Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 98.0%

Time: 24.8s

Alternatives: 11

Speedup: 60.1×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

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
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Initial Program: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin normAngle}\\ \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot t_0\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot t_0\right) \cdot n1_i \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sin normAngle))))
   (+
    (* (* (sin (* (- 1.0 u) normAngle)) t_0) n0_i)
    (* (* (sin (* u normAngle)) t_0) n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float t_0 = 1.0f / sinf(normAngle);
	return ((sinf(((1.0f - u) * normAngle)) * t_0) * n0_i) + ((sinf((u * normAngle)) * t_0) * n1_i);
}

Fortran

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
    real(4) :: t_0
    t_0 = 1.0e0 / sin(normangle)
    code = ((sin(((1.0e0 - u) * normangle)) * t_0) * n0_i) + ((sin((u * normangle)) * t_0) * n1_i)
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	t_0 = Float32(Float32(1.0) / sin(normAngle))
	return Float32(Float32(Float32(sin(Float32(Float32(Float32(1.0) - u) * normAngle)) * t_0) * n0_i) + Float32(Float32(sin(Float32(u * normAngle)) * t_0) * n1_i))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	t_0 = single(1.0) / sin(normAngle);
	tmp = ((sin(((single(1.0) - u) * normAngle)) * t_0) * n0_i) + ((sin((u * normAngle)) * t_0) * n1_i);
end

Alternative 1: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(normAngle - normAngle \cdot u\right)}{\sin normAngle} \cdot n0_i + u \cdot n1_i \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* (/ (sin (- normAngle (* normAngle u))) (sin normAngle)) n0_i)
  (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return ((sinf((normAngle - (normAngle * u))) / sinf(normAngle)) * n0_i) + (u * n1_i);
}

Fortran

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((normangle - (normangle * u))) / sin(normangle)) * n0_i) + (u * n1_i)
end function

Julia

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

MATLAB

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

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. Step-by-step derivation

   1. fma-def 97.3%

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

   2. associate-*r/ 97.4%

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

   3. *-rgt-identity 97.4%

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

   4. associate-*r/ 97.6%

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

   5. *-rgt-identity 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in normAngle around 0 97.9%

   \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{u} \cdot n1_i\right) \]
5. Step-by-step derivation

   1. fma-udef 97.9%

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

   2. *-commutative 97.9%

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

   3. sub-neg 97.9%

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

   4. distribute-rgt-in 98.0%

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

   5. *-un-lft-identity 98.0%

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

   6. cancel-sign-sub-inv 98.0%

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

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

Alternative 2: 98.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ u \cdot n1_i + n0_i \cdot \left(1 - \left(u - \left(-0.16666666666666666 \cdot \left({\left(1 - u\right)}^{3} - \left(1 - u\right)\right)\right) \cdot \left(normAngle \cdot normAngle\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* u n1_i)
  (*
   n0_i
   (-
    1.0
    (-
     u
     (*
      (* -0.16666666666666666 (- (pow (- 1.0 u) 3.0) (- 1.0 u)))
      (* normAngle normAngle)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (u * n1_i) + (n0_i * (1.0f - (u - ((-0.16666666666666666f * (powf((1.0f - u), 3.0f) - (1.0f - u))) * (normAngle * normAngle)))));
}

Fortran

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 = (u * n1_i) + (n0_i * (1.0e0 - (u - (((-0.16666666666666666e0) * (((1.0e0 - u) ** 3.0e0) - (1.0e0 - u))) * (normangle * normangle)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(u * n1_i) + Float32(n0_i * Float32(Float32(1.0) - Float32(u - Float32(Float32(Float32(-0.16666666666666666) * Float32((Float32(Float32(1.0) - u) ^ Float32(3.0)) - Float32(Float32(1.0) - u))) * Float32(normAngle * normAngle))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (u * n1_i) + (n0_i * (single(1.0) - (u - ((single(-0.16666666666666666) * (((single(1.0) - u) ^ single(3.0)) - (single(1.0) - u))) * (normAngle * normAngle)))));
end

