Curve intersection, scale width based on ribbon orientation

Percentage Accurate: 97.1% → 98.8%

Time: 22.6s

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: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(n0\_i - n0\_i \cdot u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (- n0_i (* n0_i u))
  (+
   (* u n1_i)
   (*
    (pow normAngle 2.0)
    (+
     (* -0.5 (* n0_i (pow u 2.0)))
     (* u (+ (* n0_i 0.5) (* 0.16666666666666666 (- n1_i n0_i)))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i - (n0_i * u)) + ((u * n1_i) + (powf(normAngle, 2.0f) * ((-0.5f * (n0_i * powf(u, 2.0f))) + (u * ((n0_i * 0.5f) + (0.16666666666666666f * (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 - (n0_i * u)) + ((u * n1_i) + ((normangle ** 2.0e0) * (((-0.5e0) * (n0_i * (u ** 2.0e0))) + (u * ((n0_i * 0.5e0) + (0.16666666666666666e0 * (n1_i - n0_i)))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n0_i - Float32(n0_i * u)) + Float32(Float32(u * n1_i) + Float32((normAngle ^ Float32(2.0)) * Float32(Float32(Float32(-0.5) * Float32(n0_i * (u ^ Float32(2.0)))) + Float32(u * Float32(Float32(n0_i * Float32(0.5)) + Float32(Float32(0.16666666666666666) * Float32(n1_i - n0_i))))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i - (n0_i * u)) + ((u * n1_i) + ((normAngle ^ single(2.0)) * ((single(-0.5) * (n0_i * (u ^ single(2.0)))) + (u * ((n0_i * single(0.5)) + (single(0.16666666666666666) * (n1_i - n0_i)))))));
end

Derivation

1. Initial program 97.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. *-commutative 97.1%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   2. associate-*l* 77.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   3. *-commutative 77.1%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

   4. associate-*l* 71.7%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   5. distribute-lft-out 71.7%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

3. Simplified 71.7%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
4. Add Preprocessing

5. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 98.2%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto n0\_i \cdot \color{blue}{\left(1 + \left(-u\right)\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   2. distribute-rgt-in 98.5%

      \[\leadsto \color{blue}{\left(1 \cdot n0\_i + \left(-u\right) \cdot n0\_i\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   3. *-un-lft-identity 98.5%

      \[\leadsto \left(\color{blue}{n0\_i} + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

8. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left(n0\_i + \left(-u\right) \cdot n0\_i\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

9. Taylor expanded in n1_i around 0 98.5%

   \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - \color{blue}{\left(-0.16666666666666666 \cdot n1\_i + 0.16666666666666666 \cdot n0\_i\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-commutative 98.5%

       \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - \color{blue}{\left(0.16666666666666666 \cdot n0\_i + -0.16666666666666666 \cdot n1\_i\right)}\right)\right)\right) \]

    2. metadata-eval 98.5%

       \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - \left(0.16666666666666666 \cdot n0\_i + \color{blue}{\left(-0.16666666666666666\right)} \cdot n1\_i\right)\right)\right)\right) \]

    3. distribute-lft-neg-in 98.5%

       \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - \left(0.16666666666666666 \cdot n0\_i + \color{blue}{\left(-0.16666666666666666 \cdot n1\_i\right)}\right)\right)\right)\right) \]

    4. distribute-rgt-neg-in 98.5%

       \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - \left(0.16666666666666666 \cdot n0\_i + \color{blue}{0.16666666666666666 \cdot \left(-n1\_i\right)}\right)\right)\right)\right) \]

    5. distribute-lft-in 98.5%

       \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - \color{blue}{0.16666666666666666 \cdot \left(n0\_i + \left(-n1\_i\right)\right)}\right)\right)\right) \]

    6. unsub-neg 98.5%

       \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - 0.16666666666666666 \cdot \color{blue}{\left(n0\_i - n1\_i\right)}\right)\right)\right) \]

11. Simplified 98.5%

    \[\leadsto \left(n0\_i + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - \color{blue}{0.16666666666666666 \cdot \left(n0\_i - n1\_i\right)}\right)\right)\right) \]

