HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 17.5s

Alternatives: 11

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: 98.5% 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, 0.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(\frac{1}{v} \cdot \frac{1}{v}\right)\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (pow (exp sinTheta_i) (/ sinTheta_O (- v)))
  (/
   (* cosTheta_O (* cosTheta_i (* (/ 1.0 v) (/ 1.0 v))))
   (* (sinh (/ 1.0 v)) 2.0))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(expf(sinTheta_i), (sinTheta_O / -v)) * ((cosTheta_O * (cosTheta_i * ((1.0f / v) * (1.0f / v)))) / (sinhf((1.0f / v)) * 2.0f));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o * (costheta_i * ((1.0e0 / v) * (1.0e0 / v)))) / (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(sinTheta_i) ^ (sinTheta_O / -v)) * ((cosTheta_O * (cosTheta_i * ((single(1.0) / v) * (single(1.0) / v)))) / (sinh((single(1.0) / v)) * single(2.0)));
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

   1. times-frac 98.6%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. distribute-frac-neg2 98.7%

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

   5. associate-/l* 98.7%

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

   6. exp-prod 98.7%

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

   7. *-commutative 98.7%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. times-frac 98.8%

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

6. Applied egg-rr 98.8%

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

   1. div-inv 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{\frac{1}{v}}{v}\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (pow (exp sinTheta_i) (/ sinTheta_O (- v)))
  (/ (* cosTheta_O (* cosTheta_i (/ (/ 1.0 v) v))) (* (sinh (/ 1.0 v)) 2.0))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(expf(sinTheta_i), (sinTheta_O / -v)) * ((cosTheta_O * (cosTheta_i * ((1.0f / v) / v))) / (sinhf((1.0f / v)) * 2.0f));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o * (costheta_i * ((1.0e0 / v) / v))) / (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(sinTheta_i) ^ (sinTheta_O / -v)) * ((cosTheta_O * (cosTheta_i * ((single(1.0) / v) / v))) / (sinh((single(1.0) / v)) * single(2.0)));
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

   1. times-frac 98.6%

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

   2. associate-*l/ 98.7%

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

   3. associate-*r/ 98.7%

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

   4. distribute-frac-neg2 98.7%

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

   5. associate-/l* 98.7%

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

   6. exp-prod 98.7%

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

   7. *-commutative 98.7%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. times-frac 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_i (sinh (/ 1.0 v)))
  (/ cosTheta_O (* (* v (* v 2.0)) (pow (exp sinTheta_i) (/ sinTheta_O v))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (costheta_i / sinh((1.0e0 / v))) * (costheta_o / ((v * (v * 2.0e0)) * (exp(sintheta_i) ** (sintheta_o / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.6%

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

   2. times-frac 98.7%

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

   3. *-commutative 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i sinTheta_O) (- v)))
   (* (/ 1.0 v) (* cosTheta_O cosTheta_i)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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)) * ((1.0e0 / v) * (costheta_o * costheta_i))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_i * sinTheta_O) / -v)) * ((single(1.0) / v) * (cosTheta_O * cosTheta_i))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
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. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 98.7%

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(\frac{cosTheta\_O}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta\_i}{v}}{2}\right) \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (exp (* sinTheta_i (/ sinTheta_O (- v))))
  (* (/ cosTheta_O (* v (sinh (/ 1.0 v)))) (/ (/ cosTheta_i v) 2.0))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o / (v * sinh((1.0e0 / v)))) * ((costheta_i / v) / 2.0e0))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_i * (sinTheta_O / -v))) * ((cosTheta_O / (v * sinh((single(1.0) / v)))) * ((cosTheta_i / v) / single(2.0)));
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

   1. associate-*r/ 98.6%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 98.3%

      \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)} \cdot 2\right)} \]

   2. expm1-undefine 98.3%

      \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)} - 1\right)} \cdot 2\right)} \]