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. Step-by-step derivation

   1. fma-def 97.3%

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

   2. associate-*r/ 97.4%

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

   3. *-rgt-identity 97.4%

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

   4. associate-*r/ 97.6%

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

   5. *-rgt-identity 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in normAngle around 0 97.9%

   \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{u} \cdot n1_i\right) \]
5. Step-by-step derivation

   1. fma-udef 97.9%

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

   2. *-commutative 97.9%

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

   3. sub-neg 97.9%

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

   4. distribute-rgt-in 98.0%

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

   5. *-un-lft-identity 98.0%

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

   6. cancel-sign-sub-inv 98.0%

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

6. Applied egg-rr 98.0%

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

7. Taylor expanded in normAngle around 0 97.9%

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

   1. associate--l+ 97.9%

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

   2. distribute-lft-out-- 97.9%

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

   3. unpow2 97.9%

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

9. Simplified 97.9%

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

10. Final simplification 97.9%

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

Alternative 3: 98.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ u \cdot n1_i + n0_i \cdot \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(u, 0.3333333333333333, u \cdot \left(u \cdot -0.5\right)\right) - u\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (* u n1_i)
  (*
   n0_i
   (+
    1.0
    (-
     (* (* normAngle normAngle) (fma u 0.3333333333333333 (* u (* u -0.5))))
     u)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (u * n1_i) + (n0_i * (1.0f + (((normAngle * normAngle) * fmaf(u, 0.3333333333333333f, (u * (u * -0.5f)))) - u)));
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(u * n1_i) + Float32(n0_i * Float32(Float32(1.0) + Float32(Float32(Float32(normAngle * normAngle) * fma(u, Float32(0.3333333333333333), Float32(u * Float32(u * Float32(-0.5))))) - u))))
end

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. Step-by-step derivation

   1. fma-def 97.3%

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

   2. associate-*r/ 97.4%

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

   3. *-rgt-identity 97.4%

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

   4. associate-*r/ 97.6%

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

   5. *-rgt-identity 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in normAngle around 0 97.9%

   \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{u} \cdot n1_i\right) \]
5. Step-by-step derivation

   1. fma-udef 97.9%

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

   2. *-commutative 97.9%

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

   3. sub-neg 97.9%

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

   4. distribute-rgt-in 98.0%

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

   5. *-un-lft-identity 98.0%

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

   6. cancel-sign-sub-inv 98.0%

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

6. Applied egg-rr 98.0%

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

7. Taylor expanded in normAngle around 0 97.9%

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

   1. associate--l+ 97.9%

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

   2. distribute-lft-out-- 97.9%

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

   3. unpow2 97.9%

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

9. Simplified 97.9%

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

10. Taylor expanded in u around 0 97.8%

    \[\leadsto \left(1 + \left(\color{blue}{\left(0.3333333333333333 \cdot u + -0.5 \cdot {u}^{2}\right)} \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]
11. Step-by-step derivation

    1. *-commutative 97.8%

       \[\leadsto \left(1 + \left(\left(\color{blue}{u \cdot 0.3333333333333333} + -0.5 \cdot {u}^{2}\right) \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]

    2. fma-def 97.8%

       \[\leadsto \left(1 + \left(\color{blue}{\mathsf{fma}\left(u, 0.3333333333333333, -0.5 \cdot {u}^{2}\right)} \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]

    3. *-commutative 97.8%

       \[\leadsto \left(1 + \left(\mathsf{fma}\left(u, 0.3333333333333333, \color{blue}{{u}^{2} \cdot -0.5}\right) \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]

    4. unpow2 97.8%

       \[\leadsto \left(1 + \left(\mathsf{fma}\left(u, 0.3333333333333333, \color{blue}{\left(u \cdot u\right)} \cdot -0.5\right) \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]

    5. associate-*l* 97.8%

       \[\leadsto \left(1 + \left(\mathsf{fma}\left(u, 0.3333333333333333, \color{blue}{u \cdot \left(u \cdot -0.5\right)}\right) \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]