12. Final simplification 98.5%

    \[\leadsto \left(n0\_i - n0\_i \cdot u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 2: 98.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(n0\_i - n0\_i \cdot u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  (- n0_i (* n0_i u))
  (+
   (* u n1_i)
   (*
    (pow normAngle 2.0)
    (* u (+ (* n0_i 0.5) (* 0.16666666666666666 (- n1_i n0_i))))))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	return (n0_i - (n0_i * u)) + ((u * n1_i) + (powf(normAngle, 2.0f) * (u * ((n0_i * 0.5f) + (0.16666666666666666f * (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 - (n0_i * u)) + ((u * n1_i) + ((normangle ** 2.0e0) * (u * ((n0_i * 0.5e0) + (0.16666666666666666e0 * (n1_i - n0_i))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(Float32(n0_i - Float32(n0_i * u)) + Float32(Float32(u * n1_i) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(n0_i * Float32(0.5)) + Float32(Float32(0.16666666666666666) * Float32(n1_i - n0_i)))))))
end

MATLAB

function tmp = code(normAngle, u, n0_i, n1_i)
	tmp = (n0_i - (n0_i * u)) + ((u * n1_i) + ((normAngle ^ single(2.0)) * (u * ((n0_i * single(0.5)) + (single(0.16666666666666666) * (n1_i - n0_i))))));
end

Derivation

1. Initial program 97.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. *-commutative 97.1%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(\left(1 - u\right) \cdot normAngle\right)\right)} \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   2. associate-*l* 77.1%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right)} + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]

   3. *-commutative 77.1%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\left(\frac{1}{\sin normAngle} \cdot \sin \left(u \cdot normAngle\right)\right)} \cdot n1\_i \]

   4. associate-*l* 71.7%

      \[\leadsto \frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i\right) + \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

   5. distribute-lft-out 71.7%

      \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]

3. Simplified 71.7%

   \[\leadsto \color{blue}{\frac{1}{\sin normAngle} \cdot \left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot n0\_i + \sin \left(u \cdot normAngle\right) \cdot n1\_i\right)} \]
4. Add Preprocessing

5. Taylor expanded in normAngle around 0 98.3%

   \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(\left(-0.16666666666666666 \cdot \left(n0\_i \cdot {\left(1 - u\right)}^{3}\right) + -0.16666666666666666 \cdot \left(n1\_i \cdot {u}^{3}\right)\right) - -0.16666666666666666 \cdot \left(n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u\right)\right)\right)} \]

6. Taylor expanded in u around 0 98.2%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv 98.2%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \color{blue}{\left(0.5 \cdot n0\_i + \left(--0.16666666666666666\right) \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)}\right)\right) \]

   2. *-commutative 98.2%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(\color{blue}{n0\_i \cdot 0.5} + \left(--0.16666666666666666\right) \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   3. metadata-eval 98.2%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + \color{blue}{0.16666666666666666} \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   4. mul-1-neg 98.2%

      \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i + \color{blue}{\left(-n0\_i\right)}\right)\right)\right)\right) \]

8. Simplified 98.2%

   \[\leadsto n0\_i \cdot \left(1 - u\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \color{blue}{\left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i + \left(-n0\_i\right)\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. sub-neg 98.2%

      \[\leadsto n0\_i \cdot \color{blue}{\left(1 + \left(-u\right)\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   2. distribute-rgt-in 98.5%

      \[\leadsto \color{blue}{\left(1 \cdot n0\_i + \left(-u\right) \cdot n0\_i\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   3. *-un-lft-identity 98.5%

      \[\leadsto \left(\color{blue}{n0\_i} + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \color{blue}{\left(n0\_i + \left(-u\right) \cdot n0\_i\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i + \left(-n0\_i\right)\right)\right)\right)\right) \]

