HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 10.1s

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: 98.7% 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

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

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)
 (/
  (/ 1.0 (* v (exp (/ (* sinTheta_O sinTheta_i) 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) * ((1.0f / (v * expf(((sinTheta_O * sinTheta_i) / 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) * ((1.0e0 / (v * exp(((sintheta_o * sintheta_i) / 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) / v) * Float32(Float32(Float32(1.0) / Float32(v * exp(Float32(Float32(sinTheta_O * sinTheta_i) / 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) * ((single(1.0) / (v * exp(((sinTheta_O * sinTheta_i) / 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. Step-by-step derivation

3. Applied rewrites 98.7%

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

Alternative 2: 98.7% accurate, 1.3× 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

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

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)
 (/ (/ 1.0 (+ v (* sinTheta_O sinTheta_i))) (* 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) * ((1.0f / (v + (sinTheta_O * sinTheta_i))) / (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) * ((1.0e0 / (v + (sintheta_o * sintheta_i))) / (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) / v) * Float32(Float32(Float32(1.0) / Float32(v + Float32(sinTheta_O * sinTheta_i))) / 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) * ((single(1.0) / (v + (sinTheta_O * sinTheta_i))) / (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. Step-by-step derivation

3. Applied rewrites 98.7%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.7%

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

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

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))
  (/
 (* (* (/ -1.0 v) (* cosTheta_O cosTheta_i)) (/ -0.5 v))
 (sinh (/ 1.0 v))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((single(-1.0) / v) * (cosTheta_O * cosTheta_i)) * (single(-0.5) / 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 98.6%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.4%

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

10. Applied rewrites 98.6%

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

Alternative 4: 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

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\left(\frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{v} \cdot \mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)\right) \cdot \frac{\frac{1}{v}}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)\right) \]

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)
  (*
   (*
    (/ (fmin (fabs cosTheta_i) (fabs cosTheta_O)) v)
    (fmax (fabs cosTheta_i) (fabs cosTheta_O)))
   (/ (/ 1.0 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 copysignf(1.0f, cosTheta_i) * (copysignf(1.0f, cosTheta_O) * (((fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / v) * fmaxf(fabsf(cosTheta_i), fabsf(cosTheta_O))) * ((1.0f / v) / (2.0f * 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(Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / v) * fmax(abs(cosTheta_i), abs(cosTheta_O))) * Float32(Float32(Float32(1.0) / 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 = (sign(cosTheta_i) * abs(single(1.0))) * ((sign(cosTheta_O) * abs(single(1.0))) * (((min(abs(cosTheta_i), abs(cosTheta_O)) / v) * max(abs(cosTheta_i), abs(cosTheta_O))) * ((single(1.0) / 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. Step-by-step derivation

3. Applied rewrites 98.7%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.5%

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

8. Applied rewrites 98.5%

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

Alternative 5: 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

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

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

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.2%

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

8. Applied rewrites 98.2%

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

10. Applied rewrites 98.6%

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

Alternative 6: 98.5% accurate, 1.1× 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

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

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))
  (*
 (copysign 1.0 cosTheta_O)
 (/
  (*
   (-
    (*
     (/ (fmin cosTheta_i (fabs cosTheta_O)) v)
     (fmax cosTheta_i (fabs cosTheta_O))))
   (/ -0.5 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_O) * ((-((fminf(cosTheta_i, fabsf(cosTheta_O)) / v) * fmaxf(cosTheta_i, fabsf(cosTheta_O))) * (-0.5f / v)) / sinhf((1.0f / v)));
}

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_O) * abs(single(1.0))) * ((-((min(cosTheta_i, abs(cosTheta_O)) / v) * max(cosTheta_i, abs(cosTheta_O))) * (single(-0.5) / 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 98.6%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.5%

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

10. Applied rewrites 98.5%

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

Alternative 7: 98.5% 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

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

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))
  (/
 (* (/ -0.5 v) (- (* cosTheta_O cosTheta_i)))
 (* (sinh (/ 1.0 v)) v)))

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)) / (sinhf((1.0f / 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 = (((-0.5e0) / v) * -(costheta_o * costheta_i)) / (sinh((1.0e0 / v)) * v)
end function

Julia

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

MATLAB

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

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.3%

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

10. Applied rewrites 98.5%

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

Alternative 8: 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

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \left(\frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{2 \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{\mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{v \cdot v}\right)\right) \]

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)
  (*
   (/
    (fmin (fabs cosTheta_i) (fabs cosTheta_O))
    (* 2.0 (sinh (/ 1.0 v))))
   (/ (fmax (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) * ((fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / (2.0f * sinhf((1.0f / v)))) * (fmaxf(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(Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v)))) * Float32(fmax(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))) * ((min(abs(cosTheta_i), abs(cosTheta_O)) / (single(2.0) * sinh((single(1.0) / v)))) * (max(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. Step-by-step derivation