12. Simplified 97.8%

    \[\leadsto \left(1 + \left(\color{blue}{\mathsf{fma}\left(u, 0.3333333333333333, u \cdot \left(u \cdot -0.5\right)\right)} \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]

13. Final simplification 97.8%

    \[\leadsto u \cdot n1_i + n0_i \cdot \left(1 + \left(\left(normAngle \cdot normAngle\right) \cdot \mathsf{fma}\left(u, 0.3333333333333333, u \cdot \left(u \cdot -0.5\right)\right) - u\right)\right) \]

Alternative 4: 98.2% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 (fma u (- n1_i n0_i) n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return fmaf(u, (n1_i - n0_i), n0_i);
}

Julia

function code(normAngle, u, n0_i, n1_i)
	return fma(u, Float32(n1_i - n0_i), n0_i)
end

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. Step-by-step derivation

   1. fma-def 97.3%

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

   2. associate-*r/ 97.4%

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

   3. *-rgt-identity 97.4%

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

   4. associate-*r/ 97.6%

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

   5. *-rgt-identity 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in normAngle around 0 97.9%

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

5. Taylor expanded in normAngle around 0 97.3%

   \[\leadsto \color{blue}{n1_i \cdot u + \left(1 - u\right) \cdot n0_i} \]
6. Step-by-step derivation

   1. *-commutative 97.3%

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

   2. *-commutative 97.3%

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

   3. sub-neg 97.3%

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

   4. distribute-lft-out 97.5%

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

   5. *-rgt-identity 97.5%

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

   6. +-commutative 97.5%

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

   7. associate-+r+ 97.6%

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

   8. *-commutative 97.6%

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

   9. distribute-lft-neg-out 97.6%

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

   10. distribute-rgt-neg-in 97.6%

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

   11. distribute-lft-in 97.6%

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

   12. neg-mul-1 97.6%

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

   13. fma-def 97.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1_i + -1 \cdot n0_i, n0_i\right)} \]

   14. neg-mul-1 97.8%

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

   15. unsub-neg 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 5: 98.0% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ u \cdot n1_i - n0_i \cdot \left(-1 + \left(u - \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (-
  (* u n1_i)
  (*
   n0_i
   (+ -1.0 (- u (* (* normAngle normAngle) (* u 0.3333333333333333)))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (u * n1_i) - (n0_i * (-1.0f + (u - ((normAngle * normAngle) * (u * 0.3333333333333333f)))));
}

Fortran

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 = (u * n1_i) - (n0_i * ((-1.0e0) + (u - ((normangle * normangle) * (u * 0.3333333333333333e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(u * n1_i) - Float32(n0_i * Float32(Float32(-1.0) + Float32(u - Float32(Float32(normAngle * normAngle) * Float32(u * Float32(0.3333333333333333)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (u * n1_i) - (n0_i * (single(-1.0) + (u - ((normAngle * normAngle) * (u * single(0.3333333333333333))))));
end

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. Step-by-step derivation

   1. fma-def 97.3%

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

   2. associate-*r/ 97.4%

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

   3. *-rgt-identity 97.4%

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

   4. associate-*r/ 97.6%

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

   5. *-rgt-identity 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in normAngle around 0 97.9%

   \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{u} \cdot n1_i\right) \]
5. Step-by-step derivation

   1. fma-udef 97.9%

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

   2. *-commutative 97.9%

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

   3. sub-neg 97.9%

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

   4. distribute-rgt-in 98.0%

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

   5. *-un-lft-identity 98.0%

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

   6. cancel-sign-sub-inv 98.0%

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

6. Applied egg-rr 98.0%

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

7. Taylor expanded in normAngle around 0 97.9%

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

   1. associate--l+ 97.9%

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

   2. distribute-lft-out-- 97.9%

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

   3. unpow2 97.9%

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

9. Simplified 97.9%

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

10. Taylor expanded in u around 0 97.6%

    \[\leadsto \left(1 + \left(\color{blue}{\left(0.3333333333333333 \cdot u\right)} \cdot \left(normAngle \cdot normAngle\right) - u\right)\right) \cdot n0_i + u \cdot n1_i \]
11. Step-by-step derivation

