 4.0d0)) + 1.0d0)
end function

Java

public static double code(double x) {
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
}

Python

def code(x):
	return 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0)

Julia

function code(x)
	return Float64(3.0 * Float64(Float64(Float64(Float64(x * 3.0) * x) - Float64(x * 4.0)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0);
end

Wolfram

code[x_] := N[(3.0 * N[(N[(N[(N[(x * 3.0), $MachinePrecision] * x), $MachinePrecision] - N[(x * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (fma x 9.0 -12.0) 3.0))

C

double code(double x) {
	return fma(x, fma(x, 9.0, -12.0), 3.0);
}

Julia

function code(x)
	return fma(x, fma(x, 9.0, -12.0), 3.0)
end

Wolfram

code[x_] := N[(x * N[(x * 9.0 + -12.0), $MachinePrecision] + 3.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-lft-out-- 99.8%

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

   4. associate-*l* 99.8%

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

   5. metadata-eval 99.8%

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

   6. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 3 - 4\right) \cdot 3, 3\right)} \]

   7. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot \left(x \cdot 3 - 4\right)}, 3\right) \]

   8. sub-neg 99.8%

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

   9. distribute-lft-in 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l* 99.9%

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

   12. fma-def 99.9%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 3 \cdot 3, 3 \cdot \left(-4\right)\right)}, 3\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{9}, 3 \cdot \left(-4\right)\right), 3\right) \]

   14. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, 3 \cdot \color{blue}{-4}\right), 3\right) \]

   15. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, \color{blue}{-12}\right), 3\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)} \]

4. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right) \]

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 3 + x \cdot \mathsf{fma}\left(x, 9, -12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (* x (fma x 9.0 -12.0))))

C

double code(double x) {
	return 3.0 + (x * fma(x, 9.0, -12.0));
}

Julia

function code(x)
	return Float64(3.0 + Float64(x * fma(x, 9.0, -12.0)))
end

Wolfram

code[x_] := N[(3.0 + N[(x * N[(x * 9.0 + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-lft-out-- 99.8%

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

   4. associate-*l* 99.8%

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

   5. metadata-eval 99.8%

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

   6. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 3 - 4\right) \cdot 3, 3\right)} \]

   7. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot \left(x \cdot 3 - 4\right)}, 3\right) \]

   8. sub-neg 99.8%

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

   9. distribute-lft-in 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l* 99.9%

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

   12. fma-def 99.9%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 3 \cdot 3, 3 \cdot \left(-4\right)\right)}, 3\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{9}, 3 \cdot \left(-4\right)\right), 3\right) \]

   14. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, 3 \cdot \color{blue}{-4}\right), 3\right) \]

   15. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, \color{blue}{-12}\right), 3\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 9, -12\right) + 3} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 9, -12\right) + 3} \]

6. Final simplification 99.9%

   \[\leadsto 3 + x \cdot \mathsf{fma}\left(x, 9, -12\right) \]

Alternative 3: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (* x (* x 9.0))
   (if (<= x 0.215) (* 3.0 (+ 1.0 (* x -4.0))) (* 9.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.215) {
		tmp = 3.0 * (1.0 + (x * -4.0));
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = x * (x * 9.0d0)
    else if (x <= 0.215d0) then
        tmp = 3.0d0 * (1.0d0 + (x * (-4.0d0)))
    else
        tmp = 9.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.215) {
		tmp = 3.0 * (1.0 + (x * -4.0));
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = x * (x * 9.0)
	elif x <= 0.215:
		tmp = 3.0 * (1.0 + (x * -4.0))
	else:
		tmp = 9.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 0.215)
		tmp = Float64(3.0 * Float64(1.0 + Float64(x * -4.0)));
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = x * (x * 9.0);
	elseif (x <= 0.215)
		tmp = 3.0 * (1.0 + (x * -4.0));
	else
		tmp = 9.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.215], N[(3.0 * N[(1.0 + N[(x * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 99.8%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 97.3%

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

      2. associate-*r* 97.3%

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

   6. Simplified 97.3%

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

   if -1.55000000000000004 < x < 0.214999999999999997

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 0.214999999999999997 < x

   1. Initial program 99.6%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 99.6%

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

      2. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 95.2%

      \[\leadsto \color{blue}{9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 95.2%

