HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 8.8s

Alternatives: 6

Speedup: 2.0×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / exp((log(v) + ((single(1.0) / v) - single(0.6931))));
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 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate-*r/ 99.9%

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

   2. associate-/l* 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-*l/ 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-udef 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in sinTheta_O around 0 99.9%

   \[\leadsto \frac{0.5}{\frac{v}{e^{\color{blue}{0.6931 - \frac{1}{v}}}}} \]
8. Step-by-step derivation

   1. add-exp-log 99.9%

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

   2. log-div 99.9%

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

   3. add-log-exp 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate-*r/ 99.9%

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

   2. associate-/l* 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-*l/ 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-udef 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in sinTheta_O around 0 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate-*r/ 99.9%

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

   2. associate-/l* 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-*l/ 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-udef 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in sinTheta_O around 0 99.9%

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

8. Taylor expanded in v around 0 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 12.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((cosTheta_i / (v / cosTheta_O)));
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. associate-+l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. associate-+l- 99.5%

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

   4. associate-+l- 99.5%

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

   5. sub-neg 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/r* 99.5%

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

   9. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in sinTheta_i around 0 99.5%

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

5. Taylor expanded in cosTheta_i around inf 14.6%

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

   1. associate-/l* 14.6%

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

7. Simplified 14.6%

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

8. Final simplification 14.6%

   \[\leadsto e^{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

Alternative 5: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / exp((single(1.0) / 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 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate-*r/ 99.9%

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

   2. associate-/l* 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-*l/ 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-udef 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in sinTheta_O around 0 99.9%

   \[\leadsto \frac{0.5}{\frac{v}{e^{\color{blue}{0.6931 - \frac{1}{v}}}}} \]
8. Step-by-step derivation

   1. add-exp-log 99.9%

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

   2. log-div 99.9%

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

   3. add-log-exp 99.9%

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

9. Applied egg-rr 99.9%

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

10. Taylor expanded in v around 0 99.4%

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

11. Final simplification 99.4%

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

Alternative 6: 4.7% accurate, 74.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / 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 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate-*r/ 99.9%

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

   2. associate-/l* 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-*l/ 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-udef 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in sinTheta_O around 0 99.9%

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

8. Taylor expanded in v around 0 99.4%

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

9. Taylor expanded in v around inf 4.5%

   \[\leadsto \color{blue}{\frac{0.5}{v}} \]

10. Final simplification 4.5%

    \[\leadsto \frac{0.5}{v} \]

Reproduce

herbie shell --seed 2023214 
(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))))))