11. Final simplification 98.4%

    \[\leadsto \left(n0\_i - n0\_i \cdot u\right) + \left(u \cdot n1\_i + {normAngle}^{2} \cdot \left(u \cdot \left(n0\_i \cdot 0.5 + 0.16666666666666666 \cdot \left(n1\_i - n0\_i\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 98.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(n0\_i \cdot -0.16666666666666666 - n0\_i \cdot -0.5\right) - n1\_i \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (*
    (pow normAngle 2.0)
    (*
     u
     (-
      (- (* n0_i -0.16666666666666666) (* n0_i -0.5))
      (* 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) * (u * (((n0_i * -0.16666666666666666f) - (n0_i * -0.5f)) - (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) * (u * (((n0_i * (-0.16666666666666666e0)) - (n0_i * (-0.5e0))) - (n1_i * (-0.16666666666666666e0))))))
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	return Float32(n0_i + Float32(Float32(u * Float32(n1_i - n0_i)) + Float32((normAngle ^ Float32(2.0)) * Float32(u * Float32(Float32(Float32(n0_i * Float32(-0.16666666666666666)) - Float32(n0_i * Float32(-0.5))) - 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)) * (u * (((n0_i * single(-0.16666666666666666)) - (n0_i * single(-0.5))) - (n1_i * single(-0.16666666666666666))))));
end

Derivation

1. Initial program 97.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. fma-define 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

   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. Add Preprocessing

5. Taylor expanded in u around 0 85.7%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)} \]

6. Taylor expanded in normAngle around 0 98.4%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right) - -0.16666666666666666 \cdot n1\_i\right)\right)\right)} \]

7. Final simplification 98.4%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(\left(n0\_i \cdot -0.16666666666666666 - n0\_i \cdot -0.5\right) - n1\_i \cdot -0.16666666666666666\right)\right)\right) \]
8. Add Preprocessing

Alternative 4: 98.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n1\_i \cdot \left(u \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (+
  n0_i
  (+
   (* u (- n1_i n0_i))
   (* (pow normAngle 2.0) (* n1_i (* u 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 * (u * 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 * (u * 0.16666666666666666e0))))
end function

Julia

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

Derivation

1. Initial program 97.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. fma-define 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

   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. Add Preprocessing

5. Taylor expanded in u around 0 85.7%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)} \]

6. Taylor expanded in normAngle around 0 98.4%

   \[\leadsto n0\_i + \color{blue}{\left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(u \cdot \left(-1 \cdot \left(-0.5 \cdot n0\_i - -0.16666666666666666 \cdot n0\_i\right) - -0.16666666666666666 \cdot n1\_i\right)\right)\right)} \]

7. Taylor expanded in n0_i around 0 98.2%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(n1\_i \cdot u\right)\right)}\right) \]
8. Step-by-step derivation

   1. associate-*r* 98.2%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(\left(0.16666666666666666 \cdot n1\_i\right) \cdot u\right)}\right) \]

   2. *-commutative 98.2%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \left(\color{blue}{\left(n1\_i \cdot 0.16666666666666666\right)} \cdot u\right)\right) \]

   3. associate-*l* 98.2%

      \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(n1\_i \cdot \left(0.16666666666666666 \cdot u\right)\right)}\right) \]

9. Simplified 98.2%

   \[\leadsto n0\_i + \left(u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + {normAngle}^{2} \cdot \color{blue}{\left(n1\_i \cdot \left(0.16666666666666666 \cdot u\right)\right)}\right) \]

10. Final simplification 98.2%

    \[\leadsto n0\_i + \left(u \cdot \left(n1\_i - n0\_i\right) + {normAngle}^{2} \cdot \left(n1\_i \cdot \left(u \cdot 0.16666666666666666\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 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 97.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. fma-define 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

   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. Add Preprocessing

5. Taylor expanded in u around 0 85.7%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)} \]

6. Taylor expanded in normAngle around 0 96.9%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} \]
7. Step-by-step derivation

   1. +-commutative 96.9%

      \[\leadsto \color{blue}{u \cdot \left(n1\_i + -1 \cdot n0\_i\right) + n0\_i} \]

   2. fma-define 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1\_i + -1 \cdot n0\_i, n0\_i\right)} \]

