HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 12.3s

Alternatives: 9

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\\ \left({e}^{\left(\frac{t_0 + \frac{0.48038761 + \frac{\frac{-1}{v}}{v}}{0.6931 + \frac{1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{t_0 + \left(0.6931 - \frac{1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (- (* cosTheta_O (/ cosTheta_i v)) (/ sinTheta_i (/ v sinTheta_O)))))
   (*
    (*
     (pow
      E
      (/ (+ t_0 (/ (+ 0.48038761 (/ (/ -1.0 v) v)) (+ 0.6931 (/ 1.0 v)))) 2.0))
     (pow E (/ (+ t_0 (- 0.6931 (/ 1.0 v))) 2.0)))
    (/ 0.5 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_O * (cosTheta_i / v)) - (sinTheta_i / (v / sinTheta_O));
	return (powf(((float) M_E), ((t_0 + ((0.48038761f + ((-1.0f / v) / v)) / (0.6931f + (1.0f / v)))) / 2.0f)) * powf(((float) M_E), ((t_0 + (0.6931f - (1.0f / v))) / 2.0f))) * (0.5f / v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - Float32(sinTheta_i / Float32(v / sinTheta_O)))
	return Float32(Float32((Float32(exp(1)) ^ Float32(Float32(t_0 + Float32(Float32(Float32(0.48038761) + Float32(Float32(Float32(-1.0) / v) / v)) / Float32(Float32(0.6931) + Float32(Float32(1.0) / v)))) / Float32(2.0))) * (Float32(exp(1)) ^ Float32(Float32(t_0 + Float32(Float32(0.6931) - Float32(Float32(1.0) / v))) / Float32(2.0)))) * Float32(Float32(0.5) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_O * (cosTheta_i / v)) - (sinTheta_i / (v / sinTheta_O));
	tmp = ((single(2.71828182845904523536) ^ ((t_0 + ((single(0.48038761) + ((single(-1.0) / v) / v)) / (single(0.6931) + (single(1.0) / v)))) / single(2.0))) * (single(2.71828182845904523536) ^ ((t_0 + (single(0.6931) - (single(1.0) / v))) / single(2.0)))) * (single(0.5) / v);
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.7%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.7%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \cdot \frac{0.5}{v} \]

   2. exp-prod 99.8%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \cdot \frac{0.5}{v} \]

   3. associate-/r/ 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \cdot \frac{0.5}{v} \]

   4. *-commutative 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \cdot \frac{0.5}{v} \]

   5. +-commutative 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}\right)} \cdot \frac{0.5}{v} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}} \cdot \frac{0.5}{v} \]
6. Step-by-step derivation

   1. exp-1-e 99.8%

      \[\leadsto {\color{blue}{e}}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   2. associate-/r/ 99.8%

      \[\leadsto {e}^{\left(\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   3. *-commutative 99.8%

      \[\leadsto {e}^{\left(\left(\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   4. *-commutative 99.8%

      \[\leadsto {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   5. associate-*l/ 99.8%

      \[\leadsto {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{{e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}} \cdot \frac{0.5}{v} \]
8. Step-by-step derivation

   1. sqr-pow 99.8%

      \[\leadsto \color{blue}{\left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right)} \cdot \frac{0.5}{v} \]

   2. associate-/l* 99.8%

      \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

   3. associate-/l* 99.8%

      \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

9. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right)} \cdot \frac{0.5}{v} \]
10. Step-by-step derivation

    1. flip-+ 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \color{blue}{\frac{0.6931 \cdot 0.6931 - \frac{-1}{v} \cdot \frac{-1}{v}}{0.6931 - \frac{-1}{v}}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    2. metadata-eval 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{\color{blue}{0.48038761} - \frac{-1}{v} \cdot \frac{-1}{v}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

11. Applied egg-rr 99.8%

    \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \color{blue}{\frac{0.48038761 - \frac{-1}{v} \cdot \frac{-1}{v}}{0.6931 - \frac{-1}{v}}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]
12. Step-by-step derivation

