HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 13.2s

Alternatives: 7

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

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (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)
    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

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (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)
    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

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\left(e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \left(cosTheta_i_m \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (*
    (exp (/ (* sinTheta_O (- sinTheta_i)) v))
    (* cosTheta_i_m (/ cosTheta_O v)))
   (/ (/ 0.5 v) (sinh (/ 1.0 v))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((expf(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_i_m * (cosTheta_O / v))) * ((0.5f / v) / sinhf((1.0f / v))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((exp(((sintheta_o * -sintheta_i) / v)) * (costheta_i_m * (costheta_o / v))) * ((0.5e0 / v) / sinh((1.0e0 / v))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(cosTheta_i_m * Float32(cosTheta_O / v))) * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_i_m * (cosTheta_O / v))) * ((single(0.5) / v) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. div-inv 98.3%

      \[\leadsto \color{blue}{\left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. distribute-neg-frac 98.3%

      \[\leadsto \left(e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-*r/ 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l* 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}} \]

3. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.2%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}\right)\right)} \]

   2. expm1-udef 95.6%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}\right)} - 1\right)} \]

   3. associate-/r* 95.6%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{2 \cdot v}}\right)} - 1\right) \]

   4. *-commutative 95.6%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{v \cdot 2}}\right)} - 1\right) \]

5. Applied egg-rr 95.6%

   \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{v \cdot 2}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 98.2%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{v \cdot 2}\right)\right)} \]

   2. expm1-log1p 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{v \cdot 2}} \]

   3. associate-/l/ 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\frac{1}{\left(v \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. associate-/r* 98.5%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\frac{\frac{1}{v \cdot 2}}{\sinh \left(\frac{1}{v}\right)}} \]

   5. *-commutative 98.5%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{\frac{1}{\color{blue}{2 \cdot v}}}{\sinh \left(\frac{1}{v}\right)} \]

   6. associate-/r* 98.5%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{\sinh \left(\frac{1}{v}\right)} \]

   7. metadata-eval 98.5%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{\frac{\color{blue}{0.5}}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Simplified 98.5%

   \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \color{blue}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}} \]

8. Final simplification 98.5%

   \[\leadsto \left(e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \]

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\frac{e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_i_m}{v} \cdot \frac{\frac{cosTheta_O}{v}}{2}\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/ (exp (/ sinTheta_O (/ v sinTheta_i))) (sinh (/ 1.0 v)))
   (* (/ cosTheta_i_m v) (/ (/ cosTheta_O v) 2.0)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((expf((sinTheta_O / (v / sinTheta_i))) / sinhf((1.0f / v))) * ((cosTheta_i_m / v) * ((cosTheta_O / v) / 2.0f)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((exp((sintheta_o / (v / sintheta_i))) / sinh((1.0e0 / v))) * ((costheta_i_m / v) * ((costheta_o / v) / 2.0e0)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(exp(Float32(sinTheta_O / Float32(v / sinTheta_i))) / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(cosTheta_i_m / v) * Float32(Float32(cosTheta_O / v) / Float32(2.0)))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp((sinTheta_O / (v / sinTheta_i))) / sinh((single(1.0) / v))) * ((cosTheta_i_m / v) * ((cosTheta_O / v) / single(2.0))));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. div-inv 98.3%

      \[\leadsto \color{blue}{\left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. distribute-neg-frac 98.3%

      \[\leadsto \left(e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-*r/ 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l* 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}} \]

3. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}\right)\right)} \]

   2. expm1-udef 47.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}\right)} - 1} \]

5. Applied egg-rr 47.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v \cdot 2}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 98.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v \cdot 2}\right)\right)} \]

   2. expm1-log1p 98.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v \cdot 2}} \]

   3. *-commutative 98.1%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v \cdot 2} \]

   4. associate-/l* 98.1%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v \cdot 2} \]

   5. times-frac 98.2%

      \[\leadsto \frac{e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot \frac{\frac{cosTheta_O}{v}}{2}\right)} \]

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_i}{v} \cdot \frac{\frac{cosTheta_O}{v}}{2}\right)} \]

8. Final simplification 98.2%

   \[\leadsto \frac{e^{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{cosTheta_i}{v} \cdot \frac{\frac{cosTheta_O}{v}}{2}\right) \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \frac{\left(cosTheta_i_m \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (* (* cosTheta_i_m (/ cosTheta_O v)) (exp (/ (* sinTheta_i sinTheta_O) v)))
   (* (sinh (/ 1.0 v)) (* v 2.0)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((cosTheta_i_m * (cosTheta_O / v)) * expf(((sinTheta_i * sinTheta_O) / v))) / (sinhf((1.0f / v)) * (v * 2.0f)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((costheta_i_m * (costheta_o / v)) * exp(((sintheta_i * sintheta_o) / v))) / (sinh((1.0e0 / v)) * (v * 2.0e0)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_i_m * Float32(cosTheta_O / v)) * exp(Float32(Float32(sinTheta_i * sinTheta_O) / v))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(2.0)))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((cosTheta_i_m * (cosTheta_O / v)) * exp(((sinTheta_i * sinTheta_O) / v))) / (sinh((single(1.0) / v)) * (v * single(2.0))));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. div-inv 98.3%

      \[\leadsto \color{blue}{\left(e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. distribute-neg-frac 98.3%

      \[\leadsto \left(e^{\color{blue}{\frac{-sinTheta_i \cdot sinTheta_O}{v}}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-*r/ 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l* 98.3%

      \[\leadsto \left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}} \]

3. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}} \]
4. Step-by-step derivation

   1. un-div-inv 98.4%

      \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)}} \]

   2. add-sqr-sqrt 67.6%

      \[\leadsto \frac{e^{\frac{\color{blue}{\sqrt{-sinTheta_i \cdot sinTheta_O} \cdot \sqrt{-sinTheta_i \cdot sinTheta_O}}}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)} \]

   3. sqrt-unprod 98.4%

      \[\leadsto \frac{e^{\frac{\color{blue}{\sqrt{\left(-sinTheta_i \cdot sinTheta_O\right) \cdot \left(-sinTheta_i \cdot sinTheta_O\right)}}}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)} \]

   4. sqr-neg 98.4%

      \[\leadsto \frac{e^{\frac{\sqrt{\color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \left(sinTheta_i \cdot sinTheta_O\right)}}}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)} \]

   5. sqrt-unprod 63.1%

      \[\leadsto \frac{e^{\frac{\color{blue}{\sqrt{sinTheta_i \cdot sinTheta_O} \cdot \sqrt{sinTheta_i \cdot sinTheta_O}}}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)} \]

   6. add-sqr-sqrt 98.3%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)} \]

   7. *-commutative 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot 2\right)}} \]

5. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]

6. Final simplification 98.3%

   \[\leadsto \frac{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i_m}{v \cdot \frac{v}{cosTheta_O}}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/ 1.0 (* (sinh (/ 1.0 v)) 2.0))
   (/ cosTheta_i_m (* v (/ v cosTheta_O))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((1.0f / (sinhf((1.0f / v)) * 2.0f)) * (cosTheta_i_m / (v * (v / cosTheta_O))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((1.0e0 / (sinh((1.0e0 / v)) * 2.0e0)) * (costheta_i_m / (v * (v / costheta_o))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(1.0) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) * Float32(cosTheta_i_m / Float32(v * Float32(v / cosTheta_O)))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((single(1.0) / (sinh((single(1.0) / v)) * single(2.0))) * (cosTheta_i_m / (v * (v / cosTheta_O))));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. *-commutative 98.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. associate-/l* 98.2%

      \[\leadsto \frac{e^{-\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-frac-neg 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/r/ 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   10. associate-/l/ 98.3%

       \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

4. Taylor expanded in sinTheta_i around 0 98.1%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \]
5. Step-by-step derivation

   1. rec-exp 98.1%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \]

   2. sinh-undef 98.2%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \]

6. Applied egg-rr 98.2%

   \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \]

7. Final simplification 98.2%

   \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}} \]

Alternative 5: 58.5% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i_m}}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* 0.5 (/ 1.0 (/ (/ v cosTheta_O) cosTheta_i_m)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f * (1.0f / ((v / cosTheta_O) / cosTheta_i_m)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 * (1.0e0 / ((v / costheta_o) / costheta_i_m)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(Float32(1.0) / Float32(Float32(v / cosTheta_O) / cosTheta_i_m))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) * (single(1.0) / ((v / cosTheta_O) / cosTheta_i_m)));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. *-commutative 98.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. associate-/l* 98.2%

      \[\leadsto \frac{e^{-\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-frac-neg 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/r/ 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   10. associate-/l/ 98.3%

       \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

4. Taylor expanded in v around inf 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. clear-num 54.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta_O \cdot cosTheta_i}}} \]

   2. inv-pow 54.5%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{v}{cosTheta_O \cdot cosTheta_i}\right)}^{-1}} \]

6. Applied egg-rr 54.5%

   \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{v}{cosTheta_O \cdot cosTheta_i}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 54.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta_O \cdot cosTheta_i}}} \]

   2. associate-/r* 54.5%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

8. Simplified 54.5%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

9. Final simplification 54.5%

   \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \]

Alternative 6: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i_m}{v}\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* 0.5 (* cosTheta_O (/ cosTheta_i_m v)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f * (cosTheta_O * (cosTheta_i_m / v)));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 * (costheta_o * (costheta_i_m / v)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(cosTheta_O * Float32(cosTheta_i_m / v))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) * (cosTheta_O * (cosTheta_i_m / v)));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. *-commutative 98.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. associate-/l* 98.2%

      \[\leadsto \frac{e^{-\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-frac-neg 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/r/ 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   10. associate-/l/ 98.3%

       \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

4. Taylor expanded in v around inf 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. associate-*r/ 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

6. Simplified 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

7. Final simplification 53.5%

   \[\leadsto 0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \]

Alternative 7: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_i_s \cdot \left(0.5 \cdot \frac{cosTheta_i_m \cdot cosTheta_O}{v}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* 0.5 (/ (* cosTheta_i_m cosTheta_O) v))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f * ((cosTheta_i_m * cosTheta_O) / v));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 * ((costheta_i_m * costheta_o) / v))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) * Float32(Float32(cosTheta_i_m * cosTheta_O) / v)))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) * ((cosTheta_i_m * cosTheta_O) / v));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. *-commutative 98.2%

      \[\leadsto \frac{e^{-\frac{\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. associate-/l* 98.2%

      \[\leadsto \frac{e^{-\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-frac-neg 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/r/ 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

   10. associate-/l/ 98.3%

       \[\leadsto \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

4. Taylor expanded in v around inf 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]

5. Final simplification 53.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \]

Reproduce

herbie shell --seed 2023318 
(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)))