6. Applied egg-rr 98.3%

   \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)} - 1\right)} \cdot 2\right)} \]
7. Step-by-step derivation

   1. expm1-define 98.3%

      \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)} \cdot 2\right)} \]

8. Simplified 98.3%

   \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right)} \cdot 2\right)} \]
9. Step-by-step derivation

   1. *-un-lft-identity 98.3%

      \[\leadsto \color{blue}{1 \cdot \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right) \cdot 2\right)}} \]

   2. associate-*r/ 98.2%

      \[\leadsto 1 \cdot \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{v \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right) \cdot 2\right)} \]

   3. associate-/l* 98.2%

      \[\leadsto 1 \cdot \color{blue}{\left(e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right) \cdot 2\right)}\right)} \]

   4. exp-prod 98.2%

      \[\leadsto 1 \cdot \left(\color{blue}{{\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)}} \cdot \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right) \cdot 2\right)}\right) \]

   5. associate-*r/ 98.3%

      \[\leadsto 1 \cdot \left({\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}{v \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right)\right)\right) \cdot 2\right)}\right) \]

   6. expm1-log1p-u 98.6%

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

10. Applied egg-rr 98.6%

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

    1. *-lft-identity 98.6%

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

    2. exp-prod 98.6%

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

    3. *-commutative 98.6%

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

    4. distribute-frac-neg2 98.6%

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

    5. distribute-lft-neg-in 98.6%

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

    6. distribute-rgt-neg-in 98.6%

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

    7. associate-*r/ 98.6%

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

    8. *-commutative 98.6%

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

    9. associate-/l* 98.6%

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

    10. associate-*r* 98.6%

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

    11. times-frac 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\frac{cosTheta\_i}{2}}{{v}^{2}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ cosTheta_O (sinh (/ 1.0 v))) (/ (/ cosTheta_i 2.0) (pow v 2.0))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / sinhf((1.0f / v))) * ((cosTheta_i / 2.0f) / powf(v, 2.0f));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (costheta_o / sinh((1.0e0 / v))) * ((costheta_i / 2.0e0) / (v ** 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.6%

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

   2. times-frac 98.7%

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

   3. *-commutative 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in v around inf 98.4%

   \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(0.5 \cdot \frac{cosTheta\_O}{{v}^{2}}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 98.4%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{0.5 \cdot cosTheta\_O}{{v}^{2}}} \]

   2. *-commutative 98.4%

      \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{\color{blue}{cosTheta\_O \cdot 0.5}}{{v}^{2}} \]

9. Simplified 98.4%

   \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{cosTheta\_O \cdot 0.5}{{v}^{2}}} \]
10. Step-by-step derivation

    1. frac-times 98.2%

       \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{\sinh \left(\frac{1}{v}\right) \cdot {v}^{2}}} \]

11. Applied egg-rr 98.2%

    \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{\sinh \left(\frac{1}{v}\right) \cdot {v}^{2}}} \]
12. Step-by-step derivation

    1. times-frac 98.4%

       \[\leadsto \color{blue}{\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O \cdot 0.5}{{v}^{2}}} \]

    2. associate-*r/ 98.3%

       \[\leadsto \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{0.5}{{v}^{2}}\right)} \]

    3. associate-*l/ 98.4%

       \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{0.5}{{v}^{2}}\right)}{\sinh \left(\frac{1}{v}\right)}} \]

    4. *-commutative 98.4%

       \[\leadsto \frac{\color{blue}{\left(cosTheta\_O \cdot \frac{0.5}{{v}^{2}}\right) \cdot cosTheta\_i}}{\sinh \left(\frac{1}{v}\right)} \]

    5. associate-*l* 98.2%

       \[\leadsto \frac{\color{blue}{cosTheta\_O \cdot \left(\frac{0.5}{{v}^{2}} \cdot cosTheta\_i\right)}}{\sinh \left(\frac{1}{v}\right)} \]

