HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.8%

Time: 4.9s

Alternatives: 9

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, 1.5× 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

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

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_O cosTheta_i) 1.0) v)
  (+ -0.6931 (log (+ v v))))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((cosTheta_O * cosTheta_i) - single(1.0)) / v) - (single(-0.6931) + log((v + 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 99.8%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 99.8%

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

Alternative 2: 99.8% 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

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

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 (- (/ (- (* cosTheta_O cosTheta_i) 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(((((cosTheta_O * cosTheta_i) - 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(((((costheta_o * costheta_i) - 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(Float32(cosTheta_O * cosTheta_i) - 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(((((cosTheta_O * cosTheta_i) - 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. 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

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

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 (- (/ (- (* cosTheta_O cosTheta_i) 1.0) v) -0.6931)) (+ v v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((cosTheta_O * cosTheta_i) - 1.0f) / v) - -0.6931f)) / (v + 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_o * costheta_i) - 1.0e0) / v) - (-0.6931e0))) / (v + v)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((cosTheta_O * cosTheta_i) - single(1.0)) / v) - single(-0.6931))) / (v + 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 99.8%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 99.8%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v} - -0.6931}}{v + v} \]
7. Add Preprocessing

Alternative 4: 98.1% accurate, 1.9× 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

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

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 (* sinTheta_O sinTheta_i)) (* cosTheta_O cosTheta_i))
   v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(-(((1.0f + (sinTheta_O * sinTheta_i)) - (cosTheta_O * cosTheta_i)) / 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 + (sintheta_o * sintheta_i)) - (costheta_o * costheta_i)) / v))
end function

Julia

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

MATLAB

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

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

4. Taylor expanded in cosTheta_i around inf

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

6. Applied rewrites 13.1%

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

8. Applied rewrites 13.1%

   \[\leadsto e^{-\left(-\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} \]

9. Taylor expanded in v around 0

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

11. Applied rewrites 98.1%

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

Alternative 5: 18.4% 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

\[\begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{e^{-\frac{cosTheta\_O \cdot cosTheta\_i}{v}}}\\ \mathbf{else}:\\ \;\;\;\;e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \]

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)))
  (if (<= (* sinTheta_i sinTheta_O) 4.999999955487895e-38)
  (/ 1.0 (exp (- (/ (* cosTheta_O cosTheta_i) v))))
  (exp (- (* sinTheta_i (/ sinTheta_O v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.999999955487895e-38f) {
		tmp = 1.0f / expf(-((cosTheta_O * cosTheta_i) / v));
	} else {
		tmp = expf(-(sinTheta_i * (sinTheta_O / v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 4.999999955487895e-38) then
        tmp = 1.0e0 / exp(-((costheta_o * costheta_i) / v))
    else
        tmp = exp(-(sintheta_i * (sintheta_o / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(4.999999955487895e-38))
		tmp = Float32(Float32(1.0) / exp(Float32(-Float32(Float32(cosTheta_O * cosTheta_i) / v))));
	else
		tmp = exp(Float32(-Float32(sinTheta_i * Float32(sinTheta_O / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(4.999999955487895e-38))
		tmp = single(1.0) / exp(-((cosTheta_O * cosTheta_i) / v));
	else
		tmp = exp(-(sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4.99999996e-38

   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%

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

   4. Taylor expanded in cosTheta_i around inf

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

   6. Applied rewrites 13.1%

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

   8. Applied rewrites 13.1%

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

   if 4.99999996e-38 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in sinTheta_i around inf

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

   6. Applied rewrites 13.1%

      \[\leadsto \frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   8. Applied rewrites 13.0%

      \[\leadsto e^{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.0%

       \[\leadsto e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 18.4% 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

\[\begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;e^{-\frac{-1}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}}}\\ \mathbf{else}:\\ \;\;\;\;e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \]