    1. metadata-eval 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 - \frac{\color{blue}{-1}}{v} \cdot \frac{-1}{v}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    2. distribute-neg-frac 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 - \color{blue}{\left(-\frac{1}{v}\right)} \cdot \frac{-1}{v}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    3. cancel-sign-sub 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{\color{blue}{0.48038761 + \frac{1}{v} \cdot \frac{-1}{v}}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    4. metadata-eval 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \frac{1}{v} \cdot \frac{\color{blue}{-1}}{v}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    5. distribute-neg-frac 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \frac{1}{v} \cdot \color{blue}{\left(-\frac{1}{v}\right)}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    6. distribute-rgt-neg-in 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \color{blue}{\left(-\frac{1}{v} \cdot \frac{1}{v}\right)}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    7. distribute-lft-neg-in 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \color{blue}{\left(-\frac{1}{v}\right) \cdot \frac{1}{v}}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    8. distribute-neg-frac 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \color{blue}{\frac{-1}{v}} \cdot \frac{1}{v}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    9. metadata-eval 99.8%

       \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \frac{\color{blue}{-1}}{v} \cdot \frac{1}{v}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    10. associate-*r/ 99.8%

        \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \color{blue}{\frac{\frac{-1}{v} \cdot 1}{v}}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

    11. *-rgt-identity 99.8%

        \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \frac{\color{blue}{\frac{-1}{v}}}{v}}{0.6931 - \frac{-1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

13. Simplified 99.8%

    \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \color{blue}{\frac{0.48038761 + \frac{\frac{-1}{v}}{v}}{0.6931 - \frac{-1}{v}}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

14. Final simplification 99.8%

    \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \frac{0.48038761 + \frac{\frac{-1}{v}}{v}}{0.6931 + \frac{1}{v}}}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 - \frac{1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 - \frac{1}{v}\right)}{2}\right)}\\ \frac{0.5}{v} \cdot \left(t_0 \cdot t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (pow
          E
          (/
           (+
            (- (* cosTheta_O (/ cosTheta_i v)) (/ sinTheta_i (/ v sinTheta_O)))
            (- 0.6931 (/ 1.0 v)))
           2.0))))
   (* (/ 0.5 v) (* t_0 t_0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = powf(((float) M_E), ((((cosTheta_O * (cosTheta_i / v)) - (sinTheta_i / (v / sinTheta_O))) + (0.6931f - (1.0f / v))) / 2.0f));
	return (0.5f / v) * (t_0 * t_0);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(exp(1)) ^ Float32(Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - Float32(sinTheta_i / Float32(v / sinTheta_O))) + Float32(Float32(0.6931) - Float32(Float32(1.0) / v))) / Float32(2.0))
	return Float32(Float32(Float32(0.5) / v) * Float32(t_0 * t_0))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(2.71828182845904523536) ^ ((((cosTheta_O * (cosTheta_i / v)) - (sinTheta_i / (v / sinTheta_O))) + (single(0.6931) - (single(1.0) / v))) / single(2.0));
	tmp = (single(0.5) / v) * (t_0 * t_0);
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.7%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.7%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \cdot \frac{0.5}{v} \]

   2. exp-prod 99.8%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \cdot \frac{0.5}{v} \]

   3. associate-/r/ 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \cdot \frac{0.5}{v} \]

   4. *-commutative 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \cdot \frac{0.5}{v} \]

   5. +-commutative 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}\right)} \cdot \frac{0.5}{v} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}} \cdot \frac{0.5}{v} \]
6. Step-by-step derivation

   1. exp-1-e 99.8%

      \[\leadsto {\color{blue}{e}}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   2. associate-/r/ 99.8%

      \[\leadsto {e}^{\left(\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   3. *-commutative 99.8%

      \[\leadsto {e}^{\left(\left(\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   4. *-commutative 99.8%

      \[\leadsto {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   5. associate-*l/ 99.8%

      \[\leadsto {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{{e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}} \cdot \frac{0.5}{v} \]
8. Step-by-step derivation

   1. sqr-pow 99.8%

      \[\leadsto \color{blue}{\left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right)} \cdot \frac{0.5}{v} \]

   2. associate-/l* 99.8%

      \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

   3. associate-/l* 99.8%

      \[\leadsto \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right) \cdot \frac{0.5}{v} \]

9. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 + \frac{-1}{v}\right)}{2}\right)}\right)} \cdot \frac{0.5}{v} \]

10. Final simplification 99.8%

    \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 - \frac{1}{v}\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right) + \left(0.6931 - \frac{1}{v}\right)}{2}\right)}\right) \]

Alternative 3: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {e}^{\left(0.6931 - \frac{1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (pow E (- 0.6931 (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(((float) M_E), (0.6931f - (1.0f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (Float32(exp(1)) ^ Float32(Float32(0.6931) - Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(2.71828182845904523536) ^ (single(0.6931) - (single(1.0) / v)));
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.7%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.7%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \cdot \frac{0.5}{v} \]

   2. exp-prod 99.8%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)}} \cdot \frac{0.5}{v} \]

   3. associate-/r/ 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \cdot \frac{0.5}{v} \]

   4. *-commutative 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)\right)} \cdot \frac{0.5}{v} \]

   5. +-commutative 99.8%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \color{blue}{\left(0.6931 + \frac{-1}{v}\right)}\right)} \cdot \frac{0.5}{v} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}} \cdot \frac{0.5}{v} \]
6. Step-by-step derivation

   1. exp-1-e 99.8%

      \[\leadsto {\color{blue}{e}}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   2. associate-/r/ 99.8%

      \[\leadsto {e}^{\left(\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   3. *-commutative 99.8%

      \[\leadsto {e}^{\left(\left(\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   4. *-commutative 99.8%

      \[\leadsto {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

   5. associate-*l/ 99.8%

      \[\leadsto {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \cdot \frac{0.5}{v} \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{{e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}} \cdot \frac{0.5}{v} \]

8. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}\right)}} \cdot \frac{0.5}{v} \]

9. Taylor expanded in cosTheta_i around 0 99.8%

   \[\leadsto {e}^{\left(\color{blue}{0.6931} - \frac{1}{v}\right)} \cdot \frac{0.5}{v} \]

10. Final simplification 99.8%

    \[\leadsto \frac{0.5}{v} \cdot {e}^{\left(0.6931 - \frac{1}{v}\right)} \]

Alternative 4: 44.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* sinTheta_i (- sinTheta_O)) v)))
   (if (<= (* sinTheta_i sinTheta_O) 1.000000023742228e-33) t_0 (exp t_0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (sinTheta_i * -sinTheta_O) / v;
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 1.000000023742228e-33f) {
		tmp = t_0;
	} else {
		tmp = expf(t_0);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (sintheta_i * -sintheta_o) / v
    if ((sintheta_i * sintheta_o) <= 1.000000023742228e-33) then
        tmp = t_0
    else
        tmp = exp(t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(1.000000023742228e-33))
		tmp = t_0;
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (sinTheta_i * -sinTheta_O) / v;
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(1.000000023742228e-33))
		tmp = t_0;
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 1.00000002e-33

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. associate-+l+ 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      2. sub-neg 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      3. associate-+l- 99.7%

         \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      4. associate-+l- 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      5. sub-neg 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      6. associate--l- 99.7%

         \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      7. associate-/l* 99.7%

         \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      8. associate-/r* 99.7%

         \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

   4. Taylor expanded in v around 0 97.9%

      \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - \left(1 + sinTheta_i \cdot sinTheta_O\right)}{v}}} \]

   5. Taylor expanded in sinTheta_i around inf 6.2%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 6.2%

         \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      2. distribute-frac-neg 6.2%

         \[\leadsto e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}} \]

      3. distribute-rgt-neg-in 6.2%

         \[\leadsto e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}} \]

   7. Simplified 6.2%

      \[\leadsto e^{\color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}} \]

   8. Taylor expanded in sinTheta_i around 0 6.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
   9. Step-by-step derivation

      1. associate-*r/ 6.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} + 1 \]

      2. neg-mul-1 6.3%

         \[\leadsto \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} + 1 \]

      3. distribute-rgt-neg-in 6.3%

         \[\leadsto \frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v} + 1 \]

      4. +-commutative 6.3%

         \[\leadsto \color{blue}{1 + \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \]

      5. distribute-rgt-neg-in 6.3%

         \[\leadsto 1 + \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} \]