    6. metadata-eval 98.2%

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

    7. associate-/r* 98.2%

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

    8. *-commutative 98.2%

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

    9. associate-*l/ 98.3%

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

    10. *-lft-identity 98.3%

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

    11. associate-*l/ 98.4%

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

    12. *-commutative 98.4%

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

    13. associate-/r* 98.4%

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

13. Simplified 98.4%

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

14. Final simplification 98.4%

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

Alternative 7: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (* sinTheta_i (/ sinTheta_O (- v))))
  (* 2.0 (/ v (* cosTheta_O cosTheta_i)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((sinTheta_i * (sinTheta_O / -v))) / (2.0f * (v / (cosTheta_O * cosTheta_i)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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))) / (2.0e0 * (v / (costheta_o * costheta_i)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(sinTheta_i * Float32(sinTheta_O / Float32(-v)))) / Float32(Float32(2.0) * Float32(v / Float32(cosTheta_O * cosTheta_i))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_i * (sinTheta_O / -v))) / (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

   1. associate-*r/ 98.6%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 93.5%

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

   2. inv-pow 93.5%

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

6. Applied egg-rr 93.5%

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

   1. unpow-1 93.5%

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

   2. associate-/l* 93.5%

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

   3. associate-*r/ 93.6%

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

8. Simplified 93.6%

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

   1. *-un-lft-identity 93.6%

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

   2. associate-/r* 93.6%

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

   3. pow-flip 93.6%

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

   4. distribute-frac-neg2 93.6%

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

   5. associate-*r/ 93.5%

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

   6. associate-*r/ 93.5%

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

10. Applied egg-rr 93.5%

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

    1. *-lft-identity 93.5%

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

    2. exp-prod 93.5%

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

    3. *-commutative 93.5%

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

    4. distribute-frac-neg2 93.5%

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

    5. distribute-lft-neg-in 93.5%

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

    6. distribute-rgt-neg-in 93.5%

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

    7. associate-/l* 93.5%

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

    8. associate-*r/ 93.6%

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

    9. *-commutative 93.6%

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

    10. associate-/l* 93.5%

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

    11. times-frac 93.5%

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

12. Simplified 93.5%

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

13. Taylor expanded in v around inf 59.9%

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

14. Final simplification 59.9%

    \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]
15. Add Preprocessing

Alternative 8: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{v \cdot \frac{\frac{2}{cosTheta\_O}}{cosTheta\_i}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (exp (* sinTheta_i (/ sinTheta_O (- v))))
  (* v (/ (/ 2.0 cosTheta_O) cosTheta_i))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((sinTheta_i * (sinTheta_O / -v))) / (v * ((2.0f / cosTheta_O) / cosTheta_i));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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))) / (v * ((2.0e0 / costheta_o) / costheta_i))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(sinTheta_i * Float32(sinTheta_O / Float32(-v)))) / Float32(v * Float32(Float32(Float32(2.0) / cosTheta_O) / cosTheta_i)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_i * (sinTheta_O / -v))) / (v * ((single(2.0) / cosTheta_O) / cosTheta_i));
end