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)))
  (if (<= (* sinTheta_i sinTheta_O) 4.999999955487895e-38)
  (exp (- (/ -1.0 (/ v (* cosTheta_O cosTheta_i)))))
  (exp (- (* sinTheta_i (/ sinTheta_O v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.999999955487895e-38f) {
		tmp = expf(-(-1.0f / (v / (cosTheta_O * cosTheta_i))));
	} else {
		tmp = expf(-(sinTheta_i * (sinTheta_O / v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 4.999999955487895e-38) then
        tmp = exp(-((-1.0e0) / (v / (costheta_o * costheta_i))))
    else
        tmp = exp(-(sintheta_i * (sintheta_o / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(4.999999955487895e-38))
		tmp = exp(Float32(-Float32(Float32(-1.0) / Float32(v / Float32(cosTheta_O * cosTheta_i)))));
	else
		tmp = exp(Float32(-Float32(sinTheta_i * Float32(sinTheta_O / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(4.999999955487895e-38))
		tmp = exp(-(single(-1.0) / (v / (cosTheta_O * cosTheta_i))));
	else
		tmp = exp(-(sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4.99999996e-38

   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%

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

   4. Taylor expanded in cosTheta_i around inf

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

   6. Applied rewrites 13.1%

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

   8. Applied rewrites 13.1%

      \[\leadsto e^{-\left(-\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.1%

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

   if 4.99999996e-38 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in sinTheta_i around inf

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

   6. Applied rewrites 13.1%

      \[\leadsto \frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   8. Applied rewrites 13.0%

      \[\leadsto e^{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.0%

       \[\leadsto e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 18.4% accurate, 1.9× 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

\[\begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;e^{-\left(-\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \]

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)))
  (if (<= (* sinTheta_i sinTheta_O) 4.999999955487895e-38)
  (exp (- (- (/ (* cosTheta_O cosTheta_i) v))))
  (exp (- (* sinTheta_i (/ sinTheta_O v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 4.999999955487895e-38f) {
		tmp = expf(-(-((cosTheta_O * cosTheta_i) / v)));
	} else {
		tmp = expf(-(sinTheta_i * (sinTheta_O / v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 4.999999955487895e-38) then
        tmp = exp(-(-((costheta_o * costheta_i) / v)))
    else
        tmp = exp(-(sintheta_i * (sintheta_o / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(4.999999955487895e-38))
		tmp = exp(Float32(-Float32(-Float32(Float32(cosTheta_O * cosTheta_i) / v))));
	else
		tmp = exp(Float32(-Float32(sinTheta_i * Float32(sinTheta_O / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(4.999999955487895e-38))
		tmp = exp(-(-((cosTheta_O * cosTheta_i) / v)));
	else
		tmp = exp(-(sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4.99999996e-38

   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%

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

   4. Taylor expanded in cosTheta_i around inf

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

   6. Applied rewrites 13.1%

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

   8. Applied rewrites 13.1%

      \[\leadsto e^{-\left(-\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} \]

   if 4.99999996e-38 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in sinTheta_i around inf

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

   6. Applied rewrites 13.1%

      \[\leadsto \frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   8. Applied rewrites 13.0%

      \[\leadsto e^{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.0%

       \[\leadsto e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 13.0% accurate, 2.7× 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

\[e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

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 (- (* sinTheta_i (/ sinTheta_O v)))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(-(sinTheta_i * (sinTheta_O / 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%

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

4. Taylor expanded in sinTheta_i around inf

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

6. Applied rewrites 13.1%

   \[\leadsto \frac{1}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

8. Applied rewrites 13.0%

   \[\leadsto e^{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 13.0%

    \[\leadsto e^{-sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]
11. Add Preprocessing

Alternative 9: 4.6% 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

\[\frac{e^{0.6931}}{v + v} \]

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) (+ v v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(0.6931f) / (v + 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) / (v + v)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(single(0.6931)) / (v + 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 v around inf

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

4. Applied rewrites 4.6%

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

6. Applied rewrites 4.6%

   \[\leadsto \frac{e^{0.6931}}{v + v} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026084 
(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))))))