?

Average Error: 0.9 → 0.2
Time: 18.7s
Precision: binary32
Cost: 3808

?

\[\left(\left(\left(0 \leq normAngle \land normAngle \leq \frac{\pi}{2}\right) \land \left(-1 \leq n0_i \land n0_i \leq 1\right)\right) \land \left(-1 \leq n1_i \land n1_i \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right)\]
\[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
↓
\[n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \left(n0_i + \left(n0_i \cdot -0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]
(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
  (*
   u
   (-
    (/ n1_i (/ (sin normAngle) normAngle))
    (+ n0_i (* (* n0_i -0.3333333333333333) (* normAngle normAngle)))))))
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 + (u * ((n1_i / (sinf(normAngle) / normAngle)) - (n0_i + ((n0_i * -0.3333333333333333f) * (normAngle * normAngle)))));
}

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 + (u * ((n1_i / (sin(normangle) / normangle)) - (n0_i + ((n0_i * (-0.3333333333333333e0)) * (normangle * normangle)))))
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(u * Float32(Float32(n1_i / Float32(sin(normAngle) / normAngle)) - Float32(n0_i + Float32(Float32(n0_i * Float32(-0.3333333333333333)) * Float32(normAngle * normAngle))))))
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 + (u * ((n1_i / (sin(normAngle) / normAngle)) - (n0_i + ((n0_i * single(-0.3333333333333333)) * (normAngle * normAngle)))));
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 + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \left(n0_i + \left(n0_i \cdot -0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.9

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

  2. Simplified8.3

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

    [Start]0.9

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

    *-commutative [=>]0.9

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

    associate-*l* [=>]6.2

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

    *-commutative [=>]6.2

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

    associate-*l* [=>]8.4

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

    distribute-lft-out [=>]8.4

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

    +-commutative [<=]8.4

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

    associate-*l/ [=>]8.3

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

    *-commutative [=>]8.3

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

    associate-/l* [=>]8.3

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

    /-rgt-identity [=>]8.3

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

  3. Taylor expanded in u around 0 3.8

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

  4. Simplified1.6

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

    [Start]3.8

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

    associate-/l* [=>]1.6

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

    associate-*r/ [=>]1.6

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

    mul-1-neg [=>]1.6

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

  5. Taylor expanded in normAngle around 0 0.2

    \[\leadsto n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} + \color{blue}{\left(-1 \cdot n0_i + -1 \cdot \left(\left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) \cdot {normAngle}^{2}\right)\right)}\right) \]

  6. Simplified0.2

    \[\leadsto n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} + \color{blue}{-1 \cdot \left(n0_i + \left(n0_i \cdot -0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right)\right)}\right) \]
    Proof

    [Start]0.2

    \[ n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} + \left(-1 \cdot n0_i + -1 \cdot \left(\left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) \cdot {normAngle}^{2}\right)\right)\right) \]

    distribute-lft-out [=>]0.2

    \[ n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} + \color{blue}{-1 \cdot \left(n0_i + \left(-0.5 \cdot n0_i - -0.16666666666666666 \cdot n0_i\right) \cdot {normAngle}^{2}\right)}\right) \]

    distribute-rgt-out-- [=>]0.2

    \[ n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} + -1 \cdot \left(n0_i + \color{blue}{\left(n0_i \cdot \left(-0.5 - -0.16666666666666666\right)\right)} \cdot {normAngle}^{2}\right)\right) \]

    metadata-eval [=>]0.2

    \[ n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} + -1 \cdot \left(n0_i + \left(n0_i \cdot \color{blue}{-0.3333333333333333}\right) \cdot {normAngle}^{2}\right)\right) \]

    unpow2 [=>]0.2

    \[ n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} + -1 \cdot \left(n0_i + \left(n0_i \cdot -0.3333333333333333\right) \cdot \color{blue}{\left(normAngle \cdot normAngle\right)}\right)\right) \]

  7. Final simplification0.2

    \[\leadsto n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} - \left(n0_i + \left(n0_i \cdot -0.3333333333333333\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right) \]

Alternatives

Alternative 1
Error0.3
Cost3552
\[n0_i + u \cdot \left(\frac{n1_i}{\frac{\sin normAngle}{normAngle}} - n0_i\right) \]
Alternative 2
Error0.6
Cost544
\[n0_i + \left(u \cdot n1_i + \left(n0_i \cdot u\right) \cdot \left(-1 + \left(normAngle \cdot normAngle\right) \cdot 0.3333333333333333\right)\right) \]
Alternative 3
Error9.3
Cost297
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -2.0000000390829628 \cdot 10^{-25} \lor \neg \left(n0_i \leq 6.000000117248888 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]
Alternative 4
Error4.4
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.0000000195414814 \cdot 10^{-25} \lor \neg \left(n1_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]
Alternative 5
Error4.4
Cost297
\[\begin{array}{l} \mathbf{if}\;n1_i \leq -1.0000000195414814 \cdot 10^{-25} \lor \neg \left(n1_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \end{array} \]
Alternative 6
Error0.6
Cost288
\[n0_i + \left(u \cdot n1_i - n0_i \cdot u\right) \]
Alternative 7
Error12.7
Cost232
\[\begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 3.499999888929329 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]
Alternative 8
Error0.6
Cost224
\[n0_i + u \cdot \left(n1_i - n0_i\right) \]
Alternative 9
Error16.9
Cost32
\[n0_i \]

Error

Reproduce?

herbie shell --seed 2023187 
(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)))