?

Average Error: 97.3% → 98.9%
Time: 19.0s
Precision: binary32
Cost: 10880.00

?

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[\left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) + -0.16666666666666666 \cdot \left(n0_i \cdot \left(u + -1\right) - n1_i \cdot u\right)\right) \cdot {normAngle}^{2} \]
(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
 (+
  (+ (* n1_i u) (* (- 1.0 u) n0_i))
  (*
   (+
    (+
     (* -0.16666666666666666 (* n1_i (pow u 3.0)))
     (* -0.16666666666666666 (* (pow (- 1.0 u) 3.0) n0_i)))
    (* -0.16666666666666666 (- (* n0_i (+ u -1.0)) (* n1_i u))))
   (pow normAngle 2.0))))
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 ((n1_i * u) + ((1.0f - u) * n0_i)) + ((((-0.16666666666666666f * (n1_i * powf(u, 3.0f))) + (-0.16666666666666666f * (powf((1.0f - u), 3.0f) * n0_i))) + (-0.16666666666666666f * ((n0_i * (u + -1.0f)) - (n1_i * u)))) * powf(normAngle, 2.0f));
}

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 = ((n1_i * u) + ((1.0e0 - u) * n0_i)) + (((((-0.16666666666666666e0) * (n1_i * (u ** 3.0e0))) + ((-0.16666666666666666e0) * (((1.0e0 - u) ** 3.0e0) * n0_i))) + ((-0.16666666666666666e0) * ((n0_i * (u + (-1.0e0))) - (n1_i * u)))) * (normangle ** 2.0e0))
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(Float32(Float32(n1_i * u) + Float32(Float32(Float32(1.0) - u) * n0_i)) + Float32(Float32(Float32(Float32(Float32(-0.16666666666666666) * Float32(n1_i * (u ^ Float32(3.0)))) + Float32(Float32(-0.16666666666666666) * Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) * n0_i))) + Float32(Float32(-0.16666666666666666) * Float32(Float32(n0_i * Float32(u + Float32(-1.0))) - Float32(n1_i * u)))) * (normAngle ^ Float32(2.0))))
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 = ((n1_i * u) + ((single(1.0) - u) * n0_i)) + ((((single(-0.16666666666666666) * (n1_i * (u ^ single(3.0)))) + (single(-0.16666666666666666) * (((single(1.0) - u) ^ single(3.0)) * n0_i))) + (single(-0.16666666666666666) * ((n0_i * (u + single(-1.0))) - (n1_i * u)))) * (normAngle ^ single(2.0)));
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
↓
\left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) + -0.16666666666666666 \cdot \left(n0_i \cdot \left(u + -1\right) - n1_i \cdot u\right)\right) \cdot {normAngle}^{2}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 97.3

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

  2. Simplified73.8

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

    [Start]97.3

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

    *-commutative [=>]97.3

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

    associate-*l* [=>]80.4

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

    *-commutative [=>]80.4

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

    associate-*l* [=>]73.8

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

    distribute-lft-out [=>]73.8

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

  3. Taylor expanded in normAngle around 0 98.9

    \[\leadsto \color{blue}{\left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) - -0.16666666666666666 \cdot \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)\right) \cdot {normAngle}^{2} + \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right)} \]

  4. Final simplification98.9

    \[\leadsto \left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) + \left(\left(-0.16666666666666666 \cdot \left(n1_i \cdot {u}^{3}\right) + -0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} \cdot n0_i\right)\right) + -0.16666666666666666 \cdot \left(n0_i \cdot \left(u + -1\right) - n1_i \cdot u\right)\right) \cdot {normAngle}^{2} \]

Alternatives

Alternative 1
Error98.7%
Cost3968.00
\[\left(n1_i \cdot u + \left(1 - u\right) \cdot n0_i\right) - {normAngle}^{2} \cdot \left(u \cdot \left(-0.16666666666666666 \cdot \left(n1_i - n0_i\right) + n0_i \cdot -0.5\right)\right) \]
Alternative 2
Error98.2%
Cost3360.00
\[\mathsf{fma}\left(u, n1_i - n0_i, n0_i\right) \]
Alternative 3
Error70.1%
Cost297.00
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999841327613 \cdot 10^{-21} \lor \neg \left(n0_i \leq 1.5000000601271012 \cdot 10^{-26}\right):\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \mathbf{else}:\\ \;\;\;\;n1_i \cdot u\\ \end{array} \]
Alternative 4
Error85.9%
Cost297.00
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0420000413828192 \cdot 10^{-26} \lor \neg \left(n1_i \leq 4.999999841327613 \cdot 10^{-21}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;\left(1 - u\right) \cdot n0_i\\ \end{array} \]
Alternative 5
Error86.0%
Cost297.00
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -2.0420000413828192 \cdot 10^{-26} \lor \neg \left(n1_i \leq 4.999999841327613 \cdot 10^{-21}\right):\\ \;\;\;\;n0_i + n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]
Alternative 6
Error60.3%
Cost232.00
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 1.5000000601271012 \cdot 10^{-26}:\\ \;\;\;\;n1_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
Alternative 7
Error98.1%
Cost224.00
\[n0_i + u \cdot \left(n1_i - n0_i\right) \]
Alternative 8
Error47.2%
Cost32.00
\[n0_i \]

Error

Reproduce?

herbie shell --seed 2023097 
(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)))