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 93.5%

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

   2. inv-pow 93.5%

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

6. Applied egg-rr 93.5%

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

   1. unpow-1 93.5%

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

   2. associate-/l* 93.5%

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

   3. associate-*r/ 93.6%

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

8. Simplified 93.6%

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

   1. *-un-lft-identity 93.6%

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

   2. associate-/r* 93.6%

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

   3. pow-flip 93.6%

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

   4. distribute-frac-neg2 93.6%

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

   5. associate-*r/ 93.5%

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

   6. associate-*r/ 93.5%

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

10. Applied egg-rr 93.5%

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

    1. *-lft-identity 93.5%

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

    2. exp-prod 93.5%

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

    3. *-commutative 93.5%

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

    4. distribute-frac-neg2 93.5%

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

    5. distribute-lft-neg-in 93.5%

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

    6. distribute-rgt-neg-in 93.5%

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

    7. associate-/l* 93.5%

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

    8. associate-*r/ 93.6%

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

    9. *-commutative 93.6%

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

    10. associate-/l* 93.5%

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

    11. times-frac 93.5%

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

12. Simplified 93.5%

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

13. Taylor expanded in v around inf 59.9%

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

    1. *-commutative 59.9%

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

    2. associate-*l/ 59.9%

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

    3. associate-*r/ 60.1%

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

    4. associate-/r* 60.1%

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

15. Simplified 60.1%

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

16. Final simplification 60.1%

    \[\leadsto \frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{v \cdot \frac{\frac{2}{cosTheta\_O}}{cosTheta\_i}} \]
17. Add Preprocessing

Alternative 9: 57.9% accurate, 14.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(0.5 + \frac{sinTheta\_i \cdot sinTheta\_O}{v} \cdot -0.5\right)\right)}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   cosTheta_O
   (* cosTheta_i (+ 0.5 (* (/ (* sinTheta_i sinTheta_O) v) -0.5))))
  v))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * (cosTheta_i * (0.5f + (((sinTheta_i * sinTheta_O) / v) * -0.5f)))) / v;
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (costheta_o * (costheta_i * (0.5e0 + (((sintheta_i * sintheta_o) / v) * (-0.5e0))))) / v
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(cosTheta_i * Float32(Float32(0.5) + Float32(Float32(Float32(sinTheta_i * sinTheta_O) / v) * Float32(-0.5))))) / v)
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * (cosTheta_i * (single(0.5) + (((sinTheta_i * sinTheta_O) / v) * single(-0.5))))) / v;
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + 0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

6. Taylor expanded in cosTheta_i around 0 59.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(-0.5 \cdot \frac{cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v} + 0.5 \cdot cosTheta\_O\right)}{v}} \]

7. Taylor expanded in cosTheta_O around 0 59.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}{v}} \]

8. Final simplification 59.4%

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

Alternative 10: 57.9% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{0.5}{v}\right) \end{array} \]

FPCore

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

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = costheta_o * (costheta_i * (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + 0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

6. Taylor expanded in cosTheta_i around 0 59.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(-0.5 \cdot \frac{cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v} + 0.5 \cdot cosTheta\_O\right)}{v}} \]

7. Taylor expanded in cosTheta_O around 0 59.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}{v}} \]
8. Step-by-step derivation

   1. associate-/l* 59.4%

      \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}} \]

   2. associate-/l* 59.4%

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \color{blue}{\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)}\right)}{v} \]

9. Simplified 59.4%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}{v}} \]

10. Taylor expanded in sinTheta_O around 0 59.4%

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

    1. associate-*r/ 59.4%

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

    2. *-commutative 59.4%

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

    3. associate-/l* 59.4%

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

12. Simplified 59.4%

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

13. Final simplification 59.4%

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

Alternative 11: 57.9% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot 0.5\right) \end{array} \]

FPCore

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

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = costheta_o * ((costheta_i / v) * 0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.4%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + 0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

6. Taylor expanded in cosTheta_i around 0 59.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(-0.5 \cdot \frac{cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v} + 0.5 \cdot cosTheta\_O\right)}{v}} \]

7. Taylor expanded in cosTheta_O around 0 59.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}{v}} \]
8. Step-by-step derivation

   1. associate-/l* 59.4%

      \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{v}} \]

   2. associate-/l* 59.4%

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \color{blue}{\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)}\right)}{v} \]

9. Simplified 59.4%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(0.5 + -0.5 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}{v}} \]

10. Taylor expanded in sinTheta_O around 0 59.4%

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

    1. *-commutative 59.4%

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

12. Simplified 59.4%

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

13. Final simplification 59.4%

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

Reproduce

herbie shell --seed 2024071 
(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)))