HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.8%

Time: 3.5s

Alternatives: 6

Speedup: 1.7×

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

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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))))))

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)
use fmin_fmax_functions
    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

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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))))))

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)
use fmin_fmax_functions
    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: 99.8% accurate, 1.3× speedup

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[e^{0.6931 - \left(-\frac{\mathsf{fma}\left(-\log \left(v + v\right), v, cosTheta\_O \cdot cosTheta\_i\right) - 1}{v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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
 (-
  0.6931
  (-
   (/
    (- (fma (- (log (+ v v))) v (* cosTheta_O cosTheta_i)) 1.0)
    v)))))

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.8%

   \[\leadsto e^{0.6931 - \left(-\frac{\mathsf{fma}\left(-\log \left(v + v\right), v, cosTheta\_O \cdot cosTheta\_i\right) - 1}{v}\right)} \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 1.5× speedup

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[e^{\frac{\mathsf{fma}\left(-0.6931 - \left(-\log \left(v + v\right)\right), v, 1\right)}{-v}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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 (/ (fma (- -0.6931 (- (log (+ v v)))) v 1.0) (- v))))

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\left(0.6931 + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \frac{1}{v}} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 1.7× speedup

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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 (- (- (- (log (+ v v))) -0.6931) (/ 1.0 v))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\left(0.6931 + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \frac{1}{v}} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 2.1× speedup

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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 (- (/ -1.0 v) -0.6931)) (/ 0.5 v)))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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) - (-0.6931e0))) * (0.5e0 / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in sinTheta_i around 0

   \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931} \cdot \frac{0.5}{v} \]
5. Step-by-step derivation

6. Applied rewrites 99.7%

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

7. Taylor expanded in cosTheta_i around 0

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

9. Applied rewrites 99.6%

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

Alternative 5: 98.0% accurate, 2.9× speedup

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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 (- 0.6931 (/ 1.0 v))))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 - (1.0e0 / v)))
end function

Julia

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in v around 0

   \[\leadsto e^{0.6931 - \frac{1 - cosTheta\_O \cdot cosTheta\_i}{v}} \]
8. Step-by-step derivation

9. Applied rewrites 98.0%

   \[\leadsto e^{0.6931 - \frac{1 - cosTheta\_O \cdot cosTheta\_i}{v}} \]

10. Taylor expanded in cosTheta_i around 0

    \[\leadsto e^{0.6931 - \frac{1}{v}} \]
11. Step-by-step derivation

12. Applied rewrites 98.0%

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

Alternative 6: 4.6% accurate, 11.0× speedup

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\frac{0.9999527931213379}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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)))
  (/ 0.9999527931213379 v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.9999527931213379f / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = 0.9999527931213379e0 / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.9999527931213379) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.9999527931213379) / v;
end

Derivation

1. Initial program 99.7%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 4.6%

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

7. Evaluated real constant 4.6%

   \[\leadsto 0.5 \cdot \frac{1.9999055862426758}{v} \]
8. Step-by-step derivation

9. Applied rewrites 4.6%

   \[\leadsto \frac{0.9999527931213379}{v} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(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))))))