HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 43.1s

Alternatives: 17

Speedup: 1.0×

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

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

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_O (* cosTheta_i (/ 1.0 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_O * (cosTheta_i * (1.0f / 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_o * (costheta_i * (1.0e0 / 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(cosTheta_O * Float32(cosTheta_i * Float32(Float32(1.0) / 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_O * (cosTheta_i * (single(1.0) / v)))) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.8%

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

Alternative 2: 98.6% accurate, 0.9× 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

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

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))
  (*
 (*
  (exp (* (- sinTheta_i) (/ sinTheta_O v)))
  (fmin cosTheta_i cosTheta_O))
 (/ (/ (fmax cosTheta_i cosTheta_O) v) (* (sinh (/ 1.0 v)) (+ v 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))) * fminf(cosTheta_i, cosTheta_O)) * ((fmaxf(cosTheta_i, cosTheta_O) / v) / (sinhf((1.0f / v)) * (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((-sintheta_i * (sintheta_o / v))) * fmin(costheta_i, costheta_o)) * ((fmax(costheta_i, costheta_o) / v) / (sinh((1.0e0 / v)) * (v + v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.6%

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

5. Applied rewrites 98.6%

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

Alternative 3: 98.5% 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

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

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))
  (/
 (* 1.0 (* cosTheta_O (* cosTheta_i (/ 1.0 v))))
 (* (sinh (/ 1.0 v)) (+ v v))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.8%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.5%

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

8. Applied rewrites 98.5%

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

Alternative 4: 98.5% 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

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

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))
  (/
 (* (* cosTheta_i cosTheta_O) (/ 1.0 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 ((cosTheta_i * cosTheta_O) * (1.0f / 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 = ((costheta_i * costheta_o) * (1.0e0 / 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(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / 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 = ((cosTheta_i * cosTheta_O) * (single(1.0) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 98.7%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.5%

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

Alternative 5: 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

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

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))
  (*
 (fmin cosTheta_i cosTheta_O)
 (/ (/ (fmax cosTheta_i cosTheta_O) v) (* (sinh (/ 1.0 v)) (+ v v)))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = min(cosTheta_i, cosTheta_O) * ((max(cosTheta_i, cosTheta_O) / v) / (sinh((single(1.0) / v)) * (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 sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.4%

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

10. Applied rewrites 98.4%

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

Alternative 6: 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

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

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) (* (sinh (/ 1.0 v)) (+ 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) / (sinhf((1.0f / v)) * (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) / (sinh((1.0e0 / v)) * (v + v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) / (sinh((single(1.0) / v)) * (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 sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.4%

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

10. Applied rewrites 98.4%

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

Alternative 7: 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

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

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))
  (* cosTheta_i (/ cosTheta_O (* v (* (sinh (/ 1.0 v)) (+ v v))))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * (cosTheta_O / (v * (sinh((single(1.0) / v)) * (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 sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.4%

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

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

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)
 (*
  (fmax (fabs cosTheta_i) cosTheta_O)
  (/
   (fmin (fabs cosTheta_i) cosTheta_O)
   (* v (* (sinh (/ 1.0 v)) (+ 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) * (fmaxf(fabsf(cosTheta_i), cosTheta_O) * (fminf(fabsf(cosTheta_i), cosTheta_O) / (v * (sinhf((1.0f / v)) * (v + 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(v * Float32(sinh(Float32(Float32(1.0) / v)) * 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))) * (max(abs(cosTheta_i), cosTheta_O) * (min(abs(cosTheta_i), cosTheta_O) / (v * (sinh((single(1.0) / v)) * (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 sinTheta_i around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.4%

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

Alternative 9: 59.2% 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

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)\\ \mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \frac{1}{v \cdot \frac{\frac{\mathsf{fma}\left(2, \frac{v}{t\_0}, 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{t\_0}\right)}{v}}{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}}\right) \end{array} \]

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))
  (let* ((t_0 (fmax (fabs cosTheta_i) (fabs cosTheta_O))))
  (*
   (copysign 1.0 cosTheta_i)
   (*
    (copysign 1.0 cosTheta_O)
    (/
     1.0
     (*
      v
      (/
       (/
        (fma 2.0 (/ v t_0) (* 2.0 (/ (* sinTheta_O sinTheta_i) t_0)))
        v)
       (fmin (fabs cosTheta_i) (fabs cosTheta_O)))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O));
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * (1.0f / (v * ((fmaf(2.0f, (v / t_0), (2.0f * ((sinTheta_O * sinTheta_i) / t_0))) / v) / fminf(fabsf(cosTheta_i), fabsf(cosTheta_O))))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = fmax(abs(cosTheta_i), abs(cosTheta_O))
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(Float32(1.0) / Float32(v * Float32(Float32(fma(Float32(2.0), Float32(v / t_0), Float32(Float32(2.0) * Float32(Float32(sinTheta_O * sinTheta_i) / t_0))) / v) / fmin(abs(cosTheta_i), abs(cosTheta_O)))))))
end

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 36.4%

   \[\leadsto \frac{1}{v \cdot \mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}, 2 \cdot \frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)} \]

