HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 8.6s

Alternatives: 17

Speedup: 1.5×

Specification

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

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/
 (*
  (exp (- (/ (* sinTheta_i sinTheta_O) v)))
  (/ (* cosTheta_i cosTheta_O) v))
 (* (* (sinh (/ 1.0 v)) 2.0) 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)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Initial Program: 98.6% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/
 (*
  (exp (- (/ (* sinTheta_i sinTheta_O) v)))
  (/ (* cosTheta_i cosTheta_O) v))
 (* (* (sinh (/ 1.0 v)) 2.0) 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)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Alternative 1: 98.8% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (* (* cosTheta_O cosTheta_i) (/ 0.5 v))
 (/
  (/ 1.0 (* v (exp (/ (* sinTheta_O sinTheta_i) v))))
  (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * cosTheta_i) * (0.5f / v)) * ((1.0f / (v * expf(((sinTheta_O * sinTheta_i) / v)))) / sinhf((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 = ((costheta_o * costheta_i) * (0.5e0 / v)) * ((1.0e0 / (v * exp(((sintheta_o * sintheta_i) / v)))) / sinh((1.0e0 / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.7%

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

5. Applied rewrites 98.8%

   \[\leadsto \left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{0.5}{v}\right) \cdot \frac{\frac{1}{v \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{\sinh \left(\frac{1}{v}\right)} \]
6. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (/ (* cosTheta_O cosTheta_i) (+ v v))
 (/
  (/ 1.0 (* v (exp (/ (* sinTheta_O sinTheta_i) v))))
  (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * cosTheta_i) / (v + v)) * ((1.0f / (v * expf(((sinTheta_O * sinTheta_i) / v)))) / sinhf((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 = ((costheta_o * costheta_i) / (v + v)) * ((1.0e0 / (v * exp(((sintheta_o * sintheta_i) / v)))) / sinh((1.0e0 / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.7%

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

Alternative 3: 98.6% accurate, 1.5× speedup

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

Math

\[\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{0.5}{v}\right) \cdot \frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (* (* cosTheta_O cosTheta_i) (/ 0.5 v))
 (/ (/ 1.0 v) (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * cosTheta_i) * (0.5f / v)) * ((1.0f / v) / sinhf((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 = ((costheta_o * costheta_i) * (0.5e0 / v)) * ((1.0e0 / v) / sinh((1.0e0 / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.7%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.5%

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

8. Applied rewrites 98.6%

   \[\leadsto \left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{0.5}{v}\right) \cdot \frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)} \]
9. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

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

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\left(\frac{0.5}{v} \cdot \mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)\right) \cdot \frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{\sinh \left(\frac{1}{v}\right) \cdot v}\right)\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (copysign 1.0 cosTheta_O)
  (*
   (* (/ 0.5 v) (fmax (fabs cosTheta_i) (fabs cosTheta_O)))
   (/
    (fmin (fabs cosTheta_i) (fabs cosTheta_O))
    (* (sinh (/ 1.0 v)) v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * (((0.5f / v) * fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O))) * (fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / (sinhf((1.0f / v)) * v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(Float32(Float32(Float32(0.5) / v) * fmax(abs(cosTheta_i), abs(cosTheta_O))) * Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / Float32(sinh(Float32(Float32(1.0) / v)) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * ((sign(cosTheta_O) * abs(single(1.0))) * (((single(0.5) / v) * max(abs(cosTheta_i), abs(cosTheta_O))) * (min(abs(cosTheta_i), abs(cosTheta_O)) / (sinh((single(1.0) / v)) * v))));
end

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.6%

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

4. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)} \]
5. Step-by-step derivation

6. Applied rewrites 98.4%

   \[\leadsto \frac{0.5}{v} \cdot \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

   \[\leadsto \frac{0.5}{v} \cdot \left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot v}\right) \]
9. Step-by-step derivation

10. Applied rewrites 98.6%

    \[\leadsto \left(\frac{0.5}{v} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot v} \]
11. Add Preprocessing

Alternative 5: 98.4% accurate, 1.6× speedup

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

Math

\[\frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot v}}{2 \cdot \sinh \left(\frac{1}{v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (/ (* cosTheta_O cosTheta_i) (* v v)) (* 2.0 (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * cosTheta_i) / (v * v)) / (2.0f * sinhf((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 = ((costheta_o * costheta_i) / (v * v)) / (2.0e0 * sinh((1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_O * cosTheta_i) / Float32(v * v)) / Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O * cosTheta_i) / (v * v)) / (single(2.0) * sinh((single(1.0) / v)));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 98.4%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot v}}{2 \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

