HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 12.9s

Alternatives: 12

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((logf(0.5f) + ((0.6931f - (1.0f / v)) - logf(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((log(0.5e0) + ((0.6931e0 - (1.0e0 / v)) - log(v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.4%

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

   2. *-commutative 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

   9. +-rgt-identity 99.4%

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

   10. sub-neg 99.4%

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

   11. associate-+l+ 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. associate--l+ 99.9%

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

   4. sub-div 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in cosTheta_O around 0 99.9%

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

   1. add-exp-log 99.9%

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

   2. log-prod 99.9%

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

   3. log-div 99.9%

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

   4. pow-exp 99.9%

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

   5. add-log-exp 99.9%

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

   6. sub-neg 99.9%

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

   7. *-un-lft-identity 99.9%

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

   8. distribute-neg-frac 99.9%

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

   9. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * expf(((0.6931f - (1.0f / v)) - logf(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 * exp(((0.6931e0 - (1.0e0 / v)) - log(v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.4%

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

   2. *-commutative 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

   9. +-rgt-identity 99.4%

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

   10. sub-neg 99.4%

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

   11. associate-+l+ 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. associate--l+ 99.9%

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

   4. sub-div 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in cosTheta_O around 0 99.9%

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

   1. add-exp-log 99.9%

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

   2. log-div 99.9%

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

   3. pow-exp 99.9%

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

   4. add-log-exp 99.9%

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

   5. sub-neg 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. distribute-neg-frac 99.9%

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

   8. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_O \leq 1.4999999853326784 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 1.4999999853326784e-10)
   (/ 1.0 (/ v (* sinTheta_O sinTheta_i)))
   (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_O <= 1.4999999853326784e-10f) {
		tmp = 1.0f / (v / (sinTheta_O * sinTheta_i));
	} 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_o <= 1.4999999853326784e-10) then
        tmp = 1.0e0 / (v / (sintheta_o * sintheta_i))
    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_O <= Float32(1.4999999853326784e-10))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_O * sinTheta_i)));
	else
		tmp = exp(Float32(sinTheta_O * Float32(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_O <= single(1.4999999853326784e-10))
		tmp = single(1.0) / (v / (sinTheta_O * sinTheta_i));
	else
		tmp = exp((sinTheta_O * (-sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 1.49999999e-10

   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. Add Preprocessing

   5. Taylor expanded in sinTheta_i around inf 11.1%

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

   6. Taylor expanded in sinTheta_O around 0 6.3%

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

      1. mul-1-neg 6.3%

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

      2. associate-/l* 6.3%

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

      3. unsub-neg 6.3%

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

      4. associate-/l* 6.3%

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

      5. associate-*r/ 6.3%

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

   8. Simplified 6.3%

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

   9. Taylor expanded in sinTheta_O around inf 48.5%

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

       1. mul-1-neg 48.5%

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

       2. associate-*r/ 23.2%

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

       3. distribute-rgt-neg-in 23.2%

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

       4. distribute-neg-frac 23.2%

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

   11. Simplified 23.2%

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

       1. add-sqr-sqrt 15.2%

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

       2. sqrt-unprod 53.0%

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

       3. swap-sqr 60.7%

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

       4. distribute-frac-neg 60.7%

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

       5. distribute-frac-neg 60.7%

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

       6. sqr-neg 60.7%

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

       7. swap-sqr 53.0%

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

       8. sqrt-unprod 16.3%

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

       9. add-sqr-sqrt 23.1%

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

       10. associate-*r/ 48.4%

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

       11. clear-num 49.0%

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

   13. Applied egg-rr 49.0%

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

   if 1.49999999e-10 < sinTheta_O

   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. Add Preprocessing

   5. Taylor expanded in sinTheta_i around inf 31.6%

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

      1. associate-*r/ 31.6%

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

      2. neg-mul-1 31.6%

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

      3. *-commutative 31.6%

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

      4. distribute-rgt-neg-in 31.6%

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

   7. Simplified 31.6%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_O \leq 1.4999999853326784 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta_O \cdot \frac{-sinTheta_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 99.5%

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

   1. exp-sum 99.4%

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

   2. *-commutative 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

   9. +-rgt-identity 99.4%

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

   10. sub-neg 99.4%

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

   11. associate-+l+ 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. associate--l+ 99.9%

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

   4. sub-div 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in cosTheta_O around 0 99.9%

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

9. Final simplification 99.9%

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

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (exp (- 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 * (expf((0.6931f - (1.0f / v))) / 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 * (exp((0.6931e0 - (1.0e0 / v))) / v)
end function