    1. *-commutative 97.6%

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

12. Simplified 97.6%

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

13. Final simplification 97.6%

    \[\leadsto u \cdot n1_i - n0_i \cdot \left(-1 + \left(u - \left(normAngle \cdot normAngle\right) \cdot \left(u \cdot 0.3333333333333333\right)\right)\right) \]

Alternative 6: 71.0% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -2.0000000390829628 \cdot 10^{-25} \lor \neg \left(n0_i \leq 6.000000117248888 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -2.0000000390829628e-25)
         (not (<= n0_i 6.000000117248888e-25)))
   (* n0_i (- 1.0 u))
   (* u n1_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n0_i <= -2.0000000390829628e-25f) || !(n0_i <= 6.000000117248888e-25f)) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = u * n1_i;
	}
	return tmp;
}

Fortran

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
    real(4) :: tmp
    if ((n0_i <= (-2.0000000390829628e-25)) .or. (.not. (n0_i <= 6.000000117248888e-25))) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = u * n1_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n0_i <= Float32(-2.0000000390829628e-25)) || !(n0_i <= Float32(6.000000117248888e-25)))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(u * n1_i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n0_i <= single(-2.0000000390829628e-25)) || ~((n0_i <= single(6.000000117248888e-25))))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = u * n1_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2.00000004e-25 or 6.00000012e-25 < n0_i

   1. Initial program 98.7%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 98.8%

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

      2. associate-*r/ 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. associate-*r/ 98.9%

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

      5. *-rgt-identity 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in normAngle around 0 98.5%

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

   5. Taylor expanded in n1_i around 0 75.2%

      \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0_i} \]

   if -2.00000004e-25 < n0_i < 6.00000012e-25

   1. Initial program 94.5%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 94.6%

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

      2. associate-*r/ 94.8%

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

      3. *-rgt-identity 94.8%

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

      4. associate-*r/ 95.2%

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

      5. *-rgt-identity 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in normAngle around 0 95.7%

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

   5. Taylor expanded in u around inf 66.8%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -2.0000000390829628 \cdot 10^{-25} \lor \neg \left(n0_i \leq 6.000000117248888 \cdot 10^{-25}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 7: 86.1% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -1.0000000195414814 \cdot 10^{-25} \lor \neg \left(n1_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.0000000195414814e-25)
         (not (<= n1_i 1.9999999774532045e-26)))
   (+ n0_i (* u n1_i))
   (* n0_i (- 1.0 u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.0000000195414814e-25f) || !(n1_i <= 1.9999999774532045e-26f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i * (1.0f - u);
	}
	return tmp;
}

Fortran

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
    real(4) :: tmp
    if ((n1_i <= (-1.0000000195414814e-25)) .or. (.not. (n1_i <= 1.9999999774532045e-26))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i * (1.0e0 - u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.0000000195414814e-25)) || !(n1_i <= Float32(1.9999999774532045e-26)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.0000000195414814e-25)) || ~((n1_i <= single(1.9999999774532045e-26))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i * (single(1.0) - u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000002e-25 or 1.99999998e-26 < n1_i

   1. Initial program 96.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 96.6%

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

      2. associate-*r/ 96.6%

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

      3. *-rgt-identity 96.6%

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

      4. associate-*r/ 96.9%

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

      5. *-rgt-identity 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in normAngle around 0 97.4%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{u} \cdot n1_i\right) \]
   5. Step-by-step derivation

      1. fma-udef 97.4%

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

      2. *-commutative 97.4%

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

      3. sub-neg 97.4%

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

      4. distribute-rgt-in 97.5%

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

      5. *-un-lft-identity 97.5%

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

      6. cancel-sign-sub-inv 97.5%

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

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in u around 0 86.0%

      \[\leadsto \color{blue}{n0_i} + u \cdot n1_i \]

   if -1.00000002e-25 < n1_i < 1.99999998e-26

   1. Initial program 98.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 98.7%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. associate-*r/ 99.1%