   3. mul-1-neg 97.1%

      \[\leadsto \mathsf{fma}\left(u, n1\_i + \color{blue}{\left(-n0\_i\right)}, n0\_i\right) \]

   4. unsub-neg 97.1%

      \[\leadsto \mathsf{fma}\left(u, \color{blue}{n1\_i - n0\_i}, n0\_i\right) \]

8. Simplified 97.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, n1\_i - n0\_i, n0\_i\right)} \]

9. Final simplification 97.1%

   \[\leadsto \mathsf{fma}\left(u, n1\_i - n0\_i, n0\_i\right) \]
10. Add Preprocessing

Alternative 6: 70.4% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.9999998245167 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 1.999999967550318 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.9999998245167e-14) (not (<= n1_i 1.999999967550318e-17)))
   (* 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 <= -9.9999998245167e-14f) || !(n1_i <= 1.999999967550318e-17f)) {
		tmp = 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 <= (-9.9999998245167e-14)) .or. (.not. (n1_i <= 1.999999967550318e-17))) then
        tmp = 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(-9.9999998245167e-14)) || !(n1_i <= Float32(1.999999967550318e-17)))
		tmp = 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(-9.9999998245167e-14)) || ~((n1_i <= single(1.999999967550318e-17))))
		tmp = 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 < -9.99999982e-14 or 1.99999997e-17 < n1_i

   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. fma-define 96.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

      2. associate-*r/ 96.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      3. *-rgt-identity 96.9%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      4. associate-*r/ 97.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1\_i\right) \]

      5. *-rgt-identity 97.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1\_i\right) \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n0_i around 0 61.9%

      \[\leadsto \color{blue}{\frac{n1\_i \cdot \sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]
   6. Step-by-step derivation

      1. associate-/l* 68.4%

         \[\leadsto \color{blue}{n1\_i \cdot \frac{\sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]

   7. Simplified 68.4%

      \[\leadsto \color{blue}{n1\_i \cdot \frac{\sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]

   8. Taylor expanded in normAngle around 0 68.0%

      \[\leadsto n1\_i \cdot \color{blue}{u} \]

   if -9.99999982e-14 < n1_i < 1.99999997e-17

   1. Initial program 97.3%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Step-by-step derivation

      1. fma-define 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/ 97.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      3. *-rgt-identity 97.6%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      4. associate-*r/ 97.8%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1\_i\right) \]

      5. *-rgt-identity 97.8%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1\_i\right) \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in normAngle around 0 97.2%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   6. Step-by-step derivation

      1. fma-define 97.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, n1\_i \cdot u\right)} \]

      2. *-commutative 97.2%

         \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \color{blue}{u \cdot n1\_i}\right) \]

   7. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)} \]

   8. Taylor expanded in n0_i around inf 72.4%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.9999998245167 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 1.999999967550318 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i \cdot \left(1 - u\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.5% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.9999998245167 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 1.999999967550318 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -9.9999998245167e-14) (not (<= n1_i 1.999999967550318e-17)))
   (* u n1_i)
   (- n0_i (* n0_i u))))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -9.9999998245167e-14f) || !(n1_i <= 1.999999967550318e-17f)) {
		tmp = u * n1_i;
	} else {
		tmp = n0_i - (n0_i * 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 <= (-9.9999998245167e-14)) .or. (.not. (n1_i <= 1.999999967550318e-17))) then
        tmp = u * n1_i
    else
        tmp = n0_i - (n0_i * u)
    end if
    code = tmp
end function

Julia

function code(normAngle, u, n0_i, n1_i)
	tmp = Float32(0.0)
	if ((n1_i <= Float32(-9.9999998245167e-14)) || !(n1_i <= Float32(1.999999967550318e-17)))
		tmp = Float32(u * n1_i);
	else
		tmp = Float32(n0_i - Float32(n0_i * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(normAngle, u, n0_i, n1_i)
	tmp = single(0.0);
	if ((n1_i <= single(-9.9999998245167e-14)) || ~((n1_i <= single(1.999999967550318e-17))))
		tmp = u * n1_i;
	else
		tmp = n0_i - (n0_i * u);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -9.99999982e-14 or 1.99999997e-17 < n1_i

