HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.3%

Time: 17.6s

Alternatives: 10

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: 99.7% 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \frac{1}{\frac{v}{e^{0.6931}}}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.7%

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

   3. exp-1-e 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/l* 99.7%

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

   6. fma-define 99.7%

      \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)}\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]
8. Step-by-step derivation

   1. exp-1-e 99.7%

      \[\leadsto {\color{blue}{\left(e^{1}\right)}}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)\right)}} \]

   3. unpow-prod-up 99.7%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]

   4. pow-exp 99.7%

      \[\leadsto \color{blue}{e^{1 \cdot \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   5. *-un-lft-identity 99.7%

      \[\leadsto e^{\color{blue}{0.6931 + \log \left(\frac{0.5}{v}\right)}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   6. +-commutative 99.7%

      \[\leadsto e^{\color{blue}{\log \left(\frac{0.5}{v}\right) + 0.6931}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   7. exp-sum 99.7%

      \[\leadsto \color{blue}{\left(e^{\log \left(\frac{0.5}{v}\right)} \cdot e^{0.6931}\right)} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   8. add-exp-log 99.7%

      \[\leadsto \left(\color{blue}{\frac{0.5}{v}} \cdot e^{0.6931}\right) \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   9. exp-1-e 99.7%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot {\color{blue}{e}}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*l/ 99.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.6931}}{v}} \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

    2. associate-*r/ 99.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{e^{0.6931}}{v}\right)} \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

    3. fma-undefine 99.7%

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

    4. +-commutative 99.7%

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

    5. *-lft-identity 99.7%

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

    6. metadata-eval 99.7%

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

    7. cancel-sign-sub-inv 99.7%

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

    8. associate-*r/ 99.7%

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

    9. associate-*r/ 99.7%

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

    10. div-sub 99.7%

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

    11. distribute-neg-frac2 99.7%

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

    12. cancel-sign-sub-inv 99.7%

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

    13. metadata-eval 99.7%

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

    14. *-lft-identity 99.7%

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

    15. +-commutative 99.7%

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

    16. fma-define 99.7%

        \[\leadsto \left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \cdot {e}^{\left(\frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{-v}\right)} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)}} \]
12. Step-by-step derivation

    1. clear-num 99.7%

       \[\leadsto \left(0.5 \cdot \color{blue}{\frac{1}{\frac{v}{e^{0.6931}}}}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \]

    2. inv-pow 99.7%

       \[\leadsto \left(0.5 \cdot \color{blue}{{\left(\frac{v}{e^{0.6931}}\right)}^{-1}}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \]

13. Applied egg-rr 99.7%

    \[\leadsto \left(0.5 \cdot \color{blue}{{\left(\frac{v}{e^{0.6931}}\right)}^{-1}}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \]
14. Step-by-step derivation

    1. unpow-1 99.7%

       \[\leadsto \left(0.5 \cdot \color{blue}{\frac{1}{\frac{v}{e^{0.6931}}}}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \]

15. Simplified 99.7%

    \[\leadsto \left(0.5 \cdot \color{blue}{\frac{1}{\frac{v}{e^{0.6931}}}}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \]

16. Final simplification 99.7%

    \[\leadsto \left(0.5 \cdot \frac{1}{\frac{v}{e^{0.6931}}}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \]
17. Add Preprocessing

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \cdot \left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.7%

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

   3. exp-1-e 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/l* 99.7%

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

   6. fma-define 99.7%

      \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)}\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]
8. Step-by-step derivation

   1. exp-1-e 99.7%

      \[\leadsto {\color{blue}{\left(e^{1}\right)}}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)\right)}} \]

   3. unpow-prod-up 99.7%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]

   4. pow-exp 99.7%

      \[\leadsto \color{blue}{e^{1 \cdot \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   5. *-un-lft-identity 99.7%

      \[\leadsto e^{\color{blue}{0.6931 + \log \left(\frac{0.5}{v}\right)}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   6. +-commutative 99.7%

      \[\leadsto e^{\color{blue}{\log \left(\frac{0.5}{v}\right) + 0.6931}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   7. exp-sum 99.7%

      \[\leadsto \color{blue}{\left(e^{\log \left(\frac{0.5}{v}\right)} \cdot e^{0.6931}\right)} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   8. add-exp-log 99.7%

      \[\leadsto \left(\color{blue}{\frac{0.5}{v}} \cdot e^{0.6931}\right) \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   9. exp-1-e 99.7%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot {\color{blue}{e}}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*l/ 99.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.6931}}{v}} \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

