Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 99.3%

Time: 14.2s

Alternatives: 10

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-*l* 74.2%

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

   5. distribute-lft-out 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in u around 0 88.3%

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

   1. fma-def 88.3%

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

   2. associate-/l* 90.9%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 98.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ n0_i + \left({normAngle}^{2} \cdot \left(n1_i \cdot \left(u \cdot 0.16666666666666666\right)\right) + u \cdot \left(n1_i - n0_i\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* (pow normAngle 2.0) (* n1_i (* u 0.16666666666666666)))
   (* u (- n1_i n0_i)))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + ((powf(normAngle, 2.0f) * (n1_i * (u * 0.16666666666666666f))) + (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 + (((normangle ** 2.0e0) * (n1_i * (u * 0.16666666666666666e0))) + (u * (n1_i - n0_i)))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32((normAngle ^ Float32(2.0)) * Float32(n1_i * Float32(u * Float32(0.16666666666666666)))) + Float32(u * Float32(n1_i - n0_i))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (((normAngle ^ single(2.0)) * (n1_i * (u * single(0.16666666666666666)))) + (u * (n1_i - n0_i)));
end

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-*l* 74.2%

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

   5. distribute-lft-out 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in u around 0 88.3%

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

   1. fma-def 88.3%

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

   2. associate-/l* 90.9%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in normAngle around 0 98.7%

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

8. Taylor expanded in n0_i around 0 98.7%

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

   1. associate-*r* 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*l* 98.7%

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

10. Simplified 98.7%

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

11. Taylor expanded in n1_i around 0 98.6%

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

    1. +-commutative 98.6%

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

    2. associate-*r* 98.6%

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

    3. distribute-rgt-in 98.7%

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

    4. mul-1-neg 98.7%

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

    5. unsub-neg 98.7%

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

13. Simplified 98.7%

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

14. Final simplification 98.7%

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

Alternative 3: 98.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ n0_i + u \cdot \left(\left(n1_i - n0_i\right) - {normAngle}^{2} \cdot \left(n1_i \cdot -0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (*
   u
   (- (- n1_i n0_i) (* (pow normAngle 2.0) (* n1_i -0.16666666666666666))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return n0_i + (u * ((n1_i - n0_i) - (powf(normAngle, 2.0f) * (n1_i * -0.16666666666666666f))));
}

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) - ((normangle ** 2.0e0) * (n1_i * (-0.16666666666666666e0)))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(u * Float32(Float32(n1_i - n0_i) - Float32((normAngle ^ Float32(2.0)) * Float32(n1_i * Float32(-0.16666666666666666))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = n0_i + (u * ((n1_i - n0_i) - ((normAngle ^ single(2.0)) * (n1_i * single(-0.16666666666666666)))));
end

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-*l* 74.2%

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

   5. distribute-lft-out 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in u around 0 88.3%

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

   1. fma-def 88.3%

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

   2. associate-/l* 90.9%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in normAngle around 0 98.7%

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

8. Taylor expanded in n0_i around 0 98.7%

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

   1. associate-*r* 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*l* 98.7%

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

10. Simplified 98.7%

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

11. Taylor expanded in u around -inf 98.7%

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

    1. mul-1-neg 98.7%

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

    2. unsub-neg 98.7%

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

    3. +-commutative 98.7%

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

    4. mul-1-neg 98.7%

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

    5. unsub-neg 98.7%

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

    6. associate-*r* 98.7%

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

    7. mul-1-neg 98.7%

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

    8. unsub-neg 98.7%

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

13. Simplified 98.7%

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

14. Final simplification 98.7%

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

Alternative 4: 98.1% 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 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-*l* 74.2%

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

   5. distribute-lft-out 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in u around 0 88.3%

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

   1. fma-def 88.3%

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

   2. associate-/l* 90.9%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in normAngle around 0 98.0%

