?

Average Error: 97.2% → 98.8%
Time: 33.3s
Precision: binary32
Cost: 480.00

?

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[n0_i + \left(\left(n1_i + \left(normAngle \cdot normAngle\right) \cdot \left(n1_i \cdot 0.16666666666666666\right)\right) - n0_i\right) \cdot u \]
(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* (* (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 binary32
 (+
  n0_i
  (*
   (- (+ n1_i (* (* normAngle normAngle) (* n1_i 0.16666666666666666))) n0_i)
   u)))
float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((sinf(((1.0f - u) * normAngle)) * (1.0f / sinf(normAngle))) * n0_i) + ((sinf((u * normAngle)) * (1.0f / sinf(normAngle))) * n1_i);
}
↓
float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (((n1_i + ((normAngle * normAngle) * (n1_i * 0.16666666666666666f))) - n0_i) * u);
}

real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = ((sin(((1.0e0 - u) * normangle)) * (1.0e0 / sin(normangle))) * n0_i) + ((sin((u * normangle)) * (1.0e0 / sin(normangle))) * n1_i)
end function
↓
real(4) function code(normangle, u, n0_i, n1_i)
    real(4), intent (in) :: normangle
    real(4), intent (in) :: u
    real(4), intent (in) :: n0_i
    real(4), intent (in) :: n1_i
    code = n0_i + (((n1_i + ((normangle * normangle) * (n1_i * 0.16666666666666666e0))) - n0_i) * u)
end function

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * Float32(Float32(1.0) / sin(normAngle))) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * Float32(Float32(1.0) / sin(normAngle))) * n1_i))
end
↓
function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(Float32(n1_i + Float32(Float32(normAngle * normAngle) * Float32(n1_i * Float32(0.16666666666666666)))) - n0_i) * u))
end

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = ((sin(((single(1.0) - u) * normAngle)) * (single(1.0) / sin(normAngle))) * n0_i) + ((sin((u * normAngle)) * (single(1.0) / sin(normAngle))) * n1_i);
end
↓
function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (((n1_i + ((normAngle * normAngle) * (n1_i * single(0.16666666666666666)))) - n0_i) * u);
end

\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
↓
n0_i + \left(\left(n1_i + \left(normAngle \cdot normAngle\right) \cdot \left(n1_i \cdot 0.16666666666666666\right)\right) - n0_i\right) \cdot u

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 97.2

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

  2. Taylor expanded in normAngle around 0 97.1

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

  3. Taylor expanded in u around 0 90.1

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

  4. Taylor expanded in normAngle around 0 98.8

    \[\leadsto \left(\color{blue}{\left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right)} + -1 \cdot n0_i\right) \cdot u + n0_i \]

  5. Simplified98.8

    \[\leadsto \left(\color{blue}{\left(n1_i + \left(normAngle \cdot normAngle\right) \cdot \left(n1_i \cdot 0.16666666666666666\right)\right)} + -1 \cdot n0_i\right) \cdot u + n0_i \]
    Proof

    [Start]98.8

    \[ \left(\left(n1_i + 0.16666666666666666 \cdot \left(n1_i \cdot {normAngle}^{2}\right)\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]

    associate-*r* [=>]98.8

    \[ \left(\left(n1_i + \color{blue}{\left(0.16666666666666666 \cdot n1_i\right) \cdot {normAngle}^{2}}\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]

    *-commutative [=>]98.8

    \[ \left(\left(n1_i + \color{blue}{{normAngle}^{2} \cdot \left(0.16666666666666666 \cdot n1_i\right)}\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]

    unpow2 [=>]98.8

    \[ \left(\left(n1_i + \color{blue}{\left(normAngle \cdot normAngle\right)} \cdot \left(0.16666666666666666 \cdot n1_i\right)\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]

    *-commutative [=>]98.8

    \[ \left(\left(n1_i + \left(normAngle \cdot normAngle\right) \cdot \color{blue}{\left(n1_i \cdot 0.16666666666666666\right)}\right) + -1 \cdot n0_i\right) \cdot u + n0_i \]

  6. Final simplification98.8

    \[\leadsto n0_i + \left(\left(n1_i + \left(normAngle \cdot normAngle\right) \cdot \left(n1_i \cdot 0.16666666666666666\right)\right) - n0_i\right) \cdot u \]

Alternatives

Alternative 1
Error70.7%
Cost297.00
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.5000000784359874 \cdot 10^{-23} \lor \neg \left(n0_i \leq 9.999999682655225 \cdot 10^{-21}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]
Alternative 2
Error86.0%
Cost297.00
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000063421537 \cdot 10^{-28} \lor \neg \left(n1_i \leq 2.250000007134923 \cdot 10^{-30}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]
Alternative 3
Error86.0%
Cost297.00
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0000000063421537 \cdot 10^{-28} \lor \neg \left(n1_i \leq 2.250000007134923 \cdot 10^{-30}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]
Alternative 4
Error60.8%
Cost232.00
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
Alternative 5
Error98.1%
Cost224.00
\[n0_i + u \cdot \left(n1_i - n0_i\right) \]
Alternative 6
Error46.6%
Cost32.00
\[n0_i \]

Error

Reproduce?

herbie shell --seed 2023093 
(FPCore (normAngle u n0_i n1_i)
  :name "Curve intersection, scale width based on ribbon orientation"
  :precision binary32
  :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)))