      6. distribute-frac-neg 6.3%

         \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \]

      7. associate-*l/ 6.3%

         \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) \]

      8. unsub-neg 6.3%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_i}{v} \cdot sinTheta_O} \]

      9. *-commutative 6.3%

         \[\leadsto 1 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

   10. Simplified 6.3%

       \[\leadsto \color{blue}{1 - sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

   11. Taylor expanded in sinTheta_O around inf 47.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
   12. Step-by-step derivation

       1. associate-*r/ 47.1%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} \]

       2. *-commutative 47.1%

          \[\leadsto \frac{-1 \cdot \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)}}{v} \]

       3. neg-mul-1 47.1%

          \[\leadsto \frac{\color{blue}{-sinTheta_O \cdot sinTheta_i}}{v} \]

       4. distribute-rgt-neg-in 47.1%

          \[\leadsto \frac{\color{blue}{sinTheta_O \cdot \left(-sinTheta_i\right)}}{v} \]

   13. Simplified 47.1%

       \[\leadsto \color{blue}{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

   if 1.00000002e-33 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Step-by-step derivation

      1. associate-+l+ 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      2. sub-neg 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      3. associate-+l- 99.7%

         \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      4. associate-+l- 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      5. sub-neg 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      6. associate--l- 99.7%

         \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      7. associate-/l* 99.7%

         \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

      8. associate-/r* 99.7%

         \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

      9. metadata-eval 99.7%

         \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

   4. Taylor expanded in sinTheta_i around inf 45.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 45.1%

         \[\leadsto e^{\color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}}} \]

      2. neg-mul-1 45.1%

         \[\leadsto e^{\frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v}} \]

   6. Simplified 45.1%

      \[\leadsto e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq 1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}\\ \end{array} \]

Alternative 5: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (- 0.6931 (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f - (1.0f / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 - (1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) - Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) - (single(1.0) / v)));
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.7%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]

4. Taylor expanded in sinTheta_i around 0 99.7%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{1}{v}}} \cdot \frac{0.5}{v} \]

5. Taylor expanded in cosTheta_i around 0 99.7%

   \[\leadsto \color{blue}{e^{0.6931 - \frac{1}{v}}} \cdot \frac{0.5}{v} \]

6. Final simplification 99.7%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \]

Alternative 6: 98.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1 + cosTheta_O \cdot cosTheta_i}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ (+ -1.0 (* cosTheta_O cosTheta_i)) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((-1.0f + (cosTheta_O * cosTheta_i)) / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((((-1.0e0) + (costheta_o * costheta_i)) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(-1.0) + Float32(cosTheta_O * cosTheta_i)) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(-1.0) + (cosTheta_O * cosTheta_i)) / v));
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.7%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in v around 0 98.0%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - \left(1 + sinTheta_i \cdot sinTheta_O\right)}{v}}} \]

5. Taylor expanded in sinTheta_i around 0 98.0%

   \[\leadsto \color{blue}{e^{\frac{cosTheta_i \cdot cosTheta_O - 1}{v}}} \]

6. Final simplification 98.0%

   \[\leadsto e^{\frac{-1 + cosTheta_O \cdot cosTheta_i}{v}} \]

Alternative 7: 19.7% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \left(-sinTheta_i\right) \cdot \frac{sinTheta_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (- sinTheta_i) (/ sinTheta_O v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return -sinTheta_i * (sinTheta_O / v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = -sintheta_i * (sintheta_o / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(-sinTheta_i) * Float32(sinTheta_O / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = -sinTheta_i * (sinTheta_O / v);
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.7%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in v around 0 98.0%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - \left(1 + sinTheta_i \cdot sinTheta_O\right)}{v}}} \]

5. Taylor expanded in sinTheta_i around inf 13.9%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
6. Step-by-step derivation

   1. mul-1-neg 13.9%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   2. distribute-frac-neg 13.9%

      \[\leadsto e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}} \]

   3. distribute-rgt-neg-in 13.9%

      \[\leadsto e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}} \]

7. Simplified 13.9%

   \[\leadsto e^{\color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}} \]

