HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 15.7s

Alternatives: 11

Speedup: 2.0×

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \frac{-1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{v \cdot 2}\right)} \end{array} \]

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 (* v 2.0))))))

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 / (v * 2.0f)))));
}

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 = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) + ((-1.0e0) / v)) + 0.6931e0) + log((1.0e0 / (v * 2.0e0)))))
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(v * Float32(2.0))))))
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) / (v * single(2.0))))));
end

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. associate--r+ 99.8%

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

   4. associate-*r/ 99.8%

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

   5. associate-*r/ 99.8%

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

   6. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-1-e 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in cosTheta_i around 0 99.8%

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

    1. *-commutative 99.8%

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

    2. *-lft-identity 99.8%

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

    3. metadata-eval 99.8%

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

    4. cancel-sign-sub-inv 99.8%

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

    5. neg-mul-1 99.8%

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

    6. distribute-neg-frac 99.8%

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

    7. div-sub 99.8%

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

    8. *-commutative 99.8%

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

    9. mul-1-neg 99.8%

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

    10. associate-*r/ 99.8%

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

    11. mul-1-neg 99.8%

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

    12. distribute-lft-neg-out 99.8%

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

    13. cancel-sign-sub 99.8%

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

    14. neg-mul-1 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (pow E (- 0.6931 (fma 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 - fmaf(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) - fma(sinTheta_O, Float32(sinTheta_i / v), Float32(Float32(1.0) / v)))))
end

Derivation

1. Initial program 99.8%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. e-exp-1 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/l* 99.7%

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

   6. fma-define 99.7%

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

7. Applied egg-rr 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((((-1.0e0) - (sintheta_i * sintheta_o)) / v)) * exp(0.6931e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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. exp-sum 99.7%

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

   2. fma-define 99.7%

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

   3. associate--r+ 99.7%

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

   4. associate-*r/ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. sub-div 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around 0 99.7%

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

   1. *-commutative 99.8%

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

   2. *-lft-identity 99.8%

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

   3. metadata-eval 99.8%

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

   4. cancel-sign-sub-inv 99.8%

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

   5. neg-mul-1 99.8%

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

   6. distribute-neg-frac 99.8%

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

   7. div-sub 99.8%

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

   8. *-commutative 99.8%

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

   9. mul-1-neg 99.8%

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

   10. associate-*r/ 99.8%

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

   11. mul-1-neg 99.8%

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

   12. distribute-lft-neg-out 99.8%

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

   13. cancel-sign-sub 99.8%

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

   14. neg-mul-1 99.8%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 5: 17.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -5.044674471569341 \cdot 10^{-44}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -5.044674471569341e-44)
   (exp (* cosTheta_O (/ cosTheta_i v)))
   (exp (/ (* sinTheta_i sinTheta_O) (- v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -5.044674471569341e-44f) {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	} else {
		tmp = expf(((sinTheta_i * sinTheta_O) / -v));
	}
	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) :: tmp
    if ((costheta_i * costheta_o) <= (-5.044674471569341e-44)) then
        tmp = exp((costheta_o * (costheta_i / v)))
    else
        tmp = exp(((sintheta_i * sintheta_o) / -v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-5.044674471569341e-44))
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	else
		tmp = exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-5.044674471569341e-44))
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	else
		tmp = exp(((sinTheta_i * sinTheta_O) / -v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -5.04467e-44

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate--l- 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in cosTheta_i around inf 77.3%

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

   6. Taylor expanded in cosTheta_i around inf 34.0%

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

      1. associate-/l* 34.0%

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

   8. Simplified 34.0%

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

   if -5.04467e-44 < (*.f32 cosTheta_i cosTheta_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. associate--l- 99.7%

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

      3. associate-/l* 99.7%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in cosTheta_i around inf 75.4%

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

   6. Taylor expanded in sinTheta_O around inf 15.5%

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

      1. associate-*r/ 15.5%

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

      2. *-commutative 15.5%

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

      3. neg-mul-1 15.5%

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

      4. distribute-rgt-neg-in 15.5%

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

   8. Simplified 15.5%

      \[\leadsto e^{\color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -5.044674471569341 \cdot 10^{-44}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. associate--r+ 99.8%

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

   4. associate-*r/ 99.8%

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

   5. associate-*r/ 99.8%

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

   6. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-1-e 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in cosTheta_i around 0 99.6%

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

    1. log-E 99.8%

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

    2. +-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. associate-*r/ 99.8%

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

    5. fma-undefine 99.8%

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

    6. associate-+r- 99.8%

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

    7. *-commutative 99.8%

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

    8. *-rgt-identity 99.8%

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

    9. exp-sum 99.7%

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

    10. rem-exp-log 99.7%

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

    11. sub-neg 99.7%

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

    12. sub-neg 99.7%

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

    13. fma-undefine 99.7%

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

    14. associate-*r/ 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 7: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. e-exp-1 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/l* 99.7%

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

   6. fma-define 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in sinTheta_O around 0 99.3%

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

   1. exp-to-pow 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 8: 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.5%

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

7. Final simplification 99.5%

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

Alternative 9: 97.8% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow E (/ (+ (* 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 powf(((float) M_E), (((cosTheta_i * cosTheta_O) + -1.0f) / v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(1)) ^ 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 = single(2.71828182845904523536) ^ (((cosTheta_i * cosTheta_O) + single(-1.0)) / v);
end

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. associate--r+ 99.8%

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

   4. associate-*r/ 99.8%

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

   5. associate-*r/ 99.8%

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

   6. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-1-e 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in v around 0 97.7%

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

    1. associate--r+ 97.7%

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

13. Simplified 97.7%

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

14. Taylor expanded in sinTheta_O around 0 97.7%

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

15. Final simplification 97.7%

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

Alternative 10: 12.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((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((costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in cosTheta_i around inf 75.9%

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

6. Taylor expanded in cosTheta_i around inf 14.3%

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

   1. associate-/l* 14.3%

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

8. Simplified 14.3%

   \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
9. Add Preprocessing

Alternative 11: 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 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in cosTheta_i around inf 75.9%

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

6. Taylor expanded in cosTheta_i around inf 14.3%

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

   1. associate-/l* 14.3%

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

8. Simplified 14.3%

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

9. Taylor expanded in cosTheta_O around 0 6.5%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))))))