3. Applied rewrites 98.6%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.4%

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

8. Applied rewrites 98.4%

   \[\leadsto \frac{cosTheta\_i}{2 \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{v \cdot v} \]
9. 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

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

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 (/ (* cosTheta_O cosTheta_i) (* v (* (sinh (/ 1.0 v)) 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 * (sinhf((1.0f / 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 = 0.5e0 * ((costheta_o * costheta_i) / (v * (sinh((1.0e0 / v)) * 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) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_O * cosTheta_i) / (v * (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 98.5%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.3%

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

10. Applied rewrites 98.3%

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

Alternative 10: 98.3% 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. Step-by-step derivation

3. Applied rewrites 98.7%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 98.5%

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

8. Applied rewrites 97.8%

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

10. Applied rewrites 98.4%

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

Alternative 11: 58.6% 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

\[\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \frac{1}{2 \cdot \left(v \cdot \frac{\frac{1}{\mathsf{max}\left(cosTheta\_i, \left|cosTheta\_O\right|\right)}}{\mathsf{min}\left(cosTheta\_i, \left|cosTheta\_O\right|\right)}\right)} \]

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_O)
 (/
  1.0
  (*
   2.0
   (*
    v
    (/
     (/ 1.0 (fmax cosTheta_i (fabs cosTheta_O)))
     (fmin 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_O) * (1.0f / (2.0f * (v * ((1.0f / fmaxf(cosTheta_i, fabsf(cosTheta_O))) / fminf(cosTheta_i, fabsf(cosTheta_O))))));
}

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sign(cosTheta_O) * abs(single(1.0))) * (single(1.0) / (single(2.0) * (v * ((single(1.0) / max(cosTheta_i, abs(cosTheta_O))) / min(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. Step-by-step derivation

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 58.6%

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 58.6%

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

Alternative 12: 58.6% accurate, 3.3× 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

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

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 v) (/ 1.0 (* 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 + v) * (1.0f / (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 + v) * (1.0e0 / (costheta_o * costheta_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 58.6%

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

   1. pow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. lift-pow.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 58.6%

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

   9. lift-pow.f32 N/A

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

   10. unpow-1 N/A

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

   11. lower-/.f32 58.6%

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

8. Applied rewrites 58.6%

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

10. Applied rewrites 58.6%

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

Alternative 13: 58.6% 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

\[\mathsf{copysign}\left(1, cosTheta\_i\right) \cdot \left(\mathsf{copysign}\left(1, cosTheta\_O\right) \cdot \frac{1}{\frac{2}{\frac{\mathsf{min}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}{v} \cdot \mathsf{max}\left(\left|cosTheta\_i\right|, \left|cosTheta\_O\right|\right)}}\right) \]

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)
  (/
   1.0
   (/
    2.0
    (*
     (/ (fmin (fabs cosTheta_i) (fabs cosTheta_O)) v)
     (fmax (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) * (1.0f / (2.0f / ((fminf(fabsf(cosTheta_i), fabsf(cosTheta_O)) / v) * fmaxf(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(1.0) / Float32(Float32(2.0) / Float32(Float32(fmin(abs(cosTheta_i), abs(cosTheta_O)) / v) * fmax(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(1.0) / (single(2.0) / ((min(abs(cosTheta_i), abs(cosTheta_O)) / v) * max(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. Step-by-step derivation

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 58.6%

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

8. Applied rewrites 58.6%

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

9. Taylor expanded in sinTheta_i around 0

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

11. Applied rewrites 58.6%

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

Alternative 14: 58.6% accurate, 3.7× 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

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

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (single(2.0) * ((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. Step-by-step derivation

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 58.6%

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

8. Applied rewrites 58.6%

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

Alternative 15: 58.6% 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

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

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 v) (* 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 + 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 = 1.0e0 / ((v + v) / (costheta_o * costheta_i))
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_O * cosTheta_i)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((v + 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. Step-by-step derivation

3. Applied rewrites 93.4%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 58.6%

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

8. Applied rewrites 58.6%

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

Alternative 16: 58.1% 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 58.0%

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

6. Applied rewrites 58.0%

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

   1. pow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. lift-pow.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 58.7%

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

   9. lift-pow.f32 N/A

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

   10. unpow-1 N/A

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

   11. lower-/.f32 58.7%

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

8. Applied rewrites 58.7%

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

10. Applied rewrites 58.1%

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

Alternative 17: 58.0% 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

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

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 (* 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(0.5) * Float32(cosTheta_i * Float32(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

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

4. Applied rewrites 58.0%

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

6. Applied rewrites 58.0%

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

Reproduce

herbie shell --seed 2026070 
(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)))