HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 9.2s

Alternatives: 4

Speedup: 1.2×

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

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{\left(0.6931 - \log \left(2 \cdot v\right)\right) - \frac{\frac{-1}{sinTheta\_i} \cdot \left(-sinTheta\_i\right)}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (- (- 0.6931 (log (* 2.0 v))) (/ (* (/ -1.0 sinTheta_i) (- sinTheta_i)) v))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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((2.0e0 * v))) - ((((-1.0e0) / sintheta_i) * -sintheta_i) / v)))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(0.6931) - log((single(2.0) * v))) - (((single(-1.0) / sinTheta_i) * -sinTheta_i) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. lower--.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   3. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   4. rem-exp-log N/A

      \[\leadsto e^{\left(\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   5. lower-log.f32 N/A

      \[\leadsto e^{\left(\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\left(\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto e^{\left(\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   8. div-add-rev N/A

      \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   10. +-commutative N/A

       \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}}{v}} \]

   11. lower-fma.f32 42.8

       \[\leadsto e^{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{v}} \]

5. Applied rewrites 42.8%

   \[\leadsto e^{\color{blue}{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]

6. Taylor expanded in sinTheta_i around -inf

   \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \frac{-1 \cdot \left(sinTheta\_i \cdot \left(-1 \cdot sinTheta\_O - \frac{1}{sinTheta\_i}\right)\right)}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.8%

   \[\leadsto e^{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) - \frac{\left(-sinTheta\_i\right) \cdot \left(\left(-sinTheta\_O\right) - \frac{1}{sinTheta\_i}\right)}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 99.8%

    \[\leadsto \color{blue}{e^{\left(0.6931 - \log \left(2 \cdot v\right)\right) - \frac{\left(\left(-sinTheta\_O\right) - \frac{1}{sinTheta\_i}\right) \cdot \left(-sinTheta\_i\right)}{v}}} \]

11. Taylor expanded in sinTheta_i around 0

    \[\leadsto e^{\left(\frac{6931}{10000} - \log \left(2 \cdot v\right)\right) - \frac{\frac{-1}{sinTheta\_i} \cdot \left(-sinTheta\_i\right)}{v}} \]
12. Step-by-step derivation

13. Applied rewrites 99.8%

    \[\leadsto e^{\left(0.6931 - \log \left(2 \cdot v\right)\right) - \frac{\frac{-1}{sinTheta\_i} \cdot \left(-sinTheta\_i\right)}{v}} \]
14. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (- (- 0.6931 (log (* 2.0 v))) (/ 1.0 v))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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((2.0e0 * v))) - (1.0e0 / v)))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(0.6931) - log((single(2.0) * v))) - (single(1.0) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. lower--.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   3. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   4. rem-exp-log N/A

      \[\leadsto e^{\left(\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   5. lower-log.f32 N/A

      \[\leadsto e^{\left(\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\left(\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto e^{\left(\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \frac{6931}{10000}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   8. div-add-rev N/A

      \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   10. +-commutative N/A

       \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}}{v}} \]

   11. lower-fma.f32 42.4

       \[\leadsto e^{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{v}} \]

5. Applied rewrites 42.8%

   \[\leadsto e^{\color{blue}{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto e^{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.8%

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

10. Applied rewrites 99.8%

    \[\leadsto e^{\left(0.6931 - \log \left(2 \cdot v\right)\right) - \color{blue}{\frac{1}{v}}} \]
11. Add Preprocessing

Alternative 3: 99.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{v} \cdot e^{0.6931 - \frac{1}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (- 0.6931 (/ 1.0 v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   8. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   9. div-add-rev N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   10. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   11. +-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}}{v}} \]

   12. lower-fma.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{1}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.8%

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

Alternative 4: 97.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{\frac{-1}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ -1.0 v)))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(-1.0) / v));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\left(\color{blue}{-1 \cdot \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto e^{\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   2. div-add-rev N/A

      \[\leadsto e^{\left(\left(\mathsf{neg}\left(\color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}\right)\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. distribute-neg-frac N/A

      \[\leadsto e^{\left(\color{blue}{\frac{\mathsf{neg}\left(\left(1 + sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   4. lower-/.f32 N/A

      \[\leadsto e^{\left(\color{blue}{\frac{\mathsf{neg}\left(\left(1 + sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   5. distribute-neg-in N/A

      \[\leadsto e^{\left(\frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)\right)}}{v} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   6. metadata-eval N/A

      \[\leadsto e^{\left(\frac{\color{blue}{-1} + \left(\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   7. unsub-neg N/A

      \[\leadsto e^{\left(\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   8. lower--.f32 N/A

      \[\leadsto e^{\left(\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v} + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

   9. lower-*.f32 99.8

      \[\leadsto e^{\left(\frac{-1 - \color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto e^{\left(\color{blue}{\frac{-1 - sinTheta\_O \cdot sinTheta\_i}{v}} + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

6. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
7. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   2. div-add-rev N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \color{blue}{\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   3. sub-neg N/A

      \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   4. +-commutative N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + \frac{1}{v}\right)}\right)\right)} \]

   5. distribute-neg-in N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)} \]

   7. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

   8. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \]

   9. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]

   10. associate--l+ N/A

       \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}} \]

   11. associate-*r/ N/A

       \[\leadsto e^{\color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)} \]

   12. div-sub N/A

       \[\leadsto e^{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}} \]

   13. div-add-rev N/A

       \[\leadsto e^{\color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + \left(cosTheta\_O \cdot cosTheta\_i - 1\right)}{v}}} \]

   14. lower-/.f32 N/A

       \[\leadsto e^{\color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + \left(cosTheta\_O \cdot cosTheta\_i - 1\right)}{v}}} \]

8. Applied rewrites 96.0%

   \[\leadsto e^{\color{blue}{\frac{\mathsf{fma}\left(-sinTheta\_i, sinTheta\_O, \mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)\right)}{v}}} \]

9. Taylor expanded in sinTheta_i around 0

   \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{\color{blue}{v}}} \]
10. Step-by-step derivation

11. Applied rewrites 98.2%

    \[\leadsto e^{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{\color{blue}{v}}} \]

12. Taylor expanded in cosTheta_i around 0

    \[\leadsto e^{\frac{-1}{v}} \]
13. Step-by-step derivation

14. Applied rewrites 98.6%

    \[\leadsto e^{\frac{-1}{v}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(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))))))