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

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\frac{\mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{v} \cdot \frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{\left(v + v\right) \cdot \sinh \left(\frac{1}{v}\right)}\right)\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (copysign 1.0 cosTheta_O)
  (*
   (/ (fmax (fabs cosTheta_i) (fabs cosTheta_O)) v)
   (/
    (fmin (fabs cosTheta_i) (fabs cosTheta_O))
    (* (+ v v) (sinh (/ 1.0 v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * ((fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / v) * (fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / ((v + v) * sinhf((1.0f / v))))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(Float32(fmax(abs(cosTheta_i), abs(cosTheta_O)) / v) * Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / Float32(Float32(v + v) * sinh(Float32(Float32(1.0) / v)))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * ((sign(cosTheta_O) * abs(single(1.0))) * ((max(abs(cosTheta_i), abs(cosTheta_O)) / v) * (min(abs(cosTheta_i), abs(cosTheta_O)) / ((v + v) * sinh((single(1.0) / v))))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 98.4%

   \[\leadsto \frac{cosTheta\_O}{v} \cdot \frac{cosTheta\_i}{\left(v + v\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Add Preprocessing

Alternative 7: 98.4% accurate, 1.4× speedup

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

Math

\[\mathsf{max}\left(cosTheta\_i, cosTheta\_O\right) \cdot \frac{\frac{\mathsf{min}\left(cosTheta\_i, cosTheta\_O\right)}{\left(v + v\right) \cdot v}}{\sinh \left(\frac{1}{v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (fmax cosTheta_i cosTheta_O)
 (/ (/ (fmin cosTheta_i cosTheta_O) (* (+ v v) v)) (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return fmaxf(cosTheta_i, cosTheta_O) * ((fminf(cosTheta_i, cosTheta_O) / ((v + v) * v)) / sinhf((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 = fmax(costheta_i, costheta_o) * ((fmin(costheta_i, costheta_o) / ((v + v) * v)) / sinh((1.0e0 / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 98.4%

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

8. Applied rewrites 98.4%

   \[\leadsto cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{\left(v + v\right) \cdot v}}{\sinh \left(\frac{1}{v}\right)} \]
9. Add Preprocessing

Alternative 8: 98.4% accurate, 1.2× speedup

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

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{max}\left(\left|cosTheta\_i\right|, cosTheta\_O\right) \cdot \frac{\mathsf{min}\left(\left|cosTheta\_i\right|, cosTheta\_O\right)}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)}\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (fmax (fabs cosTheta_i) cosTheta_O)
  (/
   (fmin (fabs cosTheta_i) cosTheta_O)
   (* (* v (+ v v)) (sinh (/ 1.0 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (fmaxf(fabsf(cosTheta_i), cosTheta_O) * (fminf(fabsf(cosTheta_i), cosTheta_O) / ((v * (v + v)) * sinhf((1.0f / v)))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(fmax(abs(cosTheta_i), cosTheta_O) * Float32(fmin(abs(cosTheta_i), cosTheta_O) / Float32(Float32(v * Float32(v + v)) * sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * (max(abs(cosTheta_i), cosTheta_O) * (min(abs(cosTheta_i), cosTheta_O) / ((v * (v + v)) * sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 98.4%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Add Preprocessing

Alternative 9: 98.4% accurate, 1.7× speedup

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

Math

\[\frac{cosTheta\_O \cdot cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (* cosTheta_O cosTheta_i) (* (* v (+ v v)) (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / ((v * (v + v)) * sinhf((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 = (costheta_o * costheta_i) / ((v * (v + v)) * sinh((1.0e0 / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in sinTheta_i around 0

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 98.4%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(v \cdot \left(v + v\right)\right) \cdot \sinh \left(\frac{1}{v}\right)} \]
7. Add Preprocessing

Alternative 10: 59.7% accurate, 3.9× speedup

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

Math

\[\frac{1}{\frac{2}{cosTheta\_O \cdot cosTheta\_i} \cdot v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ 1.0 (* (/ 2.0 (* cosTheta_O cosTheta_i)) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((2.0f / (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 = 1.0e0 / ((2.0e0 / (costheta_o * costheta_i)) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / Float32(cosTheta_O * cosTheta_i)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((single(2.0) / (cosTheta_O * cosTheta_i)) * v);
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

6. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \]
7. Step-by-step derivation

8. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \left(cosTheta\_O \cdot \frac{1}{\frac{v}{cosTheta\_i}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 59.7%

    \[\leadsto \frac{1}{\frac{2}{cosTheta\_O \cdot cosTheta\_i} \cdot v} \]
11. Add Preprocessing

Alternative 11: 59.6% accurate, 3.9× speedup

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

Math

\[\frac{0.5}{\frac{1}{cosTheta\_O \cdot cosTheta\_i} \cdot v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ 0.5 (* (/ 1.0 (* cosTheta_O cosTheta_i)) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / ((1.0f / (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 = 0.5e0 / ((1.0e0 / (costheta_o * costheta_i)) * v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

6. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \]
7. Step-by-step derivation

8. Applied rewrites 59.6%

   \[\leadsto \frac{0.5}{\frac{1}{cosTheta\_O \cdot cosTheta\_i} \cdot v} \]
9. Add Preprocessing

Alternative 12: 59.4% accurate, 1.7× speedup

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

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \frac{0.5}{\frac{\frac{v}{\mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}}{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}}\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (copysign 1.0 cosTheta_O)
  (/
   0.5
   (/
    (/ v (fmax (fabs cosTheta_i) (fabs cosTheta_O)))
    (fmin (fabs cosTheta_i) (fabs cosTheta_O)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * (0.5f / ((v / fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O))) / fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(Float32(0.5) / Float32(Float32(v / fmax(abs(cosTheta_i), abs(cosTheta_O))) / fmin(abs(cosTheta_i), abs(cosTheta_O))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * ((sign(cosTheta_O) * abs(single(1.0))) * (single(0.5) / ((v / max(abs(cosTheta_i), abs(cosTheta_O))) / min(abs(cosTheta_i), abs(cosTheta_O)))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

6. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \]
7. Step-by-step derivation

8. Applied rewrites 59.6%

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

10. Applied rewrites 59.4%

    \[\leadsto \frac{0.5}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}} \]
11. Add Preprocessing

Alternative 13: 59.4% accurate, 4.9× speedup

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

Math

\[\frac{0.5}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ 0.5 (/ v (* cosTheta_O cosTheta_i))))

C

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

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 / (costheta_o * costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

6. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \]
7. Step-by-step derivation

8. Applied rewrites 59.4%

   \[\leadsto \frac{0.5}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]
9. Add Preprocessing

Alternative 14: 59.0% accurate, 5.3× speedup

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

Math

\[\frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (* 0.5 (* cosTheta_O cosTheta_i)) v))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

6. Applied rewrites 59.0%

   \[\leadsto \frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]
7. Add Preprocessing

Alternative 15: 58.9% accurate, 5.3× speedup

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

Math

\[\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{0.5}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (* (* cosTheta_O cosTheta_i) (/ 0.5 v)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

6. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \]
7. Step-by-step derivation

8. Applied rewrites 58.9%

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{0.5}{v} \]
9. Add Preprocessing

Alternative 16: 58.9% accurate, 5.5× speedup

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

Math

\[\frac{cosTheta\_O \cdot cosTheta\_i}{v + v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (/ (* cosTheta_O cosTheta_i) (+ v v)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]

6. Applied rewrites 59.5%

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

8. Applied rewrites 58.9%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v + v} \]
9. Add Preprocessing

Alternative 17: 58.9% accurate, 1.8× speedup

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

Math

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right) \cdot \frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{v + v}\right)\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (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)))
          (< 0.1 v))
     (<= v 1.5707964))
  (*
 (copysign 1.0 cosTheta_i)
 (*
  (copysign 1.0 cosTheta_O)
  (*
   (fmax (fabs cosTheta_i) (fabs cosTheta_O))
   (/ (fmin (fabs cosTheta_i) (fabs cosTheta_O)) (+ v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * (fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O)) * (fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / (v + v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(fmax(abs(cosTheta_i), abs(cosTheta_O)) * Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / Float32(v + v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_i) * abs(single(1.0))) * ((sign(cosTheta_O) * abs(single(1.0))) * (max(abs(cosTheta_i), abs(cosTheta_O)) * (min(abs(cosTheta_i), abs(cosTheta_O)) / (v + v))));
end

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in v around inf

   \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} \]
3. Step-by-step derivation

4. Applied rewrites 58.9%

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

6. Applied rewrites 58.9%

   \[\leadsto 0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \]
7. Step-by-step derivation

8. Applied rewrites 58.9%

   \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v + v} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2026086 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))