Curve intersection, scale width based on ribbon orientation

Average Error: 0.4 → 0.4

Time: 1.6min

Precision: binary64

Cost: 26816

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

Math

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

↓

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

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
 (+
  (* (/ n0_i (sin normAngle)) (sin (- normAngle (* u normAngle))))
  (* (* (sin (* u normAngle)) (/ 1.0 (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 ((n0_i / sin(normAngle)) * sin((normAngle - (u * normAngle)))) + ((sin((u * normAngle)) * (1.0 / 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 = ((n0_i / sin(normangle)) * sin((normangle - (u * normangle)))) + ((sin((u * normangle)) * (1.0d0 / 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 ((n0_i / Math.sin(normAngle)) * Math.sin((normAngle - (u * normAngle)))) + ((Math.sin((u * normAngle)) * (1.0 / 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 ((n0_i / math.sin(normAngle)) * math.sin((normAngle - (u * normAngle)))) + ((math.sin((u * normAngle)) * (1.0 / 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(n0_i / sin(normAngle)) * sin(Float64(normAngle - Float64(u * normAngle)))) + Float64(Float64(sin(Float64(u * normAngle)) * Float64(1.0 / 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 = ((n0_i / sin(normAngle)) * sin((normAngle - (u * normAngle)))) + ((sin((u * normAngle)) * (1.0 / 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[(n0$95$i / N[Sin[normAngle], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(normAngle - N[(u * normAngle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[N[(u * normAngle), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sin[normAngle], $MachinePrecision]), $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

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

Alternatives

Alternative 1
Error	0.5
Cost	13632

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

Alternative 2
Error	0.5
Cost	6720

\[\mathsf{fma}\left(n1_i - n0_i, u, n0_i\right) \]

Alternative 3
Error	13.8
Cost	848

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ t_1 := \left(1 - u\right) \cdot n0_i\\ \mathbf{if}\;n1_i \leq -5.8 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq -4.5 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n1_i \leq -1.86 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 5.2 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.7
Cost	848

\[\begin{array}{l} t_0 := n1_i \cdot u + n0_i\\ \mathbf{if}\;n1_i \leq -9.8 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq -3.2 \cdot 10^{-173}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{elif}\;n1_i \leq -4.6 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n1_i \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;n1_i \leq -3.6 \cdot 10^{-145}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{elif}\;n1_i \leq 3.8 \cdot 10^{-131}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]

Alternative 6
Error	28.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;n0_i \leq -5 \cdot 10^{-79}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 3.7 \cdot 10^{-89}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 7
Error	0.5
Cost	448

\[n0_i - \left(n0_i - n1_i\right) \cdot u \]

Alternative 8
Error	48.8
Cost	64

\[n0_i \]

Reproduce

herbie shell --seed 2023033 
(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)))