7. Taylor expanded in cosTheta_i around 0

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

9. Applied rewrites 59.2%

   \[\leadsto \frac{1}{v \cdot \frac{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O \cdot v}, 2 \cdot \frac{1}{cosTheta\_O}\right)}{cosTheta\_i}} \]

10. Taylor expanded in v around 0

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

12. Applied rewrites 59.2%

    \[\leadsto \frac{1}{v \cdot \frac{\frac{\mathsf{fma}\left(2, \frac{v}{cosTheta\_O}, 2 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O}\right)}{v}}{cosTheta\_i}} \]
13. Add Preprocessing

Alternative 10: 59.2% 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

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)\\ \mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \frac{1}{v \cdot \frac{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{t\_0 \cdot v}, 2 \cdot \frac{1}{t\_0}\right)}{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}}\right) \end{array} \]

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))
  (let* ((t_0 (fmax (fabs cosTheta_i) (fabs cosTheta_O))))
  (*
   (copysign 1.0 cosTheta_i)
   (*
    (copysign 1.0 cosTheta_O)
    (/
     1.0
     (*
      v
      (/
       (fma
        2.0
        (/ (* sinTheta_O sinTheta_i) (* t_0 v))
        (* 2.0 (/ 1.0 t_0)))
       (fmin (fabs cosTheta_i) (fabs cosTheta_O)))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O));
	return copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * (1.0f / (v * (fmaf(2.0f, ((sinTheta_O * sinTheta_i) / (t_0 * v)), (2.0f * (1.0f / t_0))) / fminf(fabsf(cosTheta_i), fabsf(cosTheta_O))))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = fmax(abs(cosTheta_i), abs(cosTheta_O))
	return Float32(copysign(Float32(1.0), cosTheta_i) * Float32(copysign(Float32(1.0), cosTheta_O) * Float32(Float32(1.0) / Float32(v * Float32(fma(Float32(2.0), Float32(Float32(sinTheta_O * sinTheta_i) / Float32(t_0 * v)), Float32(Float32(2.0) * Float32(Float32(1.0) / t_0))) / fmin(abs(cosTheta_i), abs(cosTheta_O)))))))
end

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 36.4%

   \[\leadsto \frac{1}{v \cdot \mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}, 2 \cdot \frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)} \]

7. Taylor expanded in cosTheta_i around 0

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

9. Applied rewrites 59.2%

   \[\leadsto \frac{1}{v \cdot \frac{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O \cdot v}, 2 \cdot \frac{1}{cosTheta\_O}\right)}{cosTheta\_i}} \]
10. Add Preprocessing

Alternative 11: 59.2% 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

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

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))
  (/ 1.0 (* v (/ 2.0 (* cosTheta_O cosTheta_i)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 36.4%

   \[\leadsto \frac{1}{v \cdot \mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}, 2 \cdot \frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)} \]

7. Taylor expanded in cosTheta_i around 0

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

9. Applied rewrites 59.2%

   \[\leadsto \frac{1}{v \cdot \frac{\mathsf{fma}\left(2, \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O \cdot v}, 2 \cdot \frac{1}{cosTheta\_O}\right)}{cosTheta\_i}} \]

10. Taylor expanded in sinTheta_i around 0

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

12. Applied rewrites 59.2%

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

Alternative 12: 59.1% accurate, 4.0× 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

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

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 (/ (+ v v) (* cosTheta_i cosTheta_O))))

C

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

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 / ((v + v) / (costheta_i * costheta_o))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 59.1%

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

8. Applied rewrites 59.1%

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

Alternative 13: 58.6% accurate, 4.1× 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

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

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_i cosTheta_O)) (/ 1.0 v)))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * (cosTheta_i * cosTheta_O)) * (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 v around inf

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

4. Applied rewrites 58.5%

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

6. Applied rewrites 58.5%

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

8. Applied rewrites 58.6%

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

Alternative 14: 58.6% 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

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

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_i cosTheta_O)) v))

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 58.5%

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

6. Applied rewrites 58.6%

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

Alternative 15: 58.5% 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

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :precision binary32
  :pre (and (and (and (and (and (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(0.5) * Float32(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

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

4. Applied rewrites 58.5%

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

Alternative 16: 58.5% 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

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

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)
  (*
   0.5
   (*
    (fmin (fabs cosTheta_i) (fabs cosTheta_O))
    (/ (fmax (fabs cosTheta_i) (fabs cosTheta_O)) 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 * (fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) * (fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / 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(0.5) * Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) * Float32(fmax(abs(cosTheta_i), abs(cosTheta_O)) / 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) * (min(abs(cosTheta_i), abs(cosTheta_O)) * (max(abs(cosTheta_i), abs(cosTheta_O)) / 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

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

4. Applied rewrites 58.5%

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

6. Applied rewrites 58.5%

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

Alternative 17: 58.5% accurate, 3.4× 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

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

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))
  (*
 0.5
 (* (fmax cosTheta_i cosTheta_O) (/ (fmin cosTheta_i cosTheta_O) v))))

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 58.5%

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

6. Applied rewrites 58.5%

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

Reproduce

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