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

   1. +-commutative 98.0%

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

   2. fma-def 98.1%

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

   3. mul-1-neg 98.1%

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

   4. unsub-neg 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 5: 71.2% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.9999999996399175 \cdot 10^{-23} \lor \neg \left(n0_i \leq 1.5000000786160286 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n0_i -1.9999999996399175e-23)
         (not (<= n0_i 1.5000000786160286e-23)))
   (* 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 <= -1.9999999996399175e-23f) || !(n0_i <= 1.5000000786160286e-23f)) {
		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 <= (-1.9999999996399175e-23)) .or. (.not. (n0_i <= 1.5000000786160286e-23))) 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(-1.9999999996399175e-23)) || !(n0_i <= Float32(1.5000000786160286e-23)))
		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(-1.9999999996399175e-23)) || ~((n0_i <= single(1.5000000786160286e-23))))
		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 < -2e-23 or 1.50000008e-23 < n0_i

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 82.5%

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

      3. *-commutative 82.5%

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

      4. associate-*l* 81.4%

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

      5. distribute-lft-out 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in u around 0 94.1%

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

      1. fma-def 94.1%

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

      2. associate-/l* 97.1%

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

      3. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in normAngle around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. fma-def 98.3%

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

      3. mul-1-neg 98.3%

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

      4. unsub-neg 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in n1_i around 0 80.0%

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

       1. *-rgt-identity 80.0%

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

       2. mul-1-neg 80.0%

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

       3. distribute-rgt-neg-in 80.0%

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

       4. mul-1-neg 80.0%

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

       5. distribute-lft-in 79.9%

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

       6. mul-1-neg 79.9%

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

       7. unsub-neg 79.9%

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

   12. Simplified 79.9%

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

   if -2e-23 < n0_i < 1.50000008e-23

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-*l* 62.4%

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

      5. distribute-lft-out 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in normAngle around 0 97.5%

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

   5. Taylor expanded in n0_i around 0 68.5%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 68.5%

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

   7. Simplified 68.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.9999999996399175 \cdot 10^{-23} \lor \neg \left(n0_i \leq 1.5000000786160286 \cdot 10^{-23}\right):\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot n1_i\\ \end{array} \]

Alternative 6: 61.0% accurate, 58.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -1.9999999996399175e-23)
   n0_i
   (if (<= n0_i 1.5000000786160286e-23) (* u n1_i) n0_i)))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -1.9999999996399175e-23f) {
		tmp = n0_i;
	} else if (n0_i <= 1.5000000786160286e-23f) {
		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 <= (-1.9999999996399175e-23)) then
        tmp = n0_i
    else if (n0_i <= 1.5000000786160286e-23) 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(-1.9999999996399175e-23))
		tmp = n0_i;
	elseif (n0_i <= Float32(1.5000000786160286e-23))
		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(-1.9999999996399175e-23))
		tmp = n0_i;
	elseif (n0_i <= single(1.5000000786160286e-23))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -2e-23 or 1.50000008e-23 < n0_i

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 82.5%

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

      3. *-commutative 82.5%

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

      4. associate-*l* 81.4%

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

      5. distribute-lft-out 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in u around 0 94.1%

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

      1. fma-def 94.1%

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

      2. associate-/l* 97.1%

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

      3. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in u around 0 63.4%

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

   if -2e-23 < n0_i < 1.50000008e-23

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*l* 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-*l* 62.4%

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

      5. distribute-lft-out 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in normAngle around 0 97.5%

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

   5. Taylor expanded in n0_i around 0 68.5%

      \[\leadsto \color{blue}{n1_i \cdot u} \]
   6. Step-by-step derivation

      1. *-commutative 68.5%

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

   7. Simplified 68.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;n0_i\\ \mathbf{elif}\;n0_i \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;u \cdot n1_i\\ \mathbf{else}:\\ \;\;\;\;n0_i\\ \end{array} \]

Alternative 7: 84.0% accurate, 59.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -4.999999980020986e-13) (* n0_i (- 1.0 u)) (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -4.999999980020986e-13f) {
		tmp = n0_i * (1.0f - u);
	} else {
		tmp = n0_i + (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 <= (-4.999999980020986e-13)) then
        tmp = n0_i * (1.0e0 - u)
    else
        tmp = n0_i + (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(-4.999999980020986e-13))
		tmp = Float32(n0_i * Float32(Float32(1.0) - u));
	else
		tmp = Float32(n0_i + 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(-4.999999980020986e-13))
		tmp = n0_i * (single(1.0) - u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999998e-13