8. Taylor expanded in sinTheta_i around 0 6.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
9. Step-by-step derivation

   1. associate-*r/ 6.2%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} + 1 \]

   2. neg-mul-1 6.2%

      \[\leadsto \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} + 1 \]

   3. distribute-rgt-neg-in 6.2%

      \[\leadsto \frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v} + 1 \]

   4. +-commutative 6.2%

      \[\leadsto \color{blue}{1 + \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \]

   5. distribute-rgt-neg-in 6.2%

      \[\leadsto 1 + \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} \]

   6. distribute-frac-neg 6.2%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \]

   7. associate-*l/ 6.2%

      \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) \]

   8. unsub-neg 6.2%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_i}{v} \cdot sinTheta_O} \]

   9. *-commutative 6.2%

      \[\leadsto 1 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

10. Simplified 6.2%

    \[\leadsto \color{blue}{1 - sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 39.1%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
12. Step-by-step derivation

    1. mul-1-neg 39.1%

       \[\leadsto \color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \]

    2. *-commutative 39.1%

       \[\leadsto -\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v} \]

    3. associate-/l* 20.8%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

    4. associate-/r/ 20.8%

       \[\leadsto -\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

    5. distribute-rgt-neg-in 20.8%

       \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

13. Simplified 20.8%

    \[\leadsto \color{blue}{\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)} \]

14. Final simplification 20.8%

    \[\leadsto \left(-sinTheta_i\right) \cdot \frac{sinTheta_O}{v} \]

Alternative 8: 37.9% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* sinTheta_i (- sinTheta_O)) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (sinTheta_i * -sinTheta_O) / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (sintheta_i * -sintheta_o) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sinTheta_i * -sinTheta_O) / v;
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.7%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in v around 0 98.0%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - \left(1 + sinTheta_i \cdot sinTheta_O\right)}{v}}} \]

5. Taylor expanded in sinTheta_i around inf 13.9%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
6. Step-by-step derivation

   1. mul-1-neg 13.9%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   2. distribute-frac-neg 13.9%

      \[\leadsto e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}} \]

   3. distribute-rgt-neg-in 13.9%

      \[\leadsto e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}} \]

7. Simplified 13.9%

   \[\leadsto e^{\color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}} \]

8. Taylor expanded in sinTheta_i around 0 6.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
9. Step-by-step derivation

   1. associate-*r/ 6.2%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} + 1 \]

   2. neg-mul-1 6.2%

      \[\leadsto \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} + 1 \]

   3. distribute-rgt-neg-in 6.2%

      \[\leadsto \frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v} + 1 \]

   4. +-commutative 6.2%

      \[\leadsto \color{blue}{1 + \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \]

   5. distribute-rgt-neg-in 6.2%

      \[\leadsto 1 + \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} \]

   6. distribute-frac-neg 6.2%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)} \]

   7. associate-*l/ 6.2%

      \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) \]

   8. unsub-neg 6.2%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_i}{v} \cdot sinTheta_O} \]

   9. *-commutative 6.2%

      \[\leadsto 1 - \color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

10. Simplified 6.2%

    \[\leadsto \color{blue}{1 - sinTheta_O \cdot \frac{sinTheta_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 39.1%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
12. Step-by-step derivation

    1. associate-*r/ 39.1%

       \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} \]

    2. *-commutative 39.1%

       \[\leadsto \frac{-1 \cdot \color{blue}{\left(sinTheta_O \cdot sinTheta_i\right)}}{v} \]

    3. neg-mul-1 39.1%

       \[\leadsto \frac{\color{blue}{-sinTheta_O \cdot sinTheta_i}}{v} \]

    4. distribute-rgt-neg-in 39.1%

       \[\leadsto \frac{\color{blue}{sinTheta_O \cdot \left(-sinTheta_i\right)}}{v} \]

13. Simplified 39.1%

    \[\leadsto \color{blue}{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

14. Final simplification 39.1%

    \[\leadsto \frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v} \]

Alternative 9: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 1.0)

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.7%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in v around 0 98.0%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - \left(1 + sinTheta_i \cdot sinTheta_O\right)}{v}}} \]

5. Taylor expanded in v around inf 6.4%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023228 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))