    2. associate-*r/ 99.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{e^{0.6931}}{v}\right)} \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

    3. fma-undefine 99.7%

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

    4. +-commutative 99.7%

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

    5. *-lft-identity 99.7%

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

    6. metadata-eval 99.7%

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

    7. cancel-sign-sub-inv 99.7%

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

    8. associate-*r/ 99.7%

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

    9. associate-*r/ 99.7%

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

    10. div-sub 99.7%

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

    11. distribute-neg-frac2 99.7%

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

    12. cancel-sign-sub-inv 99.7%

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

    13. metadata-eval 99.7%

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

    14. *-lft-identity 99.7%

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

    15. +-commutative 99.7%

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

    16. fma-define 99.7%

        \[\leadsto \left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \cdot {e}^{\left(\frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{-v}\right)} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)}} \]

12. Final simplification 99.7%

    \[\leadsto {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)} \cdot \left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \]
13. Add Preprocessing

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.7%

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

   3. exp-1-e 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/l* 99.7%

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

   6. fma-define 99.7%

      \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)}\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]
8. Step-by-step derivation

   1. exp-1-e 99.7%

      \[\leadsto {\color{blue}{\left(e^{1}\right)}}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   2. sub-neg 99.7%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)\right)}} \]

   3. unpow-prod-up 99.7%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]

   4. pow-exp 99.7%

      \[\leadsto \color{blue}{e^{1 \cdot \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   5. *-un-lft-identity 99.7%

      \[\leadsto e^{\color{blue}{0.6931 + \log \left(\frac{0.5}{v}\right)}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   6. +-commutative 99.7%

      \[\leadsto e^{\color{blue}{\log \left(\frac{0.5}{v}\right) + 0.6931}} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   7. exp-sum 99.7%

      \[\leadsto \color{blue}{\left(e^{\log \left(\frac{0.5}{v}\right)} \cdot e^{0.6931}\right)} \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   8. add-exp-log 99.7%

      \[\leadsto \left(\color{blue}{\frac{0.5}{v}} \cdot e^{0.6931}\right) \cdot {\left(e^{1}\right)}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

   9. exp-1-e 99.7%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot {\color{blue}{e}}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*l/ 99.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.6931}}{v}} \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

    2. associate-*r/ 99.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{e^{0.6931}}{v}\right)} \cdot {e}^{\left(-\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_i}{v}, \frac{1}{v}\right)\right)} \]

    3. fma-undefine 99.7%

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

    4. +-commutative 99.7%

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

    5. *-lft-identity 99.7%

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

    6. metadata-eval 99.7%

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

    7. cancel-sign-sub-inv 99.7%

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

    8. associate-*r/ 99.7%

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

    9. associate-*r/ 99.7%

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

    10. div-sub 99.7%

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

    11. distribute-neg-frac2 99.7%

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

    12. cancel-sign-sub-inv 99.7%

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

    13. metadata-eval 99.7%

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

    14. *-lft-identity 99.7%

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

    15. +-commutative 99.7%

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

    16. fma-define 99.7%

        \[\leadsto \left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \cdot {e}^{\left(\frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{-v}\right)} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{e^{0.6931}}{v}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{-v}\right)}} \]

12. Taylor expanded in sinTheta_O around 0 99.7%

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

13. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (pow
   E
   (+
    0.6931
    (+
     (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_O sinTheta_i)) v)
     (/ -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 + ((((cosTheta_i * cosTheta_O) - (sinTheta_O * sinTheta_i)) / v) + (-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(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_O * sinTheta_i)) / v) + 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) + ((((cosTheta_i * cosTheta_O) - (sinTheta_O * sinTheta_i)) / v) + (single(-1.0) / v))));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.7%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.7%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.7%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.7%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{1 \cdot \left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931\right)}} \]

   2. exp-prod 99.7%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931\right)}} \]

   3. fma-define 99.7%

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

   4. associate--r+ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*r/ 99.7%

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

   7. sub-div 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{0.5}{v} \cdot {e}^{\left(0.6931 + \left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_O \cdot sinTheta\_i}{v} + \frac{-1}{v}\right)\right)} \]
8. Add Preprocessing

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ (- -1.0 (* sinTheta_O sinTheta_i)) 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 - (sinTheta_O * sinTheta_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 = (0.5e0 / v) * exp((0.6931e0 + (((-1.0e0) - (sintheta_o * sintheta_i)) / 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(Float32(-1.0) - Float32(sinTheta_O * sinTheta_i)) / 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) - (sinTheta_O * sinTheta_i)) / v)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.7%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.7%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.7%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.7%