   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. fma-define 96.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

      2. associate-*r/ 96.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      3. *-rgt-identity 96.9%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      4. associate-*r/ 97.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1\_i\right) \]

      5. *-rgt-identity 97.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1\_i\right) \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n0_i around 0 61.9%

      \[\leadsto \color{blue}{\frac{n1\_i \cdot \sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]
   6. Step-by-step derivation

      1. associate-/l* 68.4%

         \[\leadsto \color{blue}{n1\_i \cdot \frac{\sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]

   7. Simplified 68.4%

      \[\leadsto \color{blue}{n1\_i \cdot \frac{\sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]

   8. Taylor expanded in normAngle around 0 68.0%

      \[\leadsto n1\_i \cdot \color{blue}{u} \]

   if -9.99999982e-14 < n1_i < 1.99999997e-17

   1. Initial program 97.3%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Step-by-step derivation

      1. fma-define 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/ 97.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      3. *-rgt-identity 97.6%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      4. associate-*r/ 97.8%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1\_i\right) \]

      5. *-rgt-identity 97.8%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1\_i\right) \]

   3. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in normAngle around 0 97.2%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right) + n1\_i \cdot u} \]
   6. Step-by-step derivation

      1. fma-define 97.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, n1\_i \cdot u\right)} \]

      2. *-commutative 97.2%

         \[\leadsto \mathsf{fma}\left(n0\_i, 1 - u, \color{blue}{u \cdot n1\_i}\right) \]

   7. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(n0\_i, 1 - u, u \cdot n1\_i\right)} \]

   8. Taylor expanded in n0_i around inf 72.4%

      \[\leadsto \color{blue}{n0\_i \cdot \left(1 - u\right)} \]
   9. Step-by-step derivation

      1. sub-neg 98.3%

         \[\leadsto n0\_i \cdot \color{blue}{\left(1 + \left(-u\right)\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

      2. distribute-rgt-in 98.6%

         \[\leadsto \color{blue}{\left(1 \cdot n0\_i + \left(-u\right) \cdot n0\_i\right)} + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

      3. *-un-lft-identity 98.6%

         \[\leadsto \left(\color{blue}{n0\_i} + \left(-u\right) \cdot n0\_i\right) + \left(n1\_i \cdot u + {normAngle}^{2} \cdot \left(-0.5 \cdot \left(n0\_i \cdot {u}^{2}\right) + u \cdot \left(0.5 \cdot n0\_i - -0.16666666666666666 \cdot \left(n1\_i + -1 \cdot n0\_i\right)\right)\right)\right) \]

   10. Applied egg-rr 72.7%

       \[\leadsto \color{blue}{n0\_i + \left(-u\right) \cdot n0\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -9.9999998245167 \cdot 10^{-14} \lor \neg \left(n1\_i \leq 1.999999967550318 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i - n0\_i \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000045813705 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 1.999999967550318 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \end{array} \]

FPCore

(FPCore (normAngle u n0_i n1_i)
 :precision binary32
 (if (or (<= n1_i -1.000000045813705e-18)
         (not (<= n1_i 1.999999967550318e-17)))
   (* u n1_i)
   n0_i))

C

float code(float normAngle, float u, float n0_i, float n1_i) {
	float tmp;
	if ((n1_i <= -1.000000045813705e-18f) || !(n1_i <= 1.999999967550318e-17f)) {
		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 ((n1_i <= (-1.000000045813705e-18)) .or. (.not. (n1_i <= 1.999999967550318e-17))) 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 ((n1_i <= Float32(-1.000000045813705e-18)) || !(n1_i <= Float32(1.999999967550318e-17)))
		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 ((n1_i <= single(-1.000000045813705e-18)) || ~((n1_i <= single(1.999999967550318e-17))))
		tmp = u * n1_i;
	else
		tmp = n0_i;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if n1_i < -1.00000005e-18 or 1.99999997e-17 < n1_i

