HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 16.4s

Alternatives: 13

Speedup: 1.1×

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, 0.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (pow E 0.6931)
  (pow
   E
   (+
    (log (/ 0.5 v))
    (/ (+ (- (* cosTheta_O cosTheta_i) (* 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 powf(((float) M_E), 0.6931f) * powf(((float) M_E), (logf((0.5f / v)) + ((((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) + -1.0f) / v)));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. associate--r+ 99.7%

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

   4. div-inv 99.7%

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

   5. clear-num 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l/ 99.7%

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

   8. div-inv 99.7%

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

   9. clear-num 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l/ 99.7%

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

   12. sub-div 99.7%

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

5. Applied egg-rr 99.7%

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

   1. exp-1-e 99.7%

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

   2. +-commutative 99.7%

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

7. Simplified 99.7%

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

   1. associate-+l+ 99.7%

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

   2. unpow-prod-up 99.7%

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

   3. sub-div 99.7%

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

   4. cancel-sign-sub-inv 99.7%

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

   5. fma-def 99.7%

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

9. Applied egg-rr 99.7%

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

    1. sub-neg 99.7%

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

    2. fma-def 99.7%

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

    3. cancel-sign-sub-inv 99.7%

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

    4. metadata-eval 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(expf(2.0f), ((((((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) + -1.0f) / v) + (0.6931f + logf((0.5f / 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(2.0e0) ** ((((((costheta_o * costheta_i) - (sintheta_o * sintheta_i)) + (-1.0e0)) / v) + (0.6931e0 + log((0.5e0 / v)))) / 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-sum 99.2%

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

   3. +-commutative 99.2%

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

   4. exp-sum 99.3%

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

   5. add-exp-log 99.3%

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

   6. associate--r+ 99.3%

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

   7. div-inv 99.3%

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

   8. clear-num 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-*l/ 99.3%

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

   11. div-inv 99.3%

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

   12. clear-num 99.3%

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

   13. *-commutative 99.3%

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

   14. associate-*l/ 99.3%

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

   15. sub-div 99.3%

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

5. Applied egg-rr 99.3%

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

   1. add-exp-log 99.3%

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

   2. prod-exp 99.2%

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

   3. +-commutative 99.2%

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

   4. exp-sum 99.7%

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

   5. *-un-lft-identity 99.7%

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

   6. pow-exp 99.7%

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

   7. e-exp-1 99.7%

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

   8. sqr-pow 99.7%

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

   9. pow-prod-down 99.7%

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

7. Applied egg-rr 99.7%

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

   1. sub-neg 99.7%

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

   2. fma-def 99.7%

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

   3. cancel-sign-sub-inv 99.7%

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

   4. metadata-eval 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  E
  (+ (log (/ 0.5 v)) (+ 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 powf(((float) M_E), (logf((0.5f / v)) + (0.6931f + ((-1.0f - (sinTheta_O * sinTheta_i)) / v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(1)) ^ Float32(log(Float32(Float32(0.5) / v)) + 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(2.71828182845904523536) ^ (log((single(0.5) / v)) + (single(0.6931) + ((single(-1.0) - (sinTheta_O * sinTheta_i)) / v)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. associate--r+ 99.7%

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

   4. div-inv 99.7%

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

   5. clear-num 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l/ 99.7%

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

   8. div-inv 99.7%

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

   9. clear-num 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l/ 99.7%

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

   12. sub-div 99.7%

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

5. Applied egg-rr 99.7%

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

   1. exp-1-e 99.7%

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

   2. +-commutative 99.7%

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

7. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. rem-log-exp 99.7%

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

   3. log-prod 99.3%

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

   4. add-log-exp 99.2%

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

   5. log-prod 99.2%

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

   6. associate-*l* 99.6%

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

   7. prod-exp 99.6%

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

   8. sub-div 99.7%

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

   9. cancel-sign-sub-inv 99.7%

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

   10. fma-def 99.7%

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

9. Applied egg-rr 99.7%

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

    1. log-prod 99.7%

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

    2. rem-log-exp 99.7%

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

    3. sub-neg 99.7%

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

    4. fma-def 99.7%

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

    5. cancel-sign-sub-inv 99.7%

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

    6. metadata-eval 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in cosTheta_O around 0 99.7%

