Curve intersection, scale width based on ribbon orientation

Average Error: 0.4 → 0.3

Time: 24.8s

Precision: binary64

Cost: 26688

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]

Math

\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

↓

\[\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle} \cdot n0_i + \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary64
 (+
  (* (* (sin (* (- 1.0 u) normAngle)) (/ 1.0 (sin normAngle))) n0_i)
  (* (* (sin (* u normAngle)) (/ 1.0 (sin normAngle))) n1_i)))

↓

(FPCore (normAngle u n0_i n1_i)
 :precision binary64
 (+
  (* (/ (sin (* (- 1.0 u) normAngle)) (sin normAngle)) n0_i)
  (* (/ (sin (* u normAngle)) (sin normAngle)) n1_i)))

C

double code(double normAngle, double u, double n0_i, double n1_i) {
	return ((sin(((1.0 - u) * normAngle)) * (1.0 / sin(normAngle))) * n0_i) + ((sin((u * normAngle)) * (1.0 / sin(normAngle))) * n1_i);
}

↓

double code(double normAngle, double u, double n0_i, double n1_i) {
	return ((sin(((1.0 - u) * normAngle)) / sin(normAngle)) * n0_i) + ((sin((u * normAngle)) / sin(normAngle)) * n1_i);
}

Fortran

real(8) function code(normangle, u, n0_i, n1_i)
    real(8), intent (in) :: normangle
    real(8), intent (in) :: u
    real(8), intent (in) :: n0_i
    real(8), intent (in) :: n1_i
    code = ((sin(((1.0d0 - u) * normangle)) * (1.0d0 / sin(normangle))) * n0_i) + ((sin((u * normangle)) * (1.0d0 / sin(normangle))) * n1_i)
end function

↓

real(8) function code(normangle, u, n0_i, n1_i)
    real(8), intent (in) :: normangle
    real(8), intent (in) :: u
    real(8), intent (in) :: n0_i
    real(8), intent (in) :: n1_i
    code = ((sin(((1.0d0 - u) * normangle)) / sin(normangle)) * n0_i) + ((sin((u * normangle)) / sin(normangle)) * n1_i)
end function

Java

public static double code(double normAngle, double u, double n0_i, double n1_i) {
	return ((Math.sin(((1.0 - u) * normAngle)) * (1.0 / Math.sin(normAngle))) * n0_i) + ((Math.sin((u * normAngle)) * (1.0 / Math.sin(normAngle))) * n1_i);
}

↓

public static double code(double normAngle, double u, double n0_i, double n1_i) {
	return ((Math.sin(((1.0 - u) * normAngle)) / Math.sin(normAngle)) * n0_i) + ((Math.sin((u * normAngle)) / Math.sin(normAngle)) * n1_i);
}

Python

def code(normAngle, u, n0_i, n1_i):
	return ((math.sin(((1.0 - u) * normAngle)) * (1.0 / math.sin(normAngle))) * n0_i) + ((math.sin((u * normAngle)) * (1.0 / math.sin(normAngle))) * n1_i)

↓

def code(normAngle, u, n0_i, n1_i):
	return ((math.sin(((1.0 - u) * normAngle)) / math.sin(normAngle)) * n0_i) + ((math.sin((u * normAngle)) / math.sin(normAngle)) * n1_i)

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float64(Float64(Float64(sin(Float64(Float64(1.0 - u) * normAngle)) * Float64(1.0 / sin(normAngle))) * n0_i) + Float64(Float64(sin(Float64(u * normAngle)) * Float64(1.0 / sin(normAngle))) * n1_i))
end

↓

function code(normAngle, u, n0_i, n1_i)
	return Float64(Float64(Float64(sin(Float64(Float64(1.0 - u) * normAngle)) / sin(normAngle)) * n0_i) + Float64(Float64(sin(Float64(u * normAngle)) / sin(normAngle)) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((sin(((1.0 - u) * normAngle)) * (1.0 / sin(normAngle))) * n0_i) + ((sin((u * normAngle)) * (1.0 / sin(normAngle))) * n1_i);
end

