HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 14.2s

Alternatives: 8

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, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  0.5
  (* (exp 0.6931) (exp (- (/ (fma cosTheta_O cosTheta_i -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) * expf(((fmaf(cosTheta_O, cosTheta_i, -1.0f) / v) - logf(v))));
}

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. exp-sum 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. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{\left(0.6931 + \frac{cosTheta_O \cdot cosTheta_i}{v}\right) - \frac{1}{v}}}{v}} \]
5. 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 99.8%

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

   3. associate--l+ 99.8%

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

   4. associate-/l* 99.8%

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

6. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. pow-exp 99.8%

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

   3. *-un-lft-identity 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate--l+ 99.8%

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

   6. log-div 99.8%

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

   7. add-log-exp 99.8%

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

   8. associate--l+ 99.8%

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

   9. sub-div 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate--l+ 99.8%

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

   2. exp-sum 99.9%

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

   3. fma-neg 99.9%

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

   4. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

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 99.4%

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

   1. +-commutative 99.4%

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

   2. exp-sum 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. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{\left(0.6931 + \frac{cosTheta_O \cdot cosTheta_i}{v}\right) - \frac{1}{v}}}{v}} \]
5. 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 99.8%

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

   3. associate--l+ 99.8%

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

   4. associate-/l* 99.8%

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

6. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. pow-exp 99.8%

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

   3. *-un-lft-identity 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate--l+ 99.8%

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

   6. log-div 99.8%

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

   7. add-log-exp 99.8%

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

   8. associate--l+ 99.8%

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

   9. sub-div 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in cosTheta_O around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 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 99.4%

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

   1. +-commutative 99.4%

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

   2. exp-sum 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. Taylor expanded in sinTheta_i around 0 99.8%

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

5. Taylor expanded in cosTheta_O around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 35.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_O \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O -1.9999999774532045e-26)
   (exp (* (/ sinTheta_i v) (- sinTheta_O)))
   (/ (* 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 <= -1.9999999774532045e-26f) {
		tmp = expf(((sinTheta_i / v) * -sinTheta_O));
	} else {
		tmp = (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 <= (-1.9999999774532045e-26)) then
        tmp = exp(((sintheta_i / v) * -sintheta_o))
    else
        tmp = (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(-1.9999999774532045e-26))
		tmp = exp(Float32(Float32(sinTheta_i / v) * Float32(-sinTheta_O)));
	else
		tmp = 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_O <= single(-1.9999999774532045e-26))
		tmp = exp(((sinTheta_i / v) * -sinTheta_O));
	else
		tmp = (sinTheta_O * -sinTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < -1.99999998e-26

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in sinTheta_i around inf 18.6%

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

      1. mul-1-neg 18.6%

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

      2. associate-*r/ 18.6%

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

      3. distribute-rgt-neg-in 18.6%

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

   6. Simplified 18.6%

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

   if -1.99999998e-26 < sinTheta_O

   1. Initial program 99.7%

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

   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 11.9%

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

   5. Taylor expanded in sinTheta_O around 0 6.4%

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

      1. associate-/l* 6.4%

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

      2. neg-mul-1 6.4%

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

      3. unsub-neg 6.4%

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

      4. associate-/r/ 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in sinTheta_O around inf 43.5%

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

      1. metadata-eval 43.5%

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

      2. times-frac 43.5%

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

      3. neg-mul-1 43.5%

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

      4. distribute-rgt-neg-in 43.5%

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

      5. times-frac 22.2%

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

      6. /-rgt-identity 22.2%

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

   10. Simplified 22.2%

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

       1. associate-*r/ 43.5%

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

   12. Applied egg-rr 43.5%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_O \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;e^{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \]

Alternative 5: 20.4% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 14.2%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 36.9%

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

   1. metadata-eval 36.9%

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

   2. times-frac 36.9%

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

   3. neg-mul-1 36.9%

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

   4. distribute-rgt-neg-in 36.9%

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

   5. times-frac 18.8%

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

   6. /-rgt-identity 18.8%

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

10. Simplified 18.8%

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

11. Final simplification 18.8%

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

Alternative 6: 20.4% 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 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 14.2%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 36.9%

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

   1. mul-1-neg 36.9%

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

   2. associate-*l/ 18.8%

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

   3. distribute-lft-neg-out 18.8%

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

   4. *-commutative 18.8%

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

   5. distribute-neg-frac 18.8%

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

10. Simplified 18.8%

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

11. Final simplification 18.8%

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

Alternative 7: 38.3% accurate, 37.2× speedup

Math

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

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 * 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 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 14.2%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 36.9%

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

   1. metadata-eval 36.9%

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

   2. times-frac 36.9%

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

   3. neg-mul-1 36.9%

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

   4. distribute-rgt-neg-in 36.9%

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

   5. times-frac 18.8%

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

   6. /-rgt-identity 18.8%

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

10. Simplified 18.8%

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

    1. associate-*r/ 36.9%

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

12. Applied egg-rr 36.9%

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

13. Final simplification 36.9%

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

Alternative 8: 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 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in sinTheta_i around inf 14.2%

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

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 2023331 
(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))))))