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

    1. neg-mul-1 99.7%

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

    2. distribute-rgt-neg-in 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(((0.6931e0 + log((0.5e0 / v))) + ((costheta_i / (v / costheta_o)) + (((-1.0e0) / v) - (sintheta_i / (v / sintheta_o))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 5: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_O around 0 99.6%

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

5. Taylor expanded in v around 0 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto 0.5 \cdot \frac{e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta_O, sinTheta_i, 1\right)}{v}}}{v} \]

Alternative 6: 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 99.7%

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

   1. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_O around 0 99.6%

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

5. Taylor expanded in v around 0 99.6%

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

   1. mul-1-neg 99.6%

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

   2. +-commutative 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 7: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_O around 0 99.6%

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

5. Taylor expanded in sinTheta_O around 0 99.6%

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

6. Final simplification 99.6%

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

Alternative 8: 12.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -5.800000093623884 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -5.800000093623884e-30)
   (exp (* sinTheta_O (/ sinTheta_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 (sinTheta_i <= -5.800000093623884e-30f) {
		tmp = expf((sinTheta_O * (sinTheta_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 (sintheta_i <= (-5.800000093623884e-30)) then
        tmp = exp((sintheta_o * (sintheta_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 (sinTheta_i <= Float32(-5.800000093623884e-30))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	else
		tmp = exp(Float32(sinTheta_i * Float32(Float32(-sinTheta_O) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-5.800000093623884e-30))
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	else
		tmp = exp((sinTheta_i * (-sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -5.80000009e-30

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in sinTheta_i around inf 12.5%

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

      1. expm1-log1p-u 12.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]

      3. add-sqr-sqrt 3.4%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      4. sqrt-unprod 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      5. mul-1-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)} - 1 \]

      6. mul-1-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      7. sqr-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v} \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      8. sqrt-unprod 4.0%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      9. add-sqr-sqrt 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}}}\right)} - 1 \]

      10. associate-/l* 16.8%

          \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}\right)} - 1 \]

   6. Applied egg-rr 16.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 16.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)\right)} \]

      2. expm1-log1p 16.8%

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

      3. associate-/l* 16.8%

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

      4. associate-*r/ 16.8%

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

   8. Simplified 16.8%

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

   if -5.80000009e-30 < sinTheta_i

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in sinTheta_i around inf 12.9%

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

      1. mul-1-neg 12.9%

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

      2. associate-*l/ 12.9%

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

      3. distribute-rgt-neg-in 12.9%

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

   6. Simplified 12.9%

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

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -5.800000093623884 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{-sinTheta_O}{v}}\\ \end{array} \]

Alternative 9: 12.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -5.800000093623884 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -5.800000093623884e-30)
   (exp (* sinTheta_O (/ sinTheta_i v)))
   (exp (/ (- sinTheta_O) (/ v sinTheta_i)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= -5.800000093623884e-30f) {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	} else {
		tmp = expf((-sinTheta_O / (v / sinTheta_i)));
	}
	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 (sintheta_i <= (-5.800000093623884e-30)) then
        tmp = exp((sintheta_o * (sintheta_i / v)))
    else
        tmp = exp((-sintheta_o / (v / sintheta_i)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-5.800000093623884e-30))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	else
		tmp = exp(Float32(Float32(-sinTheta_O) / Float32(v / sinTheta_i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-5.800000093623884e-30))
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	else
		tmp = exp((-sinTheta_O / (v / sinTheta_i)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -5.80000009e-30

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in sinTheta_i around inf 12.5%

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

      1. expm1-log1p-u 12.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]

      3. add-sqr-sqrt 3.4%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      4. sqrt-unprod 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      5. mul-1-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)} - 1 \]

      6. mul-1-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      7. sqr-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v} \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      8. sqrt-unprod 4.0%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      9. add-sqr-sqrt 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}}}\right)} - 1 \]

      10. associate-/l* 16.8%

          \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}\right)} - 1 \]

