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

Percentage Accurate: 99.8% → 99.8%

Time: 4.1s

Alternatives: 5

Speedup: 1.2×

• 24.6% 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: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x * -12.0), $MachinePrecision] + N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $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. *-commutative 99.8%

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

   2. distribute-lft1-in 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-lft-out-- 99.9%

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

   5. associate-*l* 99.9%

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

   6. fma-def 99.9%

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

   7. *-commutative 99.9%

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

   8. sub-neg 99.9%

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

   9. distribute-lft-in 99.9%

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

   10. *-commutative 99.9%

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

   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%

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

Alternative 2: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.195\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.58) (not (<= x 0.195))) (* x (* x 9.0)) 3.0))

C

double code(double x) {
	double tmp;
	if ((x <= -0.58) || !(x <= 0.195)) {
		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 <= 0.195d0))) 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 <= 0.195)) {
		tmp = x * (x * 9.0);
	} else {
		tmp = 3.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.58) or not (x <= 0.195):
		tmp = x * (x * 9.0)
	else:
		tmp = 3.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.58) || !(x <= 0.195))
		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 <= 0.195)))
		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, 0.195]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], 3.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.57999999999999996 or 0.19500000000000001 < x

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around inf 95.5%

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

      1. unpow2 95.5%

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

      2. associate-*r* 95.5%

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

   4. Simplified 95.5%

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

   if -0.57999999999999996 < x < 0.19500000000000001

   1. Initial program 100.0%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.58 \lor \neg \left(x \leq 0.195\right):\\ \;\;\;\;x \cdot \left(x \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 3: 98.2% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 0.215))) (* x (* x 9.0)) (+ (* x -12.0) 3.0)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 0.215d0))) then
        tmp = x * (x * 9.0d0)
    else
        tmp = (x * (-12.0d0)) + 3.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 0.215):
		tmp = x * (x * 9.0)
	else:
		tmp = (x * -12.0) + 3.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 0.215]], $MachinePrecision]], N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * -12.0), $MachinePrecision] + 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 0.214999999999999997 < x

   1. Initial program 99.7%

      \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]

   2. Taylor expanded in x around inf 95.5%

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

      1. unpow2 95.5%

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

      2. associate-*r* 95.5%

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

   4. Simplified 95.5%

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

   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. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{-12 \cdot x + 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

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

Alternative 4: 97.6% 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. *-commutative 99.8%

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

   2. distribute-lft1-in 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-lft-out-- 99.9%

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

   5. associate-*l* 99.9%

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

   6. fma-def 99.9%

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

   7. *-commutative 99.9%

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

   8. sub-neg 99.9%

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

   9. distribute-lft-in 99.9%

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

   10. *-commutative 99.9%

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

   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 97.0%

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

   1. unpow2 97.0%

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

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

Alternative 5: 51.7% 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. Taylor expanded in x around 0 54.3%

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

3. Final simplification 54.3%

   \[\leadsto 3 \]

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 2023221 
(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)))