   1. Initial program 96.2%

      \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
   2. Step-by-step derivation

      1. fma-define 96.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

      2. associate-*r/ 96.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      3. *-rgt-identity 96.2%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      4. associate-*r/ 96.8%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1\_i\right) \]

      5. *-rgt-identity 96.8%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1\_i\right) \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in n0_i around 0 58.6%

      \[\leadsto \color{blue}{\frac{n1\_i \cdot \sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]
   6. Step-by-step derivation

      1. associate-/l* 65.0%

         \[\leadsto \color{blue}{n1\_i \cdot \frac{\sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]

   7. Simplified 65.0%

      \[\leadsto \color{blue}{n1\_i \cdot \frac{\sin \left(normAngle \cdot u\right)}{\sin normAngle}} \]

   8. Taylor expanded in normAngle around 0 64.8%

      \[\leadsto n1\_i \cdot \color{blue}{u} \]

   if -1.00000005e-18 < n1_i < 1.99999997e-17

   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. fma-define 97.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

      2. associate-*r/ 98.1%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot 1}{\sin normAngle}}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      3. *-rgt-identity 98.1%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sin \left(\left(1 - u\right) \cdot normAngle\right)}}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right) \]

      4. associate-*r/ 98.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \color{blue}{\frac{\sin \left(u \cdot normAngle\right) \cdot 1}{\sin normAngle}} \cdot n1\_i\right) \]

      5. *-rgt-identity 98.2%

         \[\leadsto \mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\color{blue}{\sin \left(u \cdot normAngle\right)}}{\sin normAngle} \cdot n1\_i\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(1 - u\right) \cdot normAngle\right)}{\sin normAngle}, n0\_i, \frac{\sin \left(u \cdot normAngle\right)}{\sin normAngle} \cdot n1\_i\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in u around 0 57.7%

      \[\leadsto \color{blue}{n0\_i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n1\_i \leq -1.000000045813705 \cdot 10^{-18} \lor \neg \left(n1\_i \leq 1.999999967550318 \cdot 10^{-17}\right):\\ \;\;\;\;u \cdot n1\_i\\ \mathbf{else}:\\ \;\;\;\;n0\_i\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% 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 97.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. fma-define 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

   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. Add Preprocessing

5. Taylor expanded in u around 0 85.7%

   \[\leadsto \color{blue}{n0\_i + u \cdot \left(-1 \cdot \frac{n0\_i \cdot \left(normAngle \cdot \cos normAngle\right)}{\sin normAngle} + \frac{n1\_i \cdot normAngle}{\sin normAngle}\right)} \]

6. Taylor expanded in normAngle around 0 96.9%

   \[\leadsto n0\_i + \color{blue}{u \cdot \left(n1\_i + -1 \cdot n0\_i\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 96.9%

      \[\leadsto n0\_i + u \cdot \left(n1\_i + \color{blue}{\left(-n0\_i\right)}\right) \]

   2. unsub-neg 96.9%

      \[\leadsto n0\_i + u \cdot \color{blue}{\left(n1\_i - n0\_i\right)} \]

8. Simplified 96.9%

   \[\leadsto n0\_i + \color{blue}{u \cdot \left(n1\_i - n0\_i\right)} \]

9. Final simplification 96.9%

   \[\leadsto n0\_i + u \cdot \left(n1\_i - n0\_i\right) \]
10. Add Preprocessing

Alternative 10: 47.4% 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 97.1%

   \[\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n0\_i + \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i \]
2. Step-by-step derivation

   1. fma-define 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(\left(1 - u\right) \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}, n0\_i, \left(\sin \left(u \cdot normAngle\right) \cdot \frac{1}{\sin normAngle}\right) \cdot n1\_i\right)} \]

   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. Add Preprocessing

5. Taylor expanded in u around 0 43.7%

   \[\leadsto \color{blue}{n0\_i} \]

6. Final simplification 43.7%

   \[\leadsto n0\_i \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))