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

      5. *-rgt-identity 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in normAngle around 0 97.6%

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

   5. Taylor expanded in n1_i around 0 90.2%

      \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -1.0000000195414814 \cdot 10^{-25} \lor \neg \left(n1_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \end{array} \]

Alternative 8: 86.2% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1_i \leq -1.0000000195414814 \cdot 10^{-25} \lor \neg \left(n1_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.0000000195414814e-25)
         (not (<= n1_i 1.9999999774532045e-26)))
   (+ n0_i (* u n1_i))
   (- n0_i (* u n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.0000000195414814e-25f) || !(n1_i <= 1.9999999774532045e-26f)) {
		tmp = n0_i + (u * n1_i);
	} else {
		tmp = n0_i - (u * n0_i);
	}
	return tmp;
}

Fortran

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
    real(4) :: tmp
    if ((n1_i <= (-1.0000000195414814e-25)) .or. (.not. (n1_i <= 1.9999999774532045e-26))) then
        tmp = n0_i + (u * n1_i)
    else
        tmp = n0_i - (u * n0_i)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-1.0000000195414814e-25)) || !(n1_i <= Float32(1.9999999774532045e-26)))
		tmp = Float32(n0_i + Float32(u * n1_i));
	else
		tmp = Float32(n0_i - Float32(u * n0_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-1.0000000195414814e-25)) || ~((n1_i <= single(1.9999999774532045e-26))))
		tmp = n0_i + (u * n1_i);
	else
		tmp = n0_i - (u * n0_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000002e-25 or 1.99999998e-26 < n1_i

   1. Initial program 96.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 96.6%

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

      2. associate-*r/ 96.6%

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

      3. *-rgt-identity 96.6%

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

      4. associate-*r/ 96.9%

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

      5. *-rgt-identity 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in normAngle around 0 97.4%

      \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0_i, \color{blue}{u} \cdot n1_i\right) \]
   5. Step-by-step derivation

      1. fma-udef 97.4%

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

      2. *-commutative 97.4%

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

      3. sub-neg 97.4%

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

      4. distribute-rgt-in 97.5%

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

      5. *-un-lft-identity 97.5%

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

      6. cancel-sign-sub-inv 97.5%

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

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in u around 0 86.0%

      \[\leadsto \color{blue}{n0_i} + u \cdot n1_i \]

   if -1.00000002e-25 < n1_i < 1.99999998e-26

   1. Initial program 98.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 98.7%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

      4. associate-*r/ 99.1%

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

      5. *-rgt-identity 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in normAngle around 0 97.6%

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

   5. Taylor expanded in n1_i around 0 90.2%

      \[\leadsto \color{blue}{\left(1 - u\right) \cdot n0_i} \]

   6. Taylor expanded in u around 0 90.6%

      \[\leadsto \color{blue}{-1 \cdot \left(u \cdot n0_i\right) + n0_i} \]
   7. Step-by-step derivation

      1. mul-1-neg 90.6%

         \[\leadsto \color{blue}{\left(-u \cdot n0_i\right)} + n0_i \]

      2. distribute-lft-neg-out 90.6%

         \[\leadsto \color{blue}{\left(-u\right) \cdot n0_i} + n0_i \]

      3. +-commutative 90.6%

         \[\leadsto \color{blue}{n0_i + \left(-u\right) \cdot n0_i} \]

      4. distribute-lft-neg-out 90.6%

         \[\leadsto n0_i + \color{blue}{\left(-u \cdot n0_i\right)} \]

      5. unsub-neg 90.6%

         \[\leadsto \color{blue}{n0_i - u \cdot n0_i} \]

      6. *-commutative 90.6%

         \[\leadsto n0_i - \color{blue}{n0_i \cdot u} \]