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

      2. associate-*r* 95.2%

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

   6. Simplified 95.2%

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

      1. associate-*r* 95.2%

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

      2. add-sqr-sqrt 94.9%

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

      3. sqrt-unprod 78.4%

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

      4. *-commutative 78.4%

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

      5. *-commutative 78.4%

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

      6. swap-sqr 78.4%

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

      7. pow2 78.4%

         \[\leadsto \sqrt{\left(\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)\right) \cdot \left(9 \cdot 9\right)} \]

      8. pow2 78.4%

         \[\leadsto \sqrt{\left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \left(9 \cdot 9\right)} \]

      9. pow-prod-up 78.5%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(9 \cdot 9\right)} \]

      10. metadata-eval 78.5%

          \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(9 \cdot 9\right)} \]

      11. metadata-eval 78.5%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{81}} \]

   8. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 81}} \]
   9. Step-by-step derivation

      1. sqrt-prod 78.5%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{81}} \]

      2. metadata-eval 78.5%

         \[\leadsto \sqrt{{x}^{4}} \cdot \color{blue}{9} \]

      3. sqrt-pow1 95.2%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot 9 \]

      4. metadata-eval 95.2%

         \[\leadsto {x}^{\color{blue}{2}} \cdot 9 \]

      5. pow2 95.2%

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

   10. Applied egg-rr 95.2%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 9} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 \cdot \left(1 + x \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 3 + \left(x \cdot -12 + x \cdot \left(x \cdot 9\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (+ (* x -12.0) (* x (* x 9.0)))))

C

double code(double x) {
	return 3.0 + ((x * -12.0) + (x * (x * 9.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + ((x * (-12.0d0)) + (x * (x * 9.0d0)))
end function

Java

public static double code(double x) {
	return 3.0 + ((x * -12.0) + (x * (x * 9.0)));
}

Python

def code(x):
	return 3.0 + ((x * -12.0) + (x * (x * 9.0)))

Julia

function code(x)
	return Float64(3.0 + Float64(Float64(x * -12.0) + Float64(x * Float64(x * 9.0))))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + ((x * -12.0) + (x * (x * 9.0)));
end

Wolfram

code[x_] := N[(3.0 + N[(N[(x * -12.0), $MachinePrecision] + N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-lft-out-- 99.8%

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

   4. associate-*l* 99.8%

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

   5. metadata-eval 99.8%

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

   6. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 3 - 4\right) \cdot 3, 3\right)} \]

   7. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot \left(x \cdot 3 - 4\right)}, 3\right) \]

   8. sub-neg 99.8%

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

   9. distribute-lft-in 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l* 99.9%

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

   12. fma-def 99.9%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 3 \cdot 3, 3 \cdot \left(-4\right)\right)}, 3\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{9}, 3 \cdot \left(-4\right)\right), 3\right) \]

   14. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, 3 \cdot \color{blue}{-4}\right), 3\right) \]

   15. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, \color{blue}{-12}\right), 3\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 9, -12\right) + 3} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 9, -12\right) + 3} \]
6. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot 9 + -12\right)} + 3 \]

   2. *-commutative 99.9%

      \[\leadsto x \cdot \left(\color{blue}{9 \cdot x} + -12\right) + 3 \]

   3. distribute-lft-in 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(9 \cdot x\right) + x \cdot -12\right)} + 3 \]

   4. *-commutative 99.9%

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot 9\right)} + x \cdot -12\right) + 3 \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot 9\right) + x \cdot -12\right)} + 3 \]

8. Final simplification 99.9%

   \[\leadsto 3 + \left(x \cdot -12 + x \cdot \left(x \cdot 9\right)\right) \]

Alternative 5: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.2))) (* x (* x 9.0)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.58d0)) .or. (.not. (x <= 0.2d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.2)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.2):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.2))
		tmp = Float64(x * Float64(x * 9.0));
	else
		tmp = 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.58) || ~((x <= 0.2)))
		tmp = x * (x * 9.0);
	else
		tmp = 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.58], N[Not[LessEqual[x, 0.2]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.20000000000000001 < x

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 99.7%

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

      2. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 96.2%

      \[\leadsto \color{blue}{9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 96.2%

