HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 33.8s

Alternatives: 11

Speedup: 2.1×

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: 99.6% 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: 99.7% 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

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

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_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0))
     v)
    (log 0.5)))
  (exp -0.6931))
 v))

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 1.6× 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

\[\frac{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v} + -4.7180561523418874 \cdot 10^{-5}}}{v} \]

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_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0)) v)
   -4.7180561523418874e-5))
 v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((cosTheta_i * cosTheta_O) - fmaf(sinTheta_i, sinTheta_O, 1.0f)) / v) + -4.7180561523418874e-5f)) / v;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - fma(sinTheta_i, sinTheta_O, Float32(1.0))) / v) + Float32(-4.7180561523418874e-5))) / v)
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.7%

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

8. Evaluated real constant 99.7%

   \[\leadsto \frac{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v} + -4.7180561523418874 \cdot 10^{-5}}}{v} \]
9. Add Preprocessing

Alternative 3: 99.7% 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 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in cosTheta_i around 0

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

9. Applied rewrites 99.6%

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

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

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 (- (/ -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(((-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((((-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(-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(((single(-1.0) / v) - single(-0.6931))) / (v + v);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in cosTheta_i around 0

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.7%

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

Alternative 5: 97.7% accurate, 2.5× 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

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

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)))

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));
}

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))
end function

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 97.7%

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

Alternative 6: 18.6% accurate, 2.0× 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

\[\begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -9.999999350456404 \cdot 10^{-39}:\\ \;\;\;\;e^{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_i\right) \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \]

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 (<= (* cosTheta_i cosTheta_O) -9.999999350456404e-39)
  (exp (* cosTheta_i (/ cosTheta_O 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 ((cosTheta_i * cosTheta_O) <= -9.999999350456404e-39f) {
		tmp = expf((cosTheta_i * (cosTheta_O / 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 ((costheta_i * costheta_o) <= (-9.999999350456404e-39)) then
        tmp = exp((costheta_i * (costheta_o / 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(cosTheta_i * cosTheta_O) <= Float32(-9.999999350456404e-39))
		tmp = exp(Float32(cosTheta_i * Float32(cosTheta_O / 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 ((cosTheta_i * cosTheta_O) <= single(-9.999999350456404e-39))
		tmp = exp((cosTheta_i * (cosTheta_O / v)));
	else
		tmp = exp((-sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -9.99999935e-39

   1. Initial program 99.6%

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

   2. Taylor expanded in sinTheta_i around 0

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in cosTheta_i around inf

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

   10. Applied rewrites 13.4%

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

   12. Applied rewrites 13.4%

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

   if -9.99999935e-39 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.6%

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

   2. Taylor expanded in sinTheta_i around 0

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in sinTheta_i around inf

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

   7. Applied rewrites 13.1%

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

   9. Applied rewrites 13.1%

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

Alternative 7: 13.4% accurate, 2.8× 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

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

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) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((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(((costheta_o * costheta_i) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in cosTheta_i around inf

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

10. Applied rewrites 13.4%

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

Alternative 8: 13.4% accurate, 2.8× 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

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

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 (* cosTheta_i (/ cosTheta_O 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)));
}

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)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in cosTheta_i around inf

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

10. Applied rewrites 13.4%

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

12. Applied rewrites 13.4%

    \[\leadsto e^{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} \]
13. Add Preprocessing

Alternative 9: 4.6% accurate, 6.7× 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

\[1.9999055862426758 \cdot \frac{0.5}{v} \]

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)))
  (* 1.9999055862426758 (/ 0.5 v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.9999055862426758f * (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 = 1.9999055862426758e0 * (0.5e0 / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.9999055862426758) * Float32(Float32(0.5) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.9999055862426758) * (single(0.5) / v);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in v around inf

   \[\leadsto e^{\frac{6931}{10000}} \cdot \frac{0.5}{v} \]

6. Applied rewrites 4.6%

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

7. Evaluated real constant 4.6%

   \[\leadsto 1.9999055862426758 \cdot \frac{0.5}{v} \]
8. Add Preprocessing

Alternative 10: 4.6% accurate, 6.7× 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

\[\frac{0.5}{v \cdot 0.500023603439331} \]

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)))
  (/ 0.5 (* v 0.500023603439331)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / (v * 0.500023603439331f);
}

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 = 0.5e0 / (v * 0.500023603439331e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(v * Float32(0.500023603439331)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v * single(0.500023603439331));
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{e^{\frac{6931}{10000}}}{v} \]

6. Applied rewrites 4.6%

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

8. Applied rewrites 4.6%

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

9. Evaluated real constant 4.6%

   \[\leadsto \frac{0.5}{v \cdot 0.500023603439331} \]
10. Add Preprocessing

Alternative 11: 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 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{e^{\frac{6931}{10000}}}{v} \]

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. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{2097053}{2097152}}{v} \]

10. Applied rewrites 4.6%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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))))))