Julia

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.4%

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

   2. *-commutative 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

   9. +-rgt-identity 99.4%

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

   10. sub-neg 99.4%

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

   11. associate-+l+ 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.5%

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

6. Taylor expanded in v around 0 98.9%

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

7. Final simplification 98.9%

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

Alternative 7: 37.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.7%

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

6. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. mul-1-neg 6.2%

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

   2. associate-/l* 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/l* 6.2%

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

   5. associate-*r/ 6.2%

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

8. Simplified 6.2%

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

9. Taylor expanded in sinTheta_O around inf 43.5%

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

    1. mul-1-neg 43.5%

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

    2. associate-*r/ 21.2%

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

    3. distribute-rgt-neg-in 21.2%

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

    4. distribute-neg-frac 21.2%

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

11. Simplified 21.2%

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

    1. add-sqr-sqrt 13.6%

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

    2. sqrt-unprod 48.8%

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

    3. swap-sqr 55.5%

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

    4. distribute-frac-neg 55.5%

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

    5. distribute-frac-neg 55.5%

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

    6. sqr-neg 55.5%

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

    7. swap-sqr 48.8%

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

    8. sqrt-unprod 14.7%

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

    9. add-sqr-sqrt 21.0%

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

    10. associate-*r/ 43.4%

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

    11. clear-num 43.9%

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

13. Applied egg-rr 43.9%

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

14. Final simplification 43.9%

    \[\leadsto \frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}} \]
15. Add Preprocessing

Alternative 8: 19.8% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ sinTheta_O \cdot \frac{-sinTheta_i}{v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.7%

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

6. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. mul-1-neg 6.2%

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

   2. associate-/l* 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/l* 6.2%

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

   5. associate-*r/ 6.2%

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

8. Simplified 6.2%

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

9. Taylor expanded in sinTheta_O around inf 43.5%

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

    1. mul-1-neg 43.5%

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

    2. associate-*r/ 21.2%

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

    3. distribute-rgt-neg-in 21.2%

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

    4. distribute-neg-frac 21.2%

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

11. Simplified 21.2%

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

12. Final simplification 21.2%

    \[\leadsto sinTheta_O \cdot \frac{-sinTheta_i}{v} \]
13. Add Preprocessing

Alternative 9: 19.8% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ sinTheta_i \cdot \frac{-sinTheta_O}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

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. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.7%

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

6. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. mul-1-neg 6.2%

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

   2. associate-/l* 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/l* 6.2%

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

   5. associate-*r/ 6.2%

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

8. Simplified 6.2%

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

9. Taylor expanded in sinTheta_O around inf 43.5%

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

    1. mul-1-neg 43.5%

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

    2. associate-*l/ 21.2%

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

    3. distribute-lft-neg-out 21.2%

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

    4. *-commutative 21.2%

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

    5. distribute-neg-frac 21.2%

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

11. Simplified 21.2%

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

12. Final simplification 21.2%

    \[\leadsto sinTheta_i \cdot \frac{-sinTheta_O}{v} \]
13. Add Preprocessing

Alternative 10: 37.4% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O \cdot sinTheta_i}{-v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.7%

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

6. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. mul-1-neg 6.2%

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

   2. associate-/l* 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/l* 6.2%

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

   5. associate-*r/ 6.2%

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

8. Simplified 6.2%

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

9. Taylor expanded in sinTheta_O around inf 43.5%

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

    1. mul-1-neg 43.5%

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

    2. associate-*r/ 21.2%

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

    3. distribute-rgt-neg-in 21.2%

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

    4. distribute-neg-frac 21.2%

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

11. Simplified 21.2%

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

    1. associate-*r/ 43.5%

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

    2. frac-2neg 43.5%

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

    3. add-sqr-sqrt 19.9%

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

    4. sqrt-unprod 56.5%

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

    5. sqr-neg 56.5%

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

    6. sqrt-unprod 23.6%

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

    7. add-sqr-sqrt 43.4%

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

    8. distribute-rgt-neg-out 43.4%

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

    9. add-sqr-sqrt 19.8%

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

    10. sqrt-unprod 56.6%

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

    11. sqr-neg 56.6%

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

    12. sqrt-unprod 23.7%

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

    13. add-sqr-sqrt 43.5%

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

13. Applied egg-rr 43.5%

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

14. Final simplification 43.5%

    \[\leadsto \frac{sinTheta_O \cdot sinTheta_i}{-v} \]
15. Add Preprocessing

Alternative 11: 19.7% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.7%

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

6. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. mul-1-neg 6.2%

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

   2. associate-/l* 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/l* 6.2%

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

   5. associate-*r/ 6.2%

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

8. Simplified 6.2%

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

9. Taylor expanded in sinTheta_O around inf 43.5%

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

    1. mul-1-neg 43.5%

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

    2. associate-*r/ 21.2%

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

    3. distribute-rgt-neg-in 21.2%

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

    4. distribute-neg-frac 21.2%

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

11. Simplified 21.2%

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

    1. add-sqr-sqrt 13.6%

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

    2. sqrt-unprod 48.8%

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

    3. swap-sqr 55.5%

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

    4. distribute-frac-neg 55.5%

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

    5. distribute-frac-neg 55.5%

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

    6. sqr-neg 55.5%

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

    7. swap-sqr 48.8%

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

    8. sqrt-unprod 14.7%

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

    9. add-sqr-sqrt 21.0%

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

    10. clear-num 21.0%

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

    11. un-div-inv 21.0%

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

13. Applied egg-rr 21.0%

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

14. Final simplification 21.0%

    \[\leadsto \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \]
15. Add Preprocessing

Alternative 12: 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 99.5%

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

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. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 13.7%

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

6. Taylor expanded in sinTheta_O around 0 6.4%

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

7. Final simplification 6.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))))