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

Average Error: 0.1 → 0.1

Time: 12.2s

Precision: binary64

Cost: 7040

Math

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

↓

\[3 + \left(-12 \cdot x + 9 \cdot {x}^{2}\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (+ 3.0 (+ (* -12.0 x) (* 9.0 (pow x 2.0)))))

C

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

↓

double code(double x) {
	return 3.0 + ((-12.0 * x) + (9.0 * pow(x, 2.0)));
}

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0 + (((-12.0d0) * x) + (9.0d0 * (x ** 2.0d0)))
end function

Java

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

↓

public static double code(double x) {
	return 3.0 + ((-12.0 * x) + (9.0 * Math.pow(x, 2.0)));
}

Python

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

↓

def code(x):
	return 3.0 + ((-12.0 * x) + (9.0 * math.pow(x, 2.0)))

Julia

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

↓

function code(x)
	return Float64(3.0 + Float64(Float64(-12.0 * x) + Float64(9.0 * (x ^ 2.0))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = 3.0 + ((-12.0 * x) + (9.0 * (x ^ 2.0)));
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]

↓

code[x_] := N[(3.0 + N[(N[(-12.0 * x), $MachinePrecision] + N[(9.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ 3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right) \]
   rational_best-simplify-2 [=>] 0.1	\[ 3 \cdot \left(\left(\color{blue}{x \cdot \left(x \cdot 3\right)} - x \cdot 4\right) + 1\right) \]
   rational_best-simplify-2 [=>] 0.1	\[ 3 \cdot \left(\left(x \cdot \left(x \cdot 3\right) - \color{blue}{4 \cdot x}\right) + 1\right) \]
   rational_best-simplify-48 [=>] 0.1	\[ 3 \cdot \left(\color{blue}{x \cdot \left(x \cdot 3 - 4\right)} + 1\right) \]
   rational_best-simplify-2 [=>] 0.1	\[ 3 \cdot \left(x \cdot \left(\color{blue}{3 \cdot x} - 4\right) + 1\right) \]

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{-12 \cdot x + \left(3 + 9 \cdot {x}^{2}\right)} \]

4. Simplified 0.1

   \[\leadsto \color{blue}{3 + \left(-12 \cdot x + 9 \cdot {x}^{2}\right)} \]
   Proof

   [Start] 0.1	\[ -12 \cdot x + \left(3 + 9 \cdot {x}^{2}\right) \]
   rational_best-simplify-1 [=>] 0.1	\[ -12 \cdot x + \color{blue}{\left(9 \cdot {x}^{2} + 3\right)} \]
   rational_best-simplify-43 [=>] 0.1	\[ \color{blue}{3 + \left(9 \cdot {x}^{2} + -12 \cdot x\right)} \]
   rational_best-simplify-1 [<=] 0.1	\[ 3 + \color{blue}{\left(-12 \cdot x + 9 \cdot {x}^{2}\right)} \]

5. Final simplification 0.1

   \[\leadsto 3 + \left(-12 \cdot x + 9 \cdot {x}^{2}\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	832

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

Alternative 2
Error	0.1
Cost	704

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

Alternative 3
Error	21.1
Cost	320

\[-12 \cdot x + 3 \]

Alternative 4
Error	21.3
Cost	64

\[3 \]

Reproduce

herbie shell --seed 2023092 
(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)))