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

      2. associate-*r* 96.2%

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

   6. Simplified 96.2%

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

   if -0.57999999999999996 < x < 0.20000000000000001

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 6: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.58) (* x (* x 9.0)) (if (<= x 0.2) 3.0 (* 9.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.58) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.2) {
		tmp = 3.0;
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.58d0)) then
        tmp = x * (x * 9.0d0)
    else if (x <= 0.2d0) then
        tmp = 3.0d0
    else
        tmp = 9.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.58) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.2) {
		tmp = 3.0;
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.58:
		tmp = x * (x * 9.0)
	elif x <= 0.2:
		tmp = 3.0
	else:
		tmp = 9.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.58)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 0.2)
		tmp = 3.0;
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.58)
		tmp = x * (x * 9.0);
	elseif (x <= 0.2)
		tmp = 3.0;
	else
		tmp = 9.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.58], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.2], 3.0, N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.57999999999999996

   1. Initial program 99.8%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 97.3%

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

      2. associate-*r* 97.3%

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

   6. Simplified 97.3%

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

   if -0.57999999999999996 < x < 0.20000000000000001

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{3} \]

   if 0.20000000000000001 < x

   1. Initial program 99.6%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 99.6%

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

      2. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 95.2%

      \[\leadsto \color{blue}{9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 95.2%

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

      2. associate-*r* 95.2%

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

   6. Simplified 95.2%

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

      1. associate-*r* 95.2%

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

      2. add-sqr-sqrt 94.9%

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

      3. sqrt-unprod 78.4%

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

      4. *-commutative 78.4%

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

      5. *-commutative 78.4%

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

      6. swap-sqr 78.4%

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

      7. pow2 78.4%

         \[\leadsto \sqrt{\left(\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)\right) \cdot \left(9 \cdot 9\right)} \]

      8. pow2 78.4%

         \[\leadsto \sqrt{\left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \left(9 \cdot 9\right)} \]

      9. pow-prod-up 78.5%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(9 \cdot 9\right)} \]

      10. metadata-eval 78.5%

          \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(9 \cdot 9\right)} \]

      11. metadata-eval 78.5%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{81}} \]

   8. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 81}} \]
   9. Step-by-step derivation

      1. sqrt-prod 78.5%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{81}} \]

      2. metadata-eval 78.5%

         \[\leadsto \sqrt{{x}^{4}} \cdot \color{blue}{9} \]

      3. sqrt-pow1 95.2%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot 9 \]

      4. metadata-eval 95.2%

         \[\leadsto {x}^{\color{blue}{2}} \cdot 9 \]

      5. pow2 95.2%

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

   10. Applied egg-rr 95.2%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 9} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.2:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (* x (* x 9.0))
   (if (<= x 0.215) (+ 3.0 (* x -12.0)) (* 9.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.215) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = x * (x * 9.0d0)
    else if (x <= 0.215d0) then
        tmp = 3.0d0 + (x * (-12.0d0))
    else
        tmp = 9.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = x * (x * 9.0);
	} else if (x <= 0.215) {
		tmp = 3.0 + (x * -12.0);
	} else {
		tmp = 9.0 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = x * (x * 9.0)
	elif x <= 0.215:
		tmp = 3.0 + (x * -12.0)
	else:
		tmp = 9.0 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(x * Float64(x * 9.0));
	elseif (x <= 0.215)
		tmp = Float64(3.0 + Float64(x * -12.0));
	else
		tmp = Float64(9.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.55)
		tmp = x * (x * 9.0);
	elseif (x <= 0.215)
		tmp = 3.0 + (x * -12.0);
	else
		tmp = 9.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.215], N[(3.0 + N[(x * -12.0), $MachinePrecision]), $MachinePrecision], N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 99.8%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 97.3%

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

      2. associate-*r* 97.3%

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

   6. Simplified 97.3%

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

   if -1.55000000000000004 < x < 0.214999999999999997

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{-12 \cdot x + 3} \]

   if 0.214999999999999997 < x

   1. Initial program 99.6%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   2. Step-by-step derivation

      1. associate-+l- 99.6%

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

      2. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 95.2%

      \[\leadsto \color{blue}{9 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 95.2%

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

      2. associate-*r* 95.2%

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

   6. Simplified 95.2%

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

      1. associate-*r* 95.2%

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

      2. add-sqr-sqrt 94.9%

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

      3. sqrt-unprod 78.4%

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

      4. *-commutative 78.4%

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

      5. *-commutative 78.4%

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

      6. swap-sqr 78.4%

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

      7. pow2 78.4%

         \[\leadsto \sqrt{\left(\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)\right) \cdot \left(9 \cdot 9\right)} \]

