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

Average Error: 0.1 → 0.1

Time: 11.5s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 + (x * ((x * 9.0d0) + (-12.0d0)))
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 + (x * ((x * 9.0) + -12.0));
}

Python

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

↓

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

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(x * Float64(Float64(x * 9.0) + -12.0)))
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 + (x * ((x * 9.0) + -12.0));
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[(x * N[(N[(x * 9.0), $MachinePrecision] + -12.0), $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.json-simplify-2 [=>] 0.1	\[ 3 \cdot \left(\left(\color{blue}{x \cdot \left(x \cdot 3\right)} - x \cdot 4\right) + 1\right) \]
   rational.json-simplify-2 [=>] 0.1	\[ 3 \cdot \left(\left(x \cdot \left(x \cdot 3\right) - \color{blue}{4 \cdot x}\right) + 1\right) \]
   rational.json-simplify-52 [=>] 0.1	\[ 3 \cdot \left(\color{blue}{x \cdot \left(x \cdot 3 - 4\right)} + 1\right) \]
   rational.json-simplify-2 [=>] 0.1	\[ 3 \cdot \left(x \cdot \left(\color{blue}{3 \cdot x} - 4\right) + 1\right) \]

3. Applied egg-rr 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ x \cdot \left(3 \cdot x + -4\right) + \left(3 + \left(3 \cdot x + -4\right) \cdot \left(x + x\right)\right) \]
   rational.json-simplify-41 [=>] 0.1	\[ \color{blue}{3 + \left(\left(3 \cdot x + -4\right) \cdot \left(x + x\right) + x \cdot \left(3 \cdot x + -4\right)\right)} \]
   rational.json-simplify-51 [=>] 0.2	\[ 3 + \color{blue}{\left(3 \cdot x + -4\right) \cdot \left(x + \left(x + x\right)\right)} \]
   rational.json-simplify-1 [<=] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \color{blue}{\left(\left(x + x\right) + x\right)} \]
   rational.json-simplify-7 [<=] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \left(\left(x + x\right) + \color{blue}{\frac{x}{1}}\right) \]
   rational.json-simplify-35 [=>] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \left(\left(x + x\right) + \color{blue}{\frac{x + x}{1 + 1}}\right) \]
   metadata-eval [=>] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \left(\left(x + x\right) + \frac{x + x}{\color{blue}{2}}\right) \]
   rational.json-simplify-30 [<=] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \frac{x + x}{2}\right)} \]
   metadata-eval [=>] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \left(\color{blue}{3} \cdot \frac{x + x}{2}\right) \]
   metadata-eval [<=] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \left(3 \cdot \frac{x + x}{\color{blue}{1 + 1}}\right) \]
   rational.json-simplify-35 [<=] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \left(3 \cdot \color{blue}{\frac{x}{1}}\right) \]
   rational.json-simplify-7 [=>] 0.2	\[ 3 + \left(3 \cdot x + -4\right) \cdot \left(3 \cdot \color{blue}{x}\right) \]
   rational.json-simplify-43 [<=] 0.1	\[ 3 + \color{blue}{x \cdot \left(\left(3 \cdot x + -4\right) \cdot 3\right)} \]
   rational.json-simplify-43 [<=] 0.1	\[ 3 + \color{blue}{3 \cdot \left(x \cdot \left(3 \cdot x + -4\right)\right)} \]
   rational.json-simplify-1 [=>] 0.1	\[ 3 + 3 \cdot \left(x \cdot \color{blue}{\left(-4 + 3 \cdot x\right)}\right) \]

5. Applied egg-rr 0.1

   \[\leadsto 3 + \color{blue}{\left(x \cdot \left(-8 + \left(-4 + x \cdot 9\right)\right) + 0\right)} \]

6. Simplified 0.1

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

   [Start] 0.1	\[ 3 + \left(x \cdot \left(-8 + \left(-4 + x \cdot 9\right)\right) + 0\right) \]
   rational.json-simplify-4 [=>] 0.1	\[ 3 + \color{blue}{x \cdot \left(-8 + \left(-4 + x \cdot 9\right)\right)} \]
   rational.json-simplify-41 [=>] 0.1	\[ 3 + x \cdot \color{blue}{\left(-4 + \left(x \cdot 9 + -8\right)\right)} \]
   rational.json-simplify-41 [=>] 0.1	\[ 3 + x \cdot \color{blue}{\left(x \cdot 9 + \left(-8 + -4\right)\right)} \]
   metadata-eval [=>] 0.1	\[ 3 + x \cdot \left(x \cdot 9 + \color{blue}{-12}\right) \]

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.9
Cost	448

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

Alternative 2
Error	1.9
Cost	448

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

Alternative 3
Error	20.4
Cost	320

\[-12 \cdot x + 3 \]

Alternative 4
Error	20.6
Cost	64

\[3 \]

Reproduce

herbie shell --seed 2023068 
(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)))