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

6. Taylor expanded in v around inf 99.7%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{0.6931 + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]
7. Step-by-step derivation

   1. mul-1-neg 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1 - sinTheta\_O \cdot sinTheta\_i}{v}} \]
10. Add Preprocessing

Alternative 6: 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 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.7%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.7%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.7%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.7%

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

6. Taylor expanded in sinTheta_O around 0 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
8. Add Preprocessing

Alternative 7: 98.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{cosTheta\_i \cdot cosTheta\_O + -1}{v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((cosTheta_i * cosTheta_O) + -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 = exp((((costheta_i * costheta_o) + (-1.0e0)) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 97.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
6. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right)}}{v}} \]

7. Simplified 97.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(sinTheta\_O \cdot sinTheta\_i + 1\right)}{v}}} \]

8. Taylor expanded in sinTheta_O around 0 97.2%

   \[\leadsto \color{blue}{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}} \]

9. Final simplification 97.2%

   \[\leadsto e^{\frac{cosTheta\_i \cdot cosTheta\_O + -1}{v}} \]
10. Add Preprocessing

Alternative 8: 12.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((sinTheta_O * (sinTheta_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((sintheta_o * (sintheta_i / -v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.9%

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

   1. mul-1-neg 13.9%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   2. associate-/l* 13.9%

      \[\leadsto e^{-\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \]

   3. distribute-rgt-neg-in 13.9%

      \[\leadsto e^{\color{blue}{sinTheta\_O \cdot \left(-\frac{sinTheta\_i}{v}\right)}} \]

   4. distribute-frac-neg2 13.9%

      \[\leadsto e^{sinTheta\_O \cdot \color{blue}{\frac{sinTheta\_i}{-v}}} \]

7. Simplified 13.9%

   \[\leadsto e^{\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}} \]

8. Final simplification 13.9%

   \[\leadsto e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}} \]
9. Add Preprocessing

Alternative 9: 13.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}} \end{array} \]

FPCore

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

C

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

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((sintheta_o / (v / sintheta_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.9%

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

   1. mul-1-neg 13.9%

      \[\leadsto e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   2. associate-/l* 13.9%

      \[\leadsto e^{-\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \]

   3. distribute-rgt-neg-in 13.9%

      \[\leadsto e^{\color{blue}{sinTheta\_O \cdot \left(-\frac{sinTheta\_i}{v}\right)}} \]

   4. distribute-frac-neg2 13.9%

      \[\leadsto e^{sinTheta\_O \cdot \color{blue}{\frac{sinTheta\_i}{-v}}} \]

7. Simplified 13.9%

   \[\leadsto e^{\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}} \]
8. Step-by-step derivation

   1. clear-num 13.9%

      \[\leadsto e^{sinTheta\_O \cdot \color{blue}{\frac{1}{\frac{-v}{sinTheta\_i}}}} \]

   2. un-div-inv 13.9%

      \[\leadsto e^{\color{blue}{\frac{sinTheta\_O}{\frac{-v}{sinTheta\_i}}}} \]

   3. add-sqr-sqrt -0.0%

      \[\leadsto e^{\frac{sinTheta\_O}{\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{sinTheta\_i}}} \]

   4. sqrt-unprod 26.6%

      \[\leadsto e^{\frac{sinTheta\_O}{\frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{sinTheta\_i}}} \]

   5. sqr-neg 26.6%

      \[\leadsto e^{\frac{sinTheta\_O}{\frac{\sqrt{\color{blue}{v \cdot v}}}{sinTheta\_i}}} \]

   6. sqrt-unprod 12.8%

      \[\leadsto e^{\frac{sinTheta\_O}{\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{sinTheta\_i}}} \]

   7. add-sqr-sqrt 12.8%

      \[\leadsto e^{\frac{sinTheta\_O}{\frac{\color{blue}{v}}{sinTheta\_i}}} \]

9. Applied egg-rr 12.8%

   \[\leadsto e^{\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}} \]

10. Final simplification 12.8%

    \[\leadsto e^{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}} \]
11. Add Preprocessing

Alternative 10: 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 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.6%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 97.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
6. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right)}}{v}} \]

7. Simplified 97.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(sinTheta\_O \cdot sinTheta\_i + 1\right)}{v}}} \]

8. Taylor expanded in v around inf 6.5%

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

9. Final simplification 6.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))))))