HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 16.3s

Alternatives: 8

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around 0 99.9%

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

5. Taylor expanded in v around inf 99.9%

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

   1. log-rec 99.9%

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

   2. sub-neg 99.9%

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

   3. log-div 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in sinTheta_O around 0 99.8%

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

5. Taylor expanded in cosTheta_O around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 98.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 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 = exp(((-1.0e0) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around 0 99.9%

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

5. Taylor expanded in v around inf 99.9%

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

   1. log-rec 99.9%

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

   2. sub-neg 99.9%

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

   3. log-div 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in v around 0 98.4%

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

9. Final simplification 98.4%

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

Alternative 4: 70.8% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around inf 10.9%

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

   1. mul-1-neg 10.9%

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

   2. associate-*r/ 10.9%

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

   3. distribute-lft-neg-in 10.9%

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

   4. *-commutative 10.9%

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

6. Simplified 10.9%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. neg-mul-1 6.3%

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

   2. *-commutative 6.3%

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

   3. unsub-neg 6.3%

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

   4. associate-*l/ 6.3%

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

   5. *-commutative 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.8%

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

    1. expm1-log1p-u 38.3%

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

    2. expm1-udef 71.3%

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

    3. log1p-udef 71.3%

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

    4. add-exp-log 71.7%

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

    5. associate-/l* 71.7%

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

12. Applied egg-rr 71.7%

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

    1. associate--l+ 72.1%

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

    2. associate-/l* 72.1%

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

    3. associate-*r/ 72.1%

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

14. Simplified 72.1%

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

15. Final simplification 72.1%

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

Alternative 5: 38.6% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \left(sinTheta_O \cdot sinTheta_i\right) \cdot \frac{-1}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (sinTheta_O * sinTheta_i) * (-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 = (sintheta_o * sintheta_i) * ((-1.0e0) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around inf 10.9%

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

   1. mul-1-neg 10.9%

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

   2. associate-*r/ 10.9%

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

   3. distribute-lft-neg-in 10.9%

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

   4. *-commutative 10.9%

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

6. Simplified 10.9%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. neg-mul-1 6.3%

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

   2. *-commutative 6.3%

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

   3. unsub-neg 6.3%

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

   4. associate-*l/ 6.3%

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

   5. *-commutative 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.8%

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

    1. div-inv 38.8%

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

    2. *-commutative 38.8%

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

12. Applied egg-rr 38.8%

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

13. Final simplification 38.8%

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

Alternative 6: 38.6% accurate, 37.2× speedup

Math

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

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(Float32(sinTheta_i * 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 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around inf 10.9%

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

   1. mul-1-neg 10.9%

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

   2. associate-*r/ 10.9%

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

   3. distribute-lft-neg-in 10.9%

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

   4. *-commutative 10.9%

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

6. Simplified 10.9%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. neg-mul-1 6.3%

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

   2. *-commutative 6.3%

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

   3. unsub-neg 6.3%

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

   4. associate-*l/ 6.3%

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

   5. *-commutative 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.8%

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

11. Final simplification 38.8%

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

Alternative 7: 20.1% accurate, 44.6× 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(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 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around inf 10.9%

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

   1. mul-1-neg 10.9%

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

   2. associate-*r/ 10.9%

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

   3. distribute-lft-neg-in 10.9%

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

   4. *-commutative 10.9%

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

6. Simplified 10.9%

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

7. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. neg-mul-1 6.3%

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

   2. *-commutative 6.3%

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

   3. unsub-neg 6.3%

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

   4. associate-*l/ 6.3%

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

   5. *-commutative 6.3%

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

9. Simplified 6.3%

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

10. Taylor expanded in sinTheta_O around inf 38.8%

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

    1. mul-1-neg 38.8%

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

    2. associate-*r/ 21.7%

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

    3. distribute-rgt-neg-in 21.7%

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

12. Simplified 21.7%

    \[\leadsto \color{blue}{sinTheta_O \cdot \left(-\frac{sinTheta_i}{v}\right)} \]
13. Step-by-step derivation

    1. expm1-log1p-u 21.4%

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

    2. expm1-udef 71.8%

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

    3. add-sqr-sqrt 37.0%

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

    4. sqrt-unprod 70.4%

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

    5. sqr-neg 70.4%

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

    6. sqrt-unprod 34.7%

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

    7. add-sqr-sqrt 71.3%

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

14. Applied egg-rr 71.3%

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

    1. expm1-def 21.3%

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

    2. expm1-log1p 21.7%

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

16. Simplified 21.7%

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

17. Final simplification 21.7%

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

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 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around inf 10.9%

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

   1. mul-1-neg 10.9%

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

   2. associate-*r/ 10.9%

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

   3. distribute-lft-neg-in 10.9%

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

   4. *-commutative 10.9%

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

6. Simplified 10.9%

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

7. Taylor expanded in sinTheta_i around 0 6.4%

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

8. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023287 
(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))))))