↓

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((sin(((1.0 - u) * normAngle)) / sin(normAngle)) * n0_i) + ((sin((u * normAngle)) / sin(normAngle)) * n1_i);
end

Wolfram

code[normAngle_, u_, n0$95$i_, n1$95$i_] := N[(N[(N[(N[Sin[N[(N[(1.0 - u), $MachinePrecision] * normAngle), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sin[normAngle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n0$95$i), $MachinePrecision] + N[(N[(N[Sin[N[(u * normAngle), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sin[normAngle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * n1$95$i), $MachinePrecision]), $MachinePrecision]

↓

code[normAngle_, u_, n0$95$i_, n1$95$i_] := N[(N[(N[(N[Sin[N[(N[(1.0 - u), $MachinePrecision] * normAngle), $MachinePrecision]], $MachinePrecision] / N[Sin[normAngle], $MachinePrecision]), $MachinePrecision] * n0$95$i), $MachinePrecision] + N[(N[(N[Sin[N[(u * normAngle), $MachinePrecision]], $MachinePrecision] / N[Sin[normAngle], $MachinePrecision]), $MachinePrecision] * n1$95$i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.4

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

2. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{-\sin normAngle}\right)} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]

3. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}} \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   Proof

4. Applied egg-rr 0.3

   \[\leadsto \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle} \cdot n0_i + \color{blue}{\left(-1 \cdot \frac{\sin \left(u \cdot normAngle\right)}{-\sin normAngle}\right)} \cdot n1_i \]

5. Simplified 0.3

   \[\leadsto \frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle} \cdot n0_i + \color{blue}{\frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle}} \cdot n1_i \]
   Proof

Alternatives

Alternative 1
Error	0.3
Cost	26688

\[\frac{n0_i}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right) + \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i \]

Alternative 2
Error	12.8
Cost	13896

\[\begin{array}{l} t_0 := n0_i + \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\\ \mathbf{if}\;n1_i \leq -1.316 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 5.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{n0_i}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right) + 1 \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.0
Cost	13640

\[\begin{array}{l} t_0 := n0_i + \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1_i\\ \mathbf{if}\;n1_i \leq -1.316 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 2.9 \cdot 10^{-176}:\\ \;\;\;\;\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{normAngle}\right) \cdot n0_i + 1 \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.2
Cost	13508

\[\begin{array}{l} t_0 := \sin \left(u \cdot normAngle\right)\\ \mathbf{if}\;n1_i \leq -1.316 \cdot 10^{-191}:\\ \;\;\;\;n0_i + \frac{n1_i}{\sin normAngle} \cdot t_0\\ \mathbf{elif}\;n1_i \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{normAngle}\right) \cdot n0_i + 1 \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i + \left(t_0 \cdot \frac{1}{normAngle}\right) \cdot n1_i\\ \end{array} \]

Alternative 5
Error	13.2
Cost	7624

\[\begin{array}{l} t_0 := n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{normAngle}\right) \cdot n1_i\\ \mathbf{if}\;n1_i \leq -1.316 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 8.5 \cdot 10^{-178}:\\ \;\;\;\;\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{normAngle}\right) \cdot n0_i + 1 \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.0
Cost	7104

\[n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{normAngle}\right) \cdot n1_i \]

Alternative 7
Error	46.3
Cost	192

\[n1_i + n0_i \]

Alternative 8
Error	48.5
Cost	64

\[n0_i \]

Reproduce

herbie shell --seed 2023010 
(FPCore (normAngle u n0_i n1_i)
  :name "Curve intersection, scale width based on ribbon orientation"
  :precision binary64
  :pre (and (and (and (and (<= 0.0 normAngle) (<= normAngle (/ PI 2.0))) (and (<= -1.0 n0_i) (<= n0_i 1.0))) (and (<= -1.0 n1_i) (<= n1_i 1.0))) (and (<= 2.328306437e-10 u) (<= u 1.0)))
  (+ (* (* (sin (* (- 1.0 u) normAngle)) (/ 1.0 (sin normAngle))) n0_i) (* (* (sin (* u normAngle)) (/ 1.0 (sin normAngle))) n1_i)))