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 94.8%

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

      3. *-commutative 94.8%

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

      4. associate-*l* 94.8%

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

      5. distribute-lft-out 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in u around 0 99.5%

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

      1. fma-def 99.5%

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

      2. associate-/l* 99.5%

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

      3. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in normAngle around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in n1_i around 0 96.2%

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

       1. *-rgt-identity 96.2%

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

       2. mul-1-neg 96.2%

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

       3. distribute-rgt-neg-in 96.2%

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

       4. mul-1-neg 96.2%

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

       5. distribute-lft-in 96.0%

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

       6. mul-1-neg 96.0%

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

       7. unsub-neg 96.0%

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

   12. Simplified 96.0%

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

   if -4.99999998e-13 < n0_i

   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. *-commutative 97.2%

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

      2. associate-*l* 77.5%

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

      3. *-commutative 77.5%

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

      4. associate-*l* 70.1%

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

      5. distribute-lft-out 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in u around 0 86.0%

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

      1. fma-def 86.0%

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

      2. associate-/l* 89.2%

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

      3. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in n0_i around 0 73.9%

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

      1. associate-/l* 82.7%

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

      2. *-commutative 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in normAngle around 0 82.9%

       \[\leadsto \color{blue}{n0_i + n1_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i \cdot \left(1 - u\right)\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \]

Alternative 8: 84.0% accurate, 59.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (<= n0_i -4.999999980020986e-13) (- n0_i (* n0_i u)) (+ n0_i (* u n1_i))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if (n0_i <= -4.999999980020986e-13f) {
		tmp = n0_i - (n0_i * u);
	} else {
		tmp = n0_i + (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 <= (-4.999999980020986e-13)) then
        tmp = n0_i - (n0_i * u)
    else
        tmp = n0_i + (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(-4.999999980020986e-13))
		tmp = Float32(n0_i - Float32(n0_i * u));
	else
		tmp = Float32(n0_i + 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(-4.999999980020986e-13))
		tmp = n0_i - (n0_i * u);
	else
		tmp = n0_i + (u * n1_i);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n0_i < -4.99999998e-13

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 94.8%

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

      3. *-commutative 94.8%

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

      4. associate-*l* 94.8%

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

      5. distribute-lft-out 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in u around 0 99.5%

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

      1. fma-def 99.5%

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

      2. associate-/l* 99.5%

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

      3. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in normAngle around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in n1_i around 0 96.2%

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

       1. associate-*r* 96.2%

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

       2. neg-mul-1 96.2%

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

   12. Simplified 96.2%

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

   if -4.99999998e-13 < n0_i

   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. *-commutative 97.2%

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

      2. associate-*l* 77.5%

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

      3. *-commutative 77.5%

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

      4. associate-*l* 70.1%

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

      5. distribute-lft-out 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in u around 0 86.0%

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

      1. fma-def 86.0%

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

      2. associate-/l* 89.2%

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

      3. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in n0_i around 0 73.9%

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

      1. associate-/l* 82.7%

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

      2. *-commutative 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in normAngle around 0 82.9%

       \[\leadsto \color{blue}{n0_i + n1_i \cdot u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n0_i \leq -4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;n0_i - n0_i \cdot u\\ \mathbf{else}:\\ \;\;\;\;n0_i + u \cdot n1_i\\ \end{array} \]

Alternative 9: 97.9% 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 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-*l* 74.2%

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

   5. distribute-lft-out 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in normAngle around 0 97.8%

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

5. Taylor expanded in u around -inf 98.0%

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

   1. mul-1-neg 98.0%

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

   2. unsub-neg 98.0%

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

   3. mul-1-neg 98.0%

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

   4. unsub-neg 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Alternative 10: 46.9% 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 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l* 80.4%

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

   3. *-commutative 80.4%

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

   4. associate-*l* 74.2%

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

   5. distribute-lft-out 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in u around 0 88.3%

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

   1. fma-def 88.3%

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

   2. associate-/l* 90.9%

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

   3. associate-/l* 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in u around 0 49.3%

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

8. Final simplification 49.3%

   \[\leadsto n0_i \]

Reproduce

herbie shell --seed 2023315 
(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)))