   6. Applied egg-rr 16.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 16.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)\right)} \]

      2. expm1-log1p 16.8%

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

      3. associate-/l* 16.8%

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

      4. associate-*r/ 16.8%

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

   8. Simplified 16.8%

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

   if -5.80000009e-30 < sinTheta_i

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in sinTheta_i around inf 12.9%

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

      1. mul-1-neg 12.9%

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

      2. exp-neg 12.9%

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

      3. associate-/l* 12.9%

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}} \]

   6. Applied egg-rr 12.9%

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

      1. rec-exp 12.9%

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

      2. distribute-neg-frac 12.9%

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

   8. Simplified 12.9%

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

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -5.800000093623884 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \end{array} \]

Alternative 10: 12.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -5.800000093623884 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -5.800000093623884e-30)
   (exp (* sinTheta_O (/ sinTheta_i v)))
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= -5.800000093623884e-30f) {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	} else {
		tmp = expf(((sinTheta_O * -sinTheta_i) / 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 (sintheta_i <= (-5.800000093623884e-30)) then
        tmp = exp((sintheta_o * (sintheta_i / v)))
    else
        tmp = exp(((sintheta_o * -sintheta_i) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-5.800000093623884e-30))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	else
		tmp = exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-5.800000093623884e-30))
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	else
		tmp = exp(((sinTheta_O * -sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -5.80000009e-30

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in sinTheta_i around inf 12.5%

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

      1. expm1-log1p-u 12.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

      2. expm1-udef 13.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]

      3. add-sqr-sqrt 3.4%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      4. sqrt-unprod 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      5. mul-1-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)} - 1 \]

      6. mul-1-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

      7. sqr-neg 5.9%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v} \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      8. sqrt-unprod 4.0%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

      9. add-sqr-sqrt 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}}}\right)} - 1 \]

      10. associate-/l* 16.8%

          \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}\right)} - 1 \]

   6. Applied egg-rr 16.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 16.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)\right)} \]

      2. expm1-log1p 16.8%

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

      3. associate-/l* 16.8%

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

      4. associate-*r/ 16.8%

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

   8. Simplified 16.8%

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

   if -5.80000009e-30 < sinTheta_i

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in sinTheta_i around inf 12.9%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -5.800000093623884 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\\ \end{array} \]

Alternative 11: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.6%

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

   2. *-commutative 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-rgt-identity 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.6%

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

   10. sub-neg 99.6%

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

   11. associate-+l+ 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_O around 0 99.6%

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

5. Taylor expanded in sinTheta_O around 0 99.6%

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

6. Taylor expanded in v around 0 97.7%

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

7. Final simplification 97.7%

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

Alternative 12: 12.9% 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 / 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 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in sinTheta_i around inf 12.8%

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

   1. expm1-log1p-u 12.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)} \]

   2. expm1-udef 13.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)} - 1} \]

   3. add-sqr-sqrt 3.8%

      \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

   4. sqrt-unprod 5.9%

      \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

   5. mul-1-neg 5.9%

      \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \cdot \left(-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}\right)} - 1 \]

   6. mul-1-neg 5.9%

      \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right) \cdot \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)}}}\right)} - 1 \]

   7. sqr-neg 5.9%

      \[\leadsto e^{\mathsf{log1p}\left(e^{\sqrt{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v} \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

   8. sqrt-unprod 4.2%

      \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}} \cdot \sqrt{\frac{sinTheta_O \cdot sinTheta_i}{v}}}}\right)} - 1 \]

   9. add-sqr-sqrt 13.9%

      \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O \cdot sinTheta_i}{v}}}\right)} - 1 \]

   10. associate-/l* 13.9%

       \[\leadsto e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}\right)} - 1 \]

6. Applied egg-rr 13.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 13.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)\right)} \]

   2. expm1-log1p 13.9%

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

   3. associate-/l* 13.9%

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

   4. associate-*r/ 13.9%

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

8. Simplified 13.9%

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

9. Final simplification 13.9%

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

Alternative 13: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in sinTheta_i around inf 12.8%

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

5. Taylor expanded in sinTheta_O around 0 6.4%

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

6. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(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))))))