      8. pow2 78.4%

         \[\leadsto \sqrt{\left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \cdot \left(9 \cdot 9\right)} \]

      9. pow-prod-up 78.5%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(9 \cdot 9\right)} \]

      10. metadata-eval 78.5%

          \[\leadsto \sqrt{{x}^{\color{blue}{4}} \cdot \left(9 \cdot 9\right)} \]

      11. metadata-eval 78.5%

          \[\leadsto \sqrt{{x}^{4} \cdot \color{blue}{81}} \]

   8. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 81}} \]
   9. Step-by-step derivation

      1. sqrt-prod 78.5%

         \[\leadsto \color{blue}{\sqrt{{x}^{4}} \cdot \sqrt{81}} \]

      2. metadata-eval 78.5%

         \[\leadsto \sqrt{{x}^{4}} \cdot \color{blue}{9} \]

      3. sqrt-pow1 95.2%

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)}} \cdot 9 \]

      4. metadata-eval 95.2%

         \[\leadsto {x}^{\color{blue}{2}} \cdot 9 \]

      5. pow2 95.2%

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

   10. Applied egg-rr 95.2%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 9} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;3 + x \cdot -12\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 8: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 3 + 9 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (* 9.0 (* x x))))

C

double code(double x) {
	return 3.0 + (9.0 * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (9.0d0 * (x * x))
end function

Java

public static double code(double x) {
	return 3.0 + (9.0 * (x * x));
}

Python

def code(x):
	return 3.0 + (9.0 * (x * x))

Julia

function code(x)
	return Float64(3.0 + Float64(9.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (9.0 * (x * x));
end

Wolfram

code[x_] := N[(3.0 + N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-lft-out-- 99.8%

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

   4. associate-*l* 99.8%

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

   5. metadata-eval 99.8%

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

   6. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot 3 - 4\right) \cdot 3, 3\right)} \]

   7. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot \left(x \cdot 3 - 4\right)}, 3\right) \]

   8. sub-neg 99.8%

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

   9. distribute-lft-in 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l* 99.9%

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

   12. fma-def 99.9%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 3 \cdot 3, 3 \cdot \left(-4\right)\right)}, 3\right) \]

   13. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{9}, 3 \cdot \left(-4\right)\right), 3\right) \]

   14. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, 3 \cdot \color{blue}{-4}\right), 3\right) \]

   15. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, \color{blue}{-12}\right), 3\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 9, -12\right) + 3} \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, 9, -12\right) + 3} \]

6. Taylor expanded in x around inf 97.5%

   \[\leadsto \color{blue}{9 \cdot {x}^{2}} + 3 \]
7. Step-by-step derivation

   1. unpow2 97.5%

      \[\leadsto 9 \cdot \color{blue}{\left(x \cdot x\right)} + 3 \]

8. Simplified 97.5%

   \[\leadsto \color{blue}{9 \cdot \left(x \cdot x\right)} + 3 \]

9. Final simplification 97.5%

   \[\leadsto 3 + 9 \cdot \left(x \cdot x\right) \]

Alternative 9: 51.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

function tmp = code(x)
	tmp = 3.0;
end

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 99.8%

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
2. Step-by-step derivation

   1. associate-+l- 99.8%

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

   2. associate-*l* 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 49.9%

   \[\leadsto \color{blue}{3} \]

5. Final simplification 49.9%

   \[\leadsto 3 \]

Developer target: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x))))

C

double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (((9.0d0 * x) * x) - (12.0d0 * x))
end function

Java

public static double code(double x) {
	return 3.0 + (((9.0 * x) * x) - (12.0 * x));
}

Python

def code(x):
	return 3.0 + (((9.0 * x) * x) - (12.0 * x))

Julia

function code(x)
	return Float64(3.0 + Float64(Float64(Float64(9.0 * x) * x) - Float64(12.0 * x)))
end

MATLAB

function tmp = code(x)
	tmp = 3.0 + (((9.0 * x) * x) - (12.0 * x));
end

Wolfram

code[x_] := N[(3.0 + N[(N[(N[(9.0 * x), $MachinePrecision] * x), $MachinePrecision] - N[(12.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))