   8. Simplified 90.6%

      \[\leadsto \color{blue}{n0_i - n0_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1_i \leq -1.0000000195414814 \cdot 10^{-25} \lor \neg \left(n1_i \leq 1.9999999774532045 \cdot 10^{-26}\right):\\ \;\;\;\;n0_i + u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i - u \cdot n0_i\\ \end{array} \]

Alternative 9: 60.2% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 3.499999888929329 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.999999936531045e-20)
   n0_i
   (if (<= n0_i 3.499999888929329e-21) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.999999936531045e-20f) {
		tmp = n0_i;
	} else if (n0_i <= 3.499999888929329e-21f) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i;
	}
	return tmp;
}

Fortran

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
    real(4) :: tmp
    if (n0_i <= (-1.999999936531045e-20)) then
        tmp = n0_i
    else if (n0_i <= 3.499999888929329e-21) then
        tmp = u * n1_i
    else
        tmp = n0_i
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if (n0_i <= Float32(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= Float32(3.499999888929329e-21))
		tmp = Float32(u * n1_i);
	else
		tmp = n0_i;
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if (n0_i <= single(-1.999999936531045e-20))
		tmp = n0_i;
	elseif (n0_i <= single(3.499999888929329e-21))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -1.99999994e-20 or 3.49999989e-21 < n0_i

   1. Initial program 98.6%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 98.7%

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

      2. associate-*r/ 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. associate-*r/ 98.9%

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

      5. *-rgt-identity 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in u around 0 64.0%

      \[\leadsto \color{blue}{n0_i} \]

   if -1.99999994e-20 < n0_i < 3.49999989e-21

   1. Initial program 95.7%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1_i \]
   2. Step-by-step derivation

      1. fma-def 95.8%

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

      2. associate-*r/ 95.9%

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

      3. *-rgt-identity 95.9%

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

      4. associate-*r/ 96.2%

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

      5. *-rgt-identity 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in normAngle around 0 96.7%

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

   5. Taylor expanded in u around inf 61.6%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 3.499999888929329 \cdot 10^{-21}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 10: 98.0% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ n0_i + u \cdot \left(n1_i - n0_i\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+ n0_i (* u (- n1_i n0_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * (n1_i - n0_i));
}

Fortran

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 - n0_i))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(n1_i - n0_i)))
end

MATLAB

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

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. Step-by-step derivation

   1. fma-def 97.3%

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

   2. associate-*r/ 97.4%

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

   3. *-rgt-identity 97.4%

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

   4. associate-*r/ 97.6%

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

   5. *-rgt-identity 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in normAngle around 0 97.3%

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

5. Taylor expanded in u around -inf 97.6%

   \[\leadsto \color{blue}{-1 \cdot \left(u \cdot \left(-1 \cdot n1_i + n0_i\right)\right) + n0_i} \]
6. Step-by-step derivation

   1. +-commutative 97.6%

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

   2. mul-1-neg 97.6%

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

   3. unsub-neg 97.6%

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

   4. +-commutative 97.6%

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

   5. mul-1-neg 97.6%

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

   6. unsub-neg 97.6%

      \[\leadsto n0_i - u \cdot \color{blue}{\left(n0_i - n1_i\right)} \]

7. Simplified 97.6%

   \[\leadsto \color{blue}{n0_i - u \cdot \left(n0_i - n1_i\right)} \]

8. Final simplification 97.6%

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

Alternative 11: 47.1% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ n0_i \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i) :precision binary32 n0_i)

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i;
}

Fortran

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
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return n0_i
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i;
end

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. Step-by-step derivation

   1. fma-def 97.3%

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

   2. associate-*r/ 97.4%

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

   3. *-rgt-identity 97.4%

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

   4. associate-*r/ 97.6%

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

   5. *-rgt-identity 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in u around 0 45.5%

   \[\leadsto \color{blue}{n0_i} \]

5. Final simplification 45.5